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Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.1
Introduction
Chapter 1
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.2
The Nature of Derivatives
A derivative is an instrument whose valuedepends on the values of other morebasic underlying variables
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.3
Examples of Derivatives
• Futures Contracts• Forward Contracts• Swaps• Options
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.4
Derivatives MarketsExchange traded
Traditionally exchanges have used the open-outcry system, but increasingly they are switchingto electronic tradingContracts are standard there is virtually no creditrisk
Over-the-counter (OTC)A computer- and telephone-linked network ofdealers at financial institutions, corporations, andfund managersContracts can be non-standard and there is somesmall amount of credit risk
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.5
Size of OTC and Exchange Markets(Figure 1.1, Page 3)
Source: Bank for International Settlements. Chart shows total principal amountsfor OTC market and value of underlying assets for exchange market
020406080
100120140160180200220240
Jun-98 Jun-99 Jun-00 Jun-01 Jun-02 Jun-03 Jun-04
Size of Market ($ trillion)
OTCExchange
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.6
Ways Derivatives are Used
To hedge risksTo speculate (take a view on thefuture direction of the market)To lock in an arbitrage profitTo change the nature of a liabilityTo change the nature of an investmentwithout incurring the costs of sellingone portfolio and buying another
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.7
Forward Contracts
Forward contracts are similar to futuresexcept that they trade in the over-the-counter marketForward contracts are particularly popularon currencies and interest rates
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.8
Foreign Exchange Quotes for GBPJune 3, 2003 (See page 4)
1.61001.60946-month forward
1.61921.61873-month forward
1.62531.62481-month forward
1.62851.6281SpotOfferBid
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.9
Forward Price
The forward price for a contract is thedelivery price that would be applicableto the contract if were negotiatedtoday (i.e., it is the delivery price thatwould make the contract worth exactlyzero)The forward price may be different forcontracts of different maturities
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.10
Terminology
The party that has agreed to buyhas what is termed a long positionThe party that has agreed to sellhas what is termed a short position
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.11
Example (page 4)
On June 3, 2003 the treasurer of acorporation enters into a long forwardcontract to buy £1 million in six months atan exchange rate of 1.6100This obligates the corporation to pay$1,610,000 for £1 million on December 3,2003What are the possible outcomes?
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.12
Profit from aLong Forward Position
Profit
Price of Underlying at Maturity, STK
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.13
Profit from aShort Forward Position
Profit
Price of Underlying at Maturity, STK
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.14
Futures Contracts (page 6)
Agreement to buy or sell an asset for acertain price at a certain timeSimilar to forward contractWhereas a forward contract is traded OTC,a futures contract is traded on anexchange
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.15
Exchanges Trading Futures
Chicago Board of TradeChicago Mercantile ExchangeLIFFE (London)Eurex (Europe)BM&F (Sao Paulo, Brazil)TIFFE (Tokyo)and many more (see list at end of book)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.16
Examples of Futures Contracts
Agreement to:buy 100 oz. of gold @ US$400/oz. inDecember (NYMEX)sell £62,500 @ 1.5000 US$/£ inMarch (CME)sell 1,000 bbl. of oil @ US$20/bbl. inApril (NYMEX)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.17
1. Gold: An ArbitrageOpportunity?
Suppose that:The spot price of gold is US$300The 1-year forward price of gold is
US$340The 1-year US$ interest rate is 5%
per annumIs there an arbitrage opportunity?
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.18
2. Gold: Another ArbitrageOpportunity?
Suppose that:- The spot price of gold is US$300- The 1-year forward price of gold
is US$300- The 1-year US$ interest rate is
5% per annumIs there an arbitrage opportunity?
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.19
The Forward Price of Gold
If the spot price of gold is S and the forwardprice for a contract deliverable in T years is F,then
F = S (1+r )T
where r is the 1-year (domestic currency) risk-free rate of interest.In our examples, S = 300, T = 1, and r =0.05 sothat
F = 300(1+0.05) = 315
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.20
1. Oil: An ArbitrageOpportunity?
Suppose that:- The spot price of oil is US$19- The quoted 1-year futures price of
oil is US$25- The 1-year US$ interest rate is 5%
per annum- The storage costs of oil are 2%
per annumIs there an arbitrage opportunity?
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.21
2. Oil: Another ArbitrageOpportunity?
Suppose that:- The spot price of oil is US$19- The quoted 1-year futures price of
oil is US$16- The 1-year US$ interest rate is 5%
per annum- The storage costs of oil are 2%
per annumIs there an arbitrage opportunity?
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.22
Options
A call option is an option to buy acertain asset by a certain date for acertain price (the strike price)A put option is an option to sell acertain asset by a certain date for acertain price (the strike price)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.23
American vs European Options
An American option can be exercised atany time during its lifeA European option can be exercised onlyat maturity
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.24
Intel Option Prices (May 29, 2003;Stock Price=20.83); See Table 1.2 page 7
2.852.201.851.150.450.2022.50
1.500.850.452.401.601.2520.00
OctPut
JulyPut
JunePut
OctCall
JulyCall
JuneCall
StrikePrice
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.25
Exchanges Trading Options
Chicago Board Options ExchangeAmerican Stock ExchangePhiladelphia Stock ExchangePacific ExchangeLIFFE (London)Eurex (Europe)and many more (see list at end of book)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.26
Options vs Futures/Forwards
A futures/forward contract gives the holderthe obligation to buy or sell at a certainpriceAn option gives the holder the right to buyor sell at a certain price
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.27
Types of Traders
• Hedgers
• Speculators
• Arbitrageurs
Some of the largest trading losses in derivatives haveoccurred because individuals who had a mandate to behedgers or arbitrageurs switched to being speculators(See for example Barings Bank, Business Snapshot 1.2,page 15)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.28
Hedging Examples (pages 10-11)
A US company will pay £10 million forimports from Britain in 3 months anddecides to hedge using a long positionin a forward contractAn investor owns 1,000 Microsoftshares currently worth $28 per share. Atwo-month put with a strike price of$27.50 costs $1. The investor decidesto hedge by buying 10 contracts
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.29
Value of Microsoft Shares withand without Hedging (Fig 1.4, page 11)
20,000
25,000
30,000
35,000
40,000
20 25 30 35 40
Stock Price ($)
Value of Holding ($)
No HedgingHedging
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.30
Speculation Example
An investor with $4,000 to invest feelsthat Amazon.com’s stock price willincrease over the next 2 months. Thecurrent stock price is $40 and the priceof a 2-month call option with a strike of45 is $2What are the alternative strategies?
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.31
Arbitrage Example
A stock price is quoted as £100 inLondon and $172 in New YorkThe current exchange rate is 1.7500What is the arbitrage opportunity?
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.32
Hedge Funds (see Business Snapshot 1.1, page 9)
Hedge funds are not subject to the same rules asmutual funds and cannot offer their securities publicly.Mutual funds must
disclose investment policies,makes shares redeemable at any time,limit use of leveragetake no short positions.
Hedge funds are not subject to these constraints.Hedge funds use complex trading strategies: big usersof derivatives for hedging, speculation and arbitrage
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 2.1
Mechanics of FuturesMarkets
Chapter 2
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 2.2
Futures Contracts
Available on a wide range of underlyingsExchange tradedSpecifications need to be defined:
What can be delivered,Where it can be delivered, &When it can be delivered
Settled daily
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 2.3
Margins
A margin is cash or marketable securitiesdeposited by an investor with his or herbrokerThe balance in the margin account isadjusted to reflect daily settlementMargins minimize the possibility of a lossthrough a default on a contract
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 2.4
Example of a Futures Trade (page 27-28)
An investor takes a long position in 2December gold futures contracts onJune 5
contract size is 100 oz.futures price is US$400margin requirement is US$2,000/contract(US$4,000 in total)maintenance margin is US$1,500/contract(US$3,000 in total)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 2.5
A Possible OutcomeTable 2.1, Page 28
Daily Cumulative MarginFutures Gain Gain Account Margin
Price (Loss) (Loss) Balance CallDay (US$) (US$) (US$) (US$) (US$)
400.00 4,000
5-Jun 397.00 (600) (600) 3,400 0. . . . . .. . . . . .. . . . . .
13-Jun 393.30 (420) (1,340) 2,660 1,340 . . . . . .. . . . .. . . . . .
19-Jun 387.00 (1,140) (2,600) 2,740 1,260 . . . . . .. . . . . .. . . . . .
26-Jun 392.30 260 (1,540) 5,060 0
+
= 4,000
3,000+
= 4,000
<
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 2.6
Other Key Points About Futures
They are settled dailyClosing out a futures positioninvolves entering into an offsettingtradeMost contracts are closed outbefore maturity
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 2.7
Collateralization in OTC Markets
It is becoming increasingly common forcontracts to be collateralized in OTCmarketsThey are then similar to futures contractsin that they are settled regularly (e.g. everyday or every week)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 2.8
Futures Prices for Gold on Feb 4, 2004: PricesIncrease with Maturity (Figure 2.2, page 35)
(a) Gold
398399400401402403404405406407408
Feb-04 Apr-04 Jun-04 Aug-04 Oct-04 Dec-04
Contract Maturity Month
Futu
res
Pric
e ($
per
oz)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 2.9
Futures Prices for Oil on February 4, 2004:Prices Decrease with Maturity (Figure 2.2, page 35)
(b) Brent Crude Oil
24
25
26
27
28
29
30
Mar-04 May-04 Jul-04 Sep-04 Nov-04 Jan-05
Contract Maturity Month
Futu
res
Pric
e ($
per
bar
rel)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 2.10
Delivery
If a futures contract is not closed out beforematurity, it is usually settled by delivering theassets underlying the contract. When there arealternatives about what is delivered, where it isdelivered, and when it is delivered, the party withthe short position chooses. A few contracts (for example, those on stockindices and Eurodollars) are settled in cash
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 2.11
Some Terminology
Open interest: the total number of contractsoutstanding
equal to number of long positions ornumber of short positions
Settlement price: the price just before thefinal bell each day
used for the daily settlement processVolume of trading: the number of trades in 1day
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 2.12
Convergence of Futures to Spot(Figure 2.1, page 26)
Time Time
(a) (b)
FuturesPrice
FuturesPrice
Spot Price
Spot Price
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 2.13
Questions
When a new trade is completed whatare the possible effects on the openinterest?Can the volume of trading in a daybe greater than the open interest?
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 2.14
Regulation of Futures
Regulation is designed toprotect the public interestRegulators try to preventquestionable trading practicesby either individuals on the floorof the exchange or outsidegroups
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 2.15
Accounting & Tax
Ideally hedging profits (losses) should berecognized at the same time as the losses(profits) on the item being hedgedIdeally profits and losses from speculationshould be recognized on a mark-to-marketbasisRoughly speaking, this is what theaccounting and tax treatment of futures inthe U.S.and many other countries attemptsto achieve
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 2.16
Forward Contracts vs FuturesContracts
Private contract between 2 parties Exchange traded
Non-standard contract Standard contract
Usually 1 specified delivery date Range of delivery dates
Settled at end of contract Settled daily
Delivery or final cashsettlement usually occurs
Contract usually closed outprior to maturity
FORWARDS FUTURES
TABLE 2.3 (p. 41)
Some credit risk Virtually no credit risk
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 2.17
Foreign Exchange Quotes
Futures exchange rates are quoted as thenumber of USD per unit of the foreign currencyForward exchange rates are quoted in the sameway as spot exchange rates. This means thatGBP, EUR, AUD, and NZD are quoted as USDper unit of foreign currency. Other currencies(e.g., CAD and JPY) are quoted as units of theforeign currency per USD.
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 3.1
Hedging Strategies UsingFutures
Chapter 3
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 3.2
Long & Short Hedges
A long futures hedge is appropriate whenyou know you will purchase an asset inthe future and want to lock in the priceA short futures hedge is appropriatewhen you know you will sell an asset inthe future & want to lock in the price
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 3.3
Arguments in Favor of Hedging
Companies should focus on the mainbusiness they are in and take steps tominimize risks arising from interest rates,exchange rates, and other marketvariables
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 3.4
Arguments against Hedging
Shareholders are usually well diversifiedand can make their own hedging decisionsIt may increase risk to hedge whencompetitors do notExplaining a situation where there is a losson the hedge and a gain on the underlyingcan be difficult
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 3.5
Convergence of Futures to Spot(Hedge initiated at time t1 and closed out at time t2)
Time
SpotPrice
FuturesPrice
t1 t2
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 3.6
Basis Risk
Basis is the difference betweenspot & futuresBasis risk arises because of theuncertainty about the basiswhen the hedge is closed out
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 3.7
Long Hedge
Suppose thatF1 : Initial Futures PriceF2 : Final Futures PriceS2 : Final Asset Price
You hedge the future purchase of anasset by entering into a long futurescontractCost of Asset=S2 – (F2 – F1) = F1 + Basis
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 3.8
Short Hedge
Suppose thatF1 : Initial Futures PriceF2 : Final Futures PriceS2 : Final Asset Price
You hedge the future sale of an asset byentering into a short futures contractPrice Realized=S2+ (F1 – F2) = F1 + Basis
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 3.9
Choice of Contract
Choose a delivery month that is as closeas possible to, but later than, the end ofthe life of the hedgeWhen there is no futures contract on theasset being hedged, choose the contractwhose futures price is most highlycorrelated with the asset price. This isknown as cross hedging.
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 3.10
Optimal Hedge Ratio
Proportion of the exposure that should optimally behedged is
whereσS is the standard deviation of ∆S, the change in thespot price during the hedging period,σF is the standard deviation of ∆F, the change in thefutures price during the hedging periodρ is the coefficient of correlation between ∆S and ∆F.
F
S
σσρ
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 3.11
Hedging Using Index Futures(Page 63)
To hedge the risk in a portfolio thenumber of contracts that should beshorted is
where P is the value of the portfolio,β is its beta, and A is the value of theassets underlying one futurescontract
βPA
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 3.12
Reasons for Hedging an EquityPortfolio
Desire to be out of the market for a shortperiod of time. (Hedging may be cheaperthan selling the portfolio and buying itback.)Desire to hedge systematic risk(Appropriate when you feel that you havepicked stocks that will outpeform themarket.)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 3.13
Example
Value of S&P 500 is 1,000Value of Portfolio is $5 millionBeta of portfolio is 1.5
What position in futures contracts on theS&P 500 is necessary to hedge theportfolio?
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 3.14
Changing Beta
What position is necessary to reduce thebeta of the portfolio to 0.75?What position is necessary to increase thebeta of the portfolio to 2.0?
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 3.15
Hedging Price of an IndividualStock
Similar to hedging a portfolioDoes not work as well because only thesystematic risk is hedgedThe unsystematic risk that is unique to thestock is not hedged
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 3.16
Why Hedge Equity Returns
May want to be out of the market for a while.Hedging avoids the costs of selling andrepurchasing the portfolioSuppose stocks in your portfolio have anaverage beta of 1.0, but you feel they have beenchosen well and will outperform the market inboth good and bad times. Hedging ensures thatthe return you earn is the risk-free return plusthe excess return of your portfolio over themarket.
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 3.17
Rolling The Hedge Forward (page 67-68)
We can use a series of futurescontracts to increase the life of ahedgeEach time we switch from 1 futurescontract to another we incur a type ofbasis risk
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.1
Interest Rates
Chapter 4
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.2
Types of Rates
Treasury ratesLIBOR ratesRepo rates
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.3
Measuring Interest Rates
The compounding frequency usedfor an interest rate is the unit ofmeasurementThe difference between quarterlyand annual compounding isanalogous to the differencebetween miles and kilometers
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.4
Continuous Compounding(Page 79)
In the limit as we compound more and morefrequently we obtain continuously compoundedinterest rates$100 grows to $100eRT when invested at acontinuously compounded rate R for time T$100 received at time T discounts to $100e-RT attime zero when the continuously compoundeddiscount rate is R
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.5
Conversion Formulas(Page 79)
DefineRc : continuously compounded rateRm: same rate with compounding m times
per year
( )R m
Rm
R m e
cm
mR mc
= +⎛⎝⎜
⎞⎠⎟
= −
ln
/
1
1
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.6
Zero Rates
A zero rate (or spot rate), for maturity T isthe rate of interest earned on aninvestment that provides a payoff only attime T
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.7
Example (Table 4.2, page 81)
M aturity(years)
Zero Rate(% cont com p)
0.5 5.0
1.0 5.8
1.5 6.4
2.0 6.8
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.8
Bond Pricing
To calculate the cash price of a bond wediscount each cash flow at the appropriate zerorateIn our example, the theoretical price of a two-year bond providing a 6% coupon semiannuallyis
3 3 3103 98 39
0 05 0 5 0 058 1 0 0 064 1 5
0 068 2 0
e e ee
− × − × − ×
− ×
+ +
+ =
. . . . . .
. . .
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.9
Bond YieldThe bond yield is the discount rate thatmakes the present value of the cash flows onthe bond equal to the market price of thebondSuppose that the market price of the bond inour example equals its theoretical price of98.39The bond yield (continuously compounded) isgiven by solving
to get y=0.0676 or 6.76%.3 3 3 103 98 390 5 1 0 1 5 2 0e e e ey y y y− × − × − × − ×+ + + =. . . . .
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.10
Par YieldThe par yield for a certain maturity is thecoupon rate that causes the bond price toequal its face value.In our example we solve
g)compoundin s.a. (with get to 876
1002
100
2220.2068.0
5.1064.00.1058.05.005.0
.c=
ec
ececec
=⎟⎠⎞
⎜⎝⎛ ++
++
×−
×−×−×−
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.11
Par Yield continued
In general if m is the number of couponpayments per year, P is the present valueof $1 received at maturity and A is thepresent value of an annuity of $1 on eachcoupon date
cP m
A=
−( )100 100
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.12
Sample Data (Table 4.3, page 82)
Bond Time to Annual Bond CashPrincipal Maturity Coupon Price(dollars) (years) (dollars) (dollars)
100 0.25 0 97.5
100 0.50 0 94.9
100 1.00 0 90.0
100 1.50 8 96.0
100 2.00 12 101.6
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.13
The Bootstrap Method
An amount 2.5 can be earned on 97.5 during 3months.The 3-month rate is 4 times 2.5/97.5 or 10.256%with quarterly compoundingThis is 10.127% with continuous compoundingSimilarly the 6 month and 1 year rates are10.469% and 10.536% with continuouscompounding
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.14
The Bootstrap Method continued
To calculate the 1.5 year rate we solve
to get R = 0.10681 or 10.681%
Similarly the two-year rate is 10.808%
9610444 5.10.110536.05.010469.0 =++ ×−×−×− Reee
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.15
Zero Curve Calculated from theData (Figure 4.1, page 84)
9
10
11
12
0 0.5 1 1.5 2 2.5
ZeroRate (%)
Maturity (yrs)
10.127
10.469 10.53610.681 10.808
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.16
Forward Rates
The forward rate is the future zero rateimplied by today’s term structure of interestrates
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.17
Calculation of Forward RatesTable 4.5, page 85
Zero Rate for Forward Ratean n -year Investment for n th Year
Year (n ) (% per annum) (% per annum)
1 3.02 4.0 5.03 4.6 5.84 5.0 6.25 5.3 6.5
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.18
Formula for Forward Rates
Suppose that the zero rates for timeperiods T1 and T2 are R1 and R2 with bothrates continuously compounded.The forward rate for the period betweentimes T1 and T2 is
R T R TT T
2 2 1 1
2 1
−−
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.19
Instantaneous Forward Rate
The instantaneous forward rate for amaturity T is the forward rate that appliesfor a very short time period starting at T. Itis
where R is the T-year rate
R T RT
+∂∂
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.20
Upward vs Downward SlopingYield Curve
For an upward sloping yield curve:Fwd Rate > Zero Rate > Par Yield
For a downward sloping yield curvePar Yield > Zero Rate > Fwd Rate
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.21
Forward Rate Agreement
A forward rate agreement (FRA) is anagreement that a certain rate will apply toa certain principal during a certain futuretime period
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.22
Forward Rate Agreementcontinued
An FRA is equivalent to an agreementwhere interest at a predetermined rate, RKis exchanged for interest at the marketrateAn FRA can be valued by assuming thatthe forward interest rate is certain to berealized
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.23
Valuation Formulas (equations 4.9 and 4.10page 88)
Value of FRA where a fixed rate RK will bereceived on a principal L between times T1 andT2 isValue of FRA where a fixed rate is paid is
RF is the forward rate for the period and R2 is thezero rate for maturity T2
What compounding frequencies are used inthese formulas for RK, RM, and R2?
22))(( 12TR
FK eTTRRL −−−
22))(( 12TR
KF eTTRRL −−−
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.24
Duration of a bond that provides cash flow c i at time t i is
where B is its price and y is its yield (continuouslycompounded)This leads to
⎥⎦
⎤⎢⎣
⎡ −
=∑ B
ectiyt
in
ii
1
yDBB
∆−=∆
Duration (page 89)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.25
Duration ContinuedWhen the yield y is expressed withcompounding m times per year
The expression
is referred to as the “modified duration”
myyBDB
+∆
−=∆1
Dy m1 +
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.26
Convexity
The convexity of a bond is defined as
2
1
2
2
2
)(21
thatso
1
yCyDBB
B
etc
yB
BC
n
i
ytii
i
∆+∆−=∆
=∂∂
=∑=
−
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 4.27
Theories of the Term StructurePage 93
Expectations Theory: forward rates equalexpected future zero ratesMarket Segmentation: short, medium andlong rates determined independently ofeach otherLiquidity Preference Theory: forwardrates higher than expected future zerorates
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 5.1
Determination of Forwardand Futures Prices
Chapter 5
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 5.2
Consumption vs Investment Assets
Investment assets are assets held bysignificant numbers of people purely forinvestment purposes (Examples: gold,silver)Consumption assets are assets heldprimarily for consumption (Examples:copper, oil)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 5.3
Short Selling (Page 99-101)
Short selling involves sellingsecurities you do not ownYour broker borrows thesecurities from another client andsells them in the market in theusual way
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 5.4
Short Selling(continued)
At some stage you must buythe securities back so theycan be replaced in theaccount of the clientYou must pay dividends andother benefits the owner ofthe securities receives
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 5.5
Notation for Valuing Futures andForward Contracts
Risk-free interest rate formaturity T
r:
Time until delivery dateT:
Futures or forward price todayF0:
Spot price todayS0:
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 5.6
1. Gold: An ArbitrageOpportunity?
Suppose that:The spot price of gold is US$390The quoted 1-year forward price ofgold is US$425The 1-year US$ interest rate is 5% perannumNo income or storage costs for gold
Is there an arbitrage opportunity?
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 5.7
2. Gold: Another ArbitrageOpportunity?
Suppose that:The spot price of gold is US$390The quoted 1-year forward price ofgold is US$390The 1-year US$ interest rate is 5%per annumNo income or storage costs for gold
Is there an arbitrage opportunity?
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 5.8
The Forward Price of Gold
If the spot price of gold is S and the futures priceis for a contract deliverable in T years is F, then
F = S (1+r )T
where r is the 1-year (domestic currency) risk-free rate of interest.In our examples, S=390, T=1, and r=0.05 so that
F = 390(1+0.05) = 409.50
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 5.9
When Interest Rates are Measuredwith Continuous Compounding
F0 = S0erT
This equation relates the forward priceand the spot price for any investmentasset that provides no income and hasno storage costs
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 5.10
When an Investment AssetProvides a Known Dollar Income(page 105, equation 5.2)
F0 = (S0 – I )erT
where I is the present value of the incomeduring life of forward contract
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 5.11
When an Investment AssetProvides a Known Yield(Page 107, equation 5.3)
F0 = S0 e(r–q )T
where q is the average yield during thelife of the contract (expressed withcontinuous compounding)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 5.12
Valuing a Forward ContractPage 108
Suppose that K is delivery price in a forward contract and F0 is forward price that would apply to the
contract todayThe value of a long forward contract, ƒ, is
ƒ = (F0 – K )e–rT
Similarly, the value of a short forward contractis
(K – F0 )e–rT
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 5.13
Forward vs Futures Prices
Forward and futures prices are usually assumedto be the same. When interest rates areuncertain they are, in theory, slightly different:A strong positive correlation between interestrates and the asset price implies the futuresprice is slightly higher than the forward priceA strong negative correlation implies thereverse
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 5.14
Stock Index (Page 110-112)
Can be viewed as an investment assetpaying a dividend yieldThe futures price and spot pricerelationship is therefore
F0 = S0 e(r–q )T where q is the average dividend yield on
the portfolio represented by the indexduring life of contract
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 5.15
Stock Index(continued)
For the formula to be true it isimportant that the index represent aninvestment assetIn other words, changes in the indexmust correspond to changes in thevalue of a tradable portfolioThe Nikkei index viewed as a dollarnumber does not represent aninvestment asset (See BusinessSnapshot 5.3, page 111)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 5.16
Index Arbitrage
When F0 > S0e(r-q)T an arbitrageur buys thestocks underlying the index and sellsfuturesWhen F0 < S0e(r-q)T an arbitrageur buysfutures and shorts or sells the stocksunderlying the index
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 5.17
Index Arbitrage(continued)
Index arbitrage involves simultaneous trades infutures and many different stocksVery often a computer is used to generate thetradesOccasionally (e.g., on Black Monday)simultaneous trades are not possible and thetheoretical no-arbitrage relationship betweenF0 and S0 does not hold
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 5.18
A foreign currency is analogous to a securityproviding a dividend yieldThe continuous dividend yield is the foreignrisk-free interest rateIt follows that if rf is the foreign risk-free interestrate
Futures and Forwards onCurrencies (Page 112-115)
F S e r r Tf0 0= −( )
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 5.19
Why the Relation Must Be TrueFigure 5.1, page 113
1000 units of foreign currency
at time zero
units of foreign currency at time T
Trfe1000
dollars at time T
TrfeF01000
1000S0 dollars at time zero
dollars at time T
rTeS01000
1000 units of foreign currency
at time zero
units of foreign currency at time T
Trfe1000
dollars at time T
TrfeF01000
1000S0 dollars at time zero
dollars at time T
rTeS01000
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 5.20
Futures on Consumption Assets(Page 117-118)
F0 ≤ S0 e(r+u )T
where u is the storage cost per unittime as a percent of the asset value.
Alternatively, F0 ≤ (S0+U )erT
where U is the present value of thestorage costs.
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 5.21
The Cost of Carry (Page 118-119)
The cost of carry, c, is the storage cost plus theinterest costs less the income earnedFor an investment asset F0 = S0ecT
For a consumption asset F0 ≤ S0ecT
The convenience yield on the consumptionasset, y, is defined so that F0 = S0 e(c–y )T
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 5.22
Futures Prices & Expected FutureSpot Prices (Page 119-121)
Suppose k is the expected return required byinvestors on an assetWe can invest F0e–r T at the risk-free rate andenter into a long futures contract so that there isa cash inflow of ST at maturityThis shows that
TkrT
TkTrT
eSEF
SEeeF
)(0
0
)(
)()(
−
−
=
=or
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 5.23
Futures Prices & Future SpotPrices (continued)
If the asset hasno systematic risk, then k = r and F0 isan unbiased estimate of ST
positive systematic risk, then k > r andF0 < E (ST )negative systematic risk, then k < r andF0 > E (ST )
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.1
Interest Rate Futures
Chapter 6
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.2
Day Count Conventionsin the U.S. (Page 129)
Actual/360Money Market Instruments:
30/360Corporate Bonds:
Actual/Actual (in period)Treasury Bonds:
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.3
Treasury Bond Price Quotesin the U.S
Cash price = Quoted price + Accrued Interest
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.4
Treasury Bond FuturesPages 133-137
Cash price received by party with shortposition =
Most Recent Settlement Price ×Conversion factor + Accrued interest
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.5
Example
Settlement price of bond delivered = 90.00Conversion factor = 1.3800Accrued interest on bond =3.00Price received for bond is1.3800×9.00)+3.00 = $127.20per $100 of principal
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.6
Conversion Factor
The conversion factor for a bond isapproximately equal to the value of thebond on the assumption that the yieldcurve is flat at 6% with semiannualcompounding
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.7
CBOTT-Bonds & T-Notes
Factors that affect the futures price:Delivery can be made any timeduring the delivery monthAny of a range of eligible bondscan be deliveredThe wild card play
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.8
A Eurodollar is a dollar deposited in a bankoutside the United StatesEurodollar futures are futures on the 3-monthEurodollar deposit rate (same as 3-monthLIBOR rate)One contract is on the rate earned on $1 millionA change of one basis point or 0.01 in aEurodollar futures quote corresponds to acontract price change of $25
Eurodollar Futures (Page 137-142)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.9
Eurodollar Futures continued
A Eurodollar futures contract is settled incashWhen it expires (on the third Wednesdayof the delivery month) the final settlementprice is 100 minus the actual three monthdeposit rate
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.10
Example
Suppose you buy (take a long position in)a contract on November 1The contract expires on December 21The prices are as shownHow much do you gain or lose a) on thefirst day, b) on the second day, c) over thewhole time until expiration?
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.11
Example
97.42Dec 21
………….
96.98Nov 3
97.23Nov 2
97.12Nov 1QuoteDate
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.12
Example continued
If on Nov. 1 you know that you will have$1 million to invest on for three months onDec 21, the contract locks in a rate of100 - 97.12 = 2.88%In the example you earn 100 – 97.42 =2.58% on $1 million for three months(=$6,450) and make a gain day by day onthe futures contract of 30×$25 =$750
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.13
Formula for Contract Value (page 138)
If Q is the quoted price of a Eurodollarfutures contract, the value of one contractis 10,000[100-0.25(100-Q)]
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.14
Forward Rates and EurodollarFutures (Page 139-142)
Eurodollar futures contracts last as long as10 yearsFor Eurodollar futures lasting beyond twoyears we cannot assume that the forwardrate equals the futures rate
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.15
There are Two Reasons
Futures is settled daily where forward issettled onceFutures is settled at the beginning of theunderlying three-month period; forward issettled at the end of the underlying three-month period
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.16
Forward Rates and EurodollarFutures continued
)012.0
21
1
2
1
212
about is (typically yearperchanges rate short the of deviation standard
the is and ) than later days (90contract futures the underlying rate the
ofmaturity the is contract, futures the ofmaturity to time the is where
rate Futures=rate Forward
is made often "adjustmentconvexity "A
σ
σ
σ−
t
tt
tt
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.17
Convexity Adjustment whenσ=0.012 (Table 6.3, page 141)
73.810
47.58
27.06
12.24
3.22
ConvexityAdjustment (bps)
Maturity ofFutures
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.18
Extending the LIBOR Zero Curve
LIBOR deposit rates define the LIBORzero curve out to one yearEurodollar futures can be used todetermine forward rates and the forwardrates can then be used to bootstrap thezero curve
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.19
Example
so that
If the 400 day LIBOR rate has been calculatedas 4.80% and the forward rate for the periodbetween 400 and 491 days is 5.30 the 491 daysrate is 4.893%
ii
iiiii TT
TRTRF−−
=+
++
1
11
1
11
)(
+
++
+−=
i
iiiiii T
TRTTFR
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.20
Duration Matching
This involves hedging against interestrate risk by matching the durations ofassets and liabilitiesIt provides protection against smallparallel shifts in the zero curve
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.21
Use of Eurodollar Futures
One contract locks in an interest rate on$1 million for a future 3-month periodHow many contracts are necessary to lockin an interest rate for a future six monthperiod?
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.22
Duration-Based Hedge Ratio
FC
P
DFPD
Duration of portfolio at hedge maturityDP
Value of portfolio being hedgedP
Duration of asset underlying futures atmaturity
DF
Contract price for interest rate futuresFC
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.23
ExampleIt is August. A fund manager has $10 million invested ina portfolio of government bonds with a duration of 6.80years and wants to hedge against interest rate movesbetween August and DecemberThe manager decides to use December T-bond futures.The futures price is 93-02 or 93.0625 and the duration ofthe cheapest to deliver bond is 9.2 yearsThe number of contracts that should be shorted is
7920.980.6
50.062,93000,000,10
=×
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.24
Limitations of Duration-BasedHedging
Assumes that only parallel shift in yieldcurve take placeAssumes that yield curve changes aresmall
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 6.25
GAP Management (Business Snapshot 6.3)
This is a more sophisticated approachused by banks to hedge interest rate. ItinvolvesBucketing the zero curveHedging exposure to situation where ratescorresponding to one bucket change andall other rates stay the same.
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.1
Swaps
Chapter 7
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.2
Nature of Swaps
A swap is an agreement toexchange cash flows at specifiedfuture times according to certainspecified rules
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.3
An Example of a “Plain Vanilla”Interest Rate Swap
An agreement by Microsoft to receive6-month LIBOR & pay a fixed rate of 5%per annum every 6 months for 3 yearson a notional principal of $100 millionNext slide illustrates cash flows
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.4
---------Millions of Dollars---------LIBOR FLOATING FIXED Net
Date Rate Cash Flow Cash Flow Cash FlowMar.5, 2004 4.2%
Sept. 5, 2004 4.8% +2.10 –2.50 –0.40Mar.5, 2005 5.3% +2.40 –2.50 –0.10
Sept. 5, 2005 5.5% +2.65 –2.50 +0.15Mar.5, 2006 5.6% +2.75 –2.50 +0.25
Sept. 5, 2006 5.9% +2.80 –2.50 +0.30Mar.5, 2007 6.4% +2.95 –2.50 +0.45
Cash Flows to Microsoft(See Table 7.1, page 151)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.5
Typical Uses of anInterest Rate Swap
Converting a liability fromfixed rate to floating ratefloating rate to fixed rate
Converting an investment fromfixed rate to floating ratefloating rate to fixed rate
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.6
Intel and Microsoft (MS)Transform a Liability(Figure 7.2, page 152)
Intel MS
LIBOR
5%
LIBOR+0.1%
5.2%
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.7
Financial Institution is Involved(Figure 7.4, page 153)
F.I.
LIBOR LIBORLIBOR+0.1%
4.985% 5.015%
5.2%Intel MS
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.8
Intel and Microsoft (MS)Transform an Asset(Figure 7.3, page 153)
Intel MS
LIBOR
5%
LIBOR-0.2%
4.7%
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.9
Financial Institution is Involved(See Figure 7.5, page 154)
Intel F.I. MS
LIBOR LIBOR
4.7%
5.015%4.985%
LIBOR-0.2%
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.10
Quotes By a Swap Market Maker(Table 7.3, page 155)
6.8506.876.8310 years
6.6656.686.657 years
6.4906.516.475 years
6.3706.396.354 years
6.2256.246.213 years
6.0456.066.032 yearsSwap Rate (%)Offer (%)Bid (%)Maturity
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.11
The Comparative AdvantageArgument (Table 7.4, page 157)
AAACorp wants to borrow floatingBBBCorp wants to borrow fixed
Fixed Floating
AAACorp 4.0% 6-month LIBOR + 0.30%
BBBCorp 5.20% 6-month LIBOR + 1.00%
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.12
The Swap (Figure 7.6, page 158)
AAACorp BBBCorp
LIBOR
LIBOR+1%
3.95%
4%
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.13
The Swap when a FinancialInstitution is Involved(Figure 7.7, page 158)
AAACorp F.I. BBBCorp4%
LIBOR LIBOR
LIBOR+1%
3.93% 3.97%
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.14
Criticism of the ComparativeAdvantage Argument
The 4.0% and 5.2% rates available to AAACorpand BBBCorp in fixed rate markets are 5-yearratesThe LIBOR+0.3% and LIBOR+1% ratesavailable in the floating rate market are six-month ratesBBBCorp’s fixed rate depends on the spreadabove LIBOR it borrows at in the future
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.15
The Nature of Swap Rates
Six-month LIBOR is a short-term AA borrowingrateThe 5-year swap rate has a risk correspondingto the situation where 10 six-month loans aremade to AA borrowers at LIBORThis is because the lender can enter into aswap where income from the LIBOR loans isexchanged for the 5-year swap rate
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.16
Using Swap Rates to Bootstrap theLIBOR/Swap Zero Curve
Consider a new swap where the fixed rate is theswap rateWhen principals are added to both sides on thefinal payment date the swap is the exchange ofa fixed rate bond for a floating rate bondThe floating-rate rate bond is worth par. Theswap is worth zero. The fixed-rate bond musttherefore also be worth parThis shows that swap rates define par yieldbonds that can be used to bootstrap the LIBOR(or LIBOR/swap) zero curve
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.17
Valuation of an Interest RateSwap that is not New
Interest rate swaps can be valued asthe difference between the value of afixed-rate bond and the value of afloating-rate bondAlternatively, they can be valued as aportfolio of forward rate agreements(FRAs)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.18
Valuation in Terms of Bonds
The fixed rate bond is valued in the usualwayThe floating rate bond is valued by notingthat it is worth par immediately after thenext payment date
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.19
Valuation in Terms of FRAs
Each exchange of payments in an interestrate swap is an FRAThe FRAs can be valued on theassumption that today’s forward rates arerealized
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.20
An Example of a Currency Swap
An agreement to pay 11% on a sterlingprincipal of £10,000,000 & receive 8% ona US$ principal of $15,000,000 everyyear for 5 years
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.21
Exchange of Principal
In an interest rate swap theprincipal is not exchangedIn a currency swap the principal isusually exchanged at thebeginning and the end of theswap’s life
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.22
The Cash Flows (Table 7.7, page 166)
Year
Dollars Pounds$
------millions------2004 –15.00 +10.002005 +0.60 –0.70 2006 +0.60 –0.70 2007 +0.60 –0.702008 +0.60 –0.70 2009 +15.60 - 10.70
£
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.23
Typical Uses of aCurrency Swap
Conversion from a liability in one currencyto a liability in another currency
Conversion from an investment in onecurrency to an investment in anothercurrency
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.24
Comparative Advantage Argumentsfor Currency Swaps (Table 7.8, page 167)
General Motors wants to borrow AUDQantas wants to borrow USD
USD AUD
General Motors 5.0% 12.6%Qantas 7.0% 13.0%
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.25
Valuation of Currency Swaps
Like interest rate swaps, currencyswaps can be valued either as thedifference between 2 bonds or as aportfolio of forward contracts
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.26
Swaps & Forwards
A swap can be regarded as a convenient wayof packaging forward contractsThe “plain vanilla” interest rate swap in ourexample (slide 7.4) consisted of 6 FRAsThe “fixed for fixed” currency swap in ourexample (slide 7.22) consisted of a cashtransaction & 5 forward contracts
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.27
Swaps & Forwards(continued)
The value of the swap is the sum of thevalues of the forward contracts underlyingthe swapSwaps are normally “at the money” initially
This means that it costs nothing to enterinto a swapIt does not mean that each forwardcontract underlying a swap is “at themoney” initially
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.28
Credit Risk
A swap is worth zero to a companyinitiallyAt a future time its value is liable to beeither positive or negativeThe company has credit risk exposureonly when its value is positive
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 7.29
Other Types of Swaps
Floating-for-floating interest rate swaps,amortizing swaps, step up swaps, forwardswaps, constant maturity swaps,compounding swaps, LIBOR-in-arrearsswaps, accrual swaps, diff swaps, crosscurrency interest rate swaps, equity swaps,extendable swaps, puttable swaps,swaptions, commodity swaps, volatilityswaps……..
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 8.1
Mechanics of OptionsMarkets
Chapter 8
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 8.2
Review of Option Types
A call is an option to buyA put is an option to sellA European option can be exercised onlyat the end of its lifeAn American option can be exercised atany time
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 8.3
Option Positions
Long callLong putShort callShort put
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 8.4
Long Call on eBay(Figure 8.1, Page 182)
Profit from buying one eBay European call option: optionprice = $5, strike price = $100, option life = 2 months
30
20
10
0-5
70 80 90 100
110 120 130
Profit ($)
Terminalstock price ($)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 8.5
Short Call on eBay(Figure 8.3, page 184)
Profit from writing one eBay European call option: optionprice = $5, strike price = $100
-30
-20
-10
05
70 80 90 100
110 120 130
Profit ($)
Terminalstock price ($)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 8.6
Long Put on IBM(Figure 8.2, page 183)
Profit from buying an IBM European put option: optionprice = $7, strike price = $70
30
20
10
0-7
70605040 80 90 100
Profit ($)
Terminalstock price ($)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 8.7
Short Put on IBM(Figure 8.4, page 184)
Profit from writing an IBM European put option: optionprice = $7, strike price = $70
-30
-20
-10
70
70
605040
80 90 100
Profit ($)Terminal
stock price ($)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 8.8
Payoffs from OptionsWhat is the Option Position in Each Case?
K = Strike price, ST = Price of asset at maturity
Payoff Payoff
ST STKK
Payoff Payoff
ST STKK
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 8.9
Assets UnderlyingExchange-Traded OptionsPage 185-186
StocksForeign CurrencyStock IndicesFutures
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 8.10
Specification ofExchange-Traded Options
Expiration dateStrike priceEuropean or AmericanCall or Put (option class)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 8.11
Terminology
Moneyness :At-the-money optionIn-the-money optionOut-of-the-money option
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 8.12
Terminology(continued)
Option class Option series Intrinsic value Time value
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 8.13
Dividends & Stock Splits(Page 188-190)
Suppose you own N options with a strikeprice of K :
No adjustments are made to the optionterms for cash dividendsWhen there is an n-for-m stock split,
the strike price is reduced to mK/nthe no. of options is increased to nN/m
Stock dividends are handled in a mannersimilar to stock splits
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 8.14
Dividends & Stock Splits(continued)
Consider a call option to buy 100shares for $20/shareHow should terms be adjusted:
for a 2-for-1 stock split?for a 5% stock dividend?
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 8.15
Market Makers
Most exchanges use market makers tofacilitate options tradingA market maker quotes both bid and askprices when requestedThe market maker does not know whetherthe individual requesting the quotes wantsto buy or sell
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 8.16
Margins (Page 194-195)
Margins are required when options are soldWhen a naked option is written the margin is thegreater of:1 A total of 100% of the proceeds of the sale plus
20% of the underlying share price less theamount (if any) by which the option is out of themoney
2 A total of 100% of the proceeds of the sale plus10% of the underlying share price
For other trading strategies there are special rules
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 8.17
Warrants
Warrants are options that are issued by acorporation or a financial institutionThe number of warrants outstanding isdetermined by the size of the originalissue and changes only when they areexercised or when they expire
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 8.18
Warrants(continued)
The issuer settles up with the holderwhen a warrant is exercisedWhen call warrants are issued by acorporation on its own stock, exercisewill lead to new treasury stock beingissued
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 8.19
Executive Stock Options
Executive stock options are a form ofremuneration issued by a company to itsexecutivesThey are usually at the money whenissuedWhen options are exercised the companyissues more stock and sells it to the optionholder for the strike price
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 8.20
Executive Stock Options continued
They become vested after a period of time(usually 1 to 4 years)They cannot be soldThey often last for as long as 10 or 15yearsAccounting standards now require theexpensing of executive stock options
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 8.21
Convertible Bonds
Convertible bonds are regular bondsthat can be exchanged for equity atcertain times in the future according toa predetermined exchange ratioVery often a convertible is callableThe call provision is a way in which theissuer can force conversion at a timeearlier than the holder might otherwisechoose
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 9.1
Properties of StockOptions
Chapter 9
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 9.2
Notation c : European calloption price p : European putoption price S0 : Stock price today K : Strike price T : Life of option σ: Volatility of stockprice
C : American Call optionprice P : American Put optionprice ST :Stock price at optionmaturity D : Present value ofdividends during option’slife r : Risk-free rate formaturity T with cont comp
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 9.3
Effect of Variables on OptionPricing (Table 9.1, page 206)
c p C PVariableS0KTσrD
+ + –+
? ? + ++ + + ++ – + –
–– – +
– + – +
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 9.4
American vs European Options
An American option is worthat least as much as thecorresponding Europeanoption
C ≥ cP ≥ p
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 9.5
Calls: An Arbitrage Opportunity?
Suppose that c = 3 S0 = 20 T = 1 r = 10% K = 18 D = 0
Is there an arbitrage opportunity?
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 9.6
Lower Bound for European CallOption Prices; No Dividends (Equation9.1, page 211)
c ≥ S0 –Ke -rT
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 9.7
Puts: An Arbitrage Opportunity?
Suppose that p = 1 S0 = 37T = 0.5 r =5%
K = 40 D = 0
Is there an arbitrageopportunity?
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 9.8
Lower Bound for European PutPrices; No Dividends(Equation 9.2, page 212)
p ≥ Ke -rT–S0
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 9.9
Put-Call Parity; No Dividends(Equation 9.3, page 212)
Consider the following 2 portfolios:Portfolio A: European call on a stock + PV of thestrike price in cashPortfolio C: European put on the stock + the stock
Both are worth max(ST , K ) at the maturity of theoptionsThey must therefore be worth the same today. Thismeans that
c + Ke -rT = p + S0
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 9.10
Arbitrage OpportunitiesSuppose that c = 3 S0 = 31T = 0.25 r = 10%K =30 D = 0What are the arbitragepossibilities when
p = 2.25 ? p = 1 ?
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 9.11
Early Exercise
Usually there is some chance that anAmerican option will be exercisedearlyAn exception is an American call on anon-dividend paying stockThis should never be exercised early
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 9.12
For an American call option:S0 = 100; T = 0.25; K = 60; D = 0
Should you exercise immediately?What should you do if
you want to hold the stock for the next 3months?you do not feel that the stock is worth holdingfor the next 3 months?
An Extreme Situation
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 9.13
Reasons For Not Exercising aCall Early (No Dividends)
No income is sacrificedPayment of the strike price isdelayedHolding the call provides insuranceagainst stock price falling belowstrike price
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 9.14
Should Puts Be ExercisedEarly ?
Are there any advantages toexercising an American put when
S0 = 60; T = 0.25; r=10% K = 100; D = 0
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 9.15
The Impact of Dividends onLower Bounds to Option Prices(Equations 9.5 and 9.6, pages 218-219)
rTKeDSc −−−≥ 0
0SKeDp rT −+≥ −
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 9.16
Extensions of Put-Call Parity
American options; D = 0S0 - K < C - P < S0 - Ke -rT
(Equation 9.4, p. 215)
European options; D > 0c + D + Ke -rT = p + S0(Equation 9.7, p. 219)
American options; D > 0S0 - D - K < C - P < S0 - Ke -rT
(Equation 9.8, p. 219)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 10.1
Trading StrategiesInvolving Options
Chapter 10
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 10.2
Types of Strategies
Take a position in the option andthe underlyingTake a position in 2 or moreoptions of the same type (A spread)Combination: Take a position in amixture of calls & puts (Acombination)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 10.3
Positions in an Option & theUnderlying (Figure 10.1, page 224)
Profit
STK
Profit
ST
K
Profit
ST
K
Profit
STK
(a) (b)
(c) (d)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 10.4
Bull Spread Using Calls(Figure 10.2, page 225)
K1 K2
Profit
ST
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 10.5
Bull Spread Using PutsFigure 10.3, page 226
K1 K2
Profit
ST
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 10.6
Bear Spread Using PutsFigure 10.4, page 227
K1 K2
Profit
ST
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 10.7
Bear Spread Using CallsFigure 10.5, page 229
K1 K2
Profit
ST
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 10.8
Box Spread
A combination of a bull call spread and abear put spreadIf all options are European a box spread isworth the present value of the differencebetween the strike pricesIf they are American this is not necessarilyso. (See Business Snapshot 10.1)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 10.9
Butterfly Spread Using CallsFigure 10.6, page 231
K1 K3
Profit
STK2
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 10.10
Butterfly Spread Using PutsFigure 10.7, page 232
K1 K3
Profit
STK2
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 10.11
Calendar Spread Using CallsFigure 10.8, page 232
Profit
ST
K
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 10.12
Calendar Spread Using PutsFigure 10.9, page 233
Profit
ST
K
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 10.13
A Straddle CombinationFigure 10.10, page 234
Profit
STK
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 10.14
Strip & StrapFigure 10.11, page 235
Profit
K ST
Profit
K ST
Strip Strap
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 10.15
A Strangle CombinationFigure 10.12, page 236
K1 K2
Profit
ST
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 11.1
Binomial Trees
Chapter 11
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 11.2
A Simple Binomial Model
A stock price is currently $20In three months it will be either $22 or $18
Stock Price = $22
Stock Price = $18
Stock price = $20
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 11.3
Stock Price = $22Option Price = $1
Stock Price = $18Option Price = $0
Stock price = $20Option Price=?
A Call Option (Figure 11.1, page 242)
A 3-month call option on the stock has a strike price of21.
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 11.4
Consider the Portfolio: long ∆ sharesshort 1 call option
Portfolio is riskless when 22∆ – 1 = 18∆ or∆ = 0.25
22∆ – 1
18∆
Setting Up a Riskless Portfolio
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 11.5
Valuing the Portfolio(Risk-Free Rate is 12%)
The riskless portfolio is:long 0.25 sharesshort 1 call option
The value of the portfolio in 3 months is22 × 0.25 – 1 = 4.50
The value of the portfolio today is 4.5e – 0.12×0.25 = 4.3670
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 11.6
Valuing the OptionThe portfolio that is
long 0.25 sharesshort 1 option
is worth 4.367The value of the shares is
5.000 (= 0.25 × 20 )The value of the option is therefore
0.633 (= 5.000 – 4.367 )
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 11.7
Generalization (Figure 11.2, page 243)
A derivative lasts for time T and isdependent on a stock
S0u ƒu
S0d ƒd
S0ƒ
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 11.8
Generalization(continued)
Consider the portfolio that is long ∆ shares and short 1derivative
The portfolio is riskless when S0u∆ – ƒu = S0d∆ – ƒd or
dSuSfdu
00 −−
=∆ƒ
S0u∆ – ƒu
S0d∆ – ƒd
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 11.9
Generalization(continued)
Value of the portfolio at time T isS0u∆ – ƒu
Value of the portfolio today is (S0u∆ – ƒu)e–rT
Another expression for theportfolio value today is S0∆ – fHence
ƒ = S0∆ – (S0u∆ – ƒu )e–rT
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 11.10
Generalization(continued)
Substituting for ∆ we obtain ƒ = [ pƒu + (1 – p)ƒd ]e–rT
where
p e du d
rT
=−
−
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 11.11
p as a Probability
It is natural to interpret p and 1-p as probabilities ofup and down movementsThe value of a derivative is then its expected payoffin a risk-neutral world discounted at the risk-free rate
S0u ƒu
S0d ƒd
S0ƒ
p
(1 – p )
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 11.12
Risk-neutral Valuation
When the probability of an up and downmovements are p and 1-p the expected stockprice at time T is S0erT
This shows that the stock price earns the risk-free rateBinomial trees illustrate the general result that tovalue a derivative we can assume that theexpected return on the underlying asset is therisk-free rate and discount at the risk-free rateThis is known as using risk-neutral valuation
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 11.13
Original Example Revisited
Since p is the probability that gives a return on thestock equal to the risk-free rate. We can find it from20e0.12 ×0.25 = 22p + 18(1 – p )which gives p = 0.6523Alternatively, we can use the formula
6523.09.01.1
9.00.250.12
=−
−=
−−
=×e
dudep
rT
S0u = 22 ƒu = 1
S0d = 18 ƒd = 0
S0 ƒ
p
(1 – p )
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 11.14
Valuing the Option Using Risk-Neutral Valuation
The value of the option is e–0.12×0.25 (0.6523×1 + 0.3477×0) = 0.633
S0u = 22 ƒu = 1
S0d = 18 ƒd = 0
S0ƒ
0.6523
0.3477
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 11.15
Irrelevance of Stock’s ExpectedReturn
When we are valuing an option in terms of thethe price of the underlying asset, theprobability of up and down movements in thereal world are irrelevantThis is an example of a more general resultstating that the expected return on theunderlying asset in the real world is irrelevant
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 11.16
A Two-Step ExampleFigure 11.3, page 246
Each time step is 3 monthsK=21, r=12%
20
22
18
24.2
19.8
16.2
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 11.17
Valuing a Call OptionFigure 11.4, page 247
Value at node B = e–0.12×0.25(0.6523×3.2 + 0.3477×0) = 2.0257
Value at node A = e–0.12×0.25(0.6523×2.0257 + 0.3477×0)
= 1.2823
201.2823
22
18
24.23.2
19.80.0
16.20.0
2.0257
0.0
A
B
C
D
E
F
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 11.18
A Put Option Example; K=52Figure 11.7, page 250
K = 52, time step = 1yrr = 5%
504.1923
60
40
720
484
3220
1.4147
9.4636
A
B
C
D
E
F
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 11.19
What Happens When an Option isAmerican (Figure 11.8, page 251)
505.0894
60
40
720
484
3220
1.4147
12.0
A
B
C
D
E
F
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 11.20
Delta
Delta (∆) is the ratio of the changein the price of a stock option to thechange in the price of theunderlying stockThe value of ∆ varies from node tonode
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 11.21
Choosing u and d
One way of matching the volatility is to set
where σ is the volatility and ∆t is the lengthof the time step. This is the approach usedby Cox, Ross, and Rubinstein
t
t
eud
eu∆σ−
∆σ
==
=
1
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 11.22
The Probability of an Up Move
contract futures a for rate free-risk
foreign the is herecurrency w a for
index the on yield dividend the is eindex wher stock a for
stock paying dnondividen a for
1
)(
)(
=
=
=
=
−−
=
∆−
∆−
∆
a
rea
qeaea
dudap
ftrr
tqr
tr
f
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Wiener Processes andItô’s Lemma
Chapter 12
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Types of Stochastic Processes
Discrete time; discrete variableDiscrete time; continuous variableContinuous time; discrete variableContinuous time; continuous variable
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Modeling Stock Prices
We can use any of the four types ofstochastic processes to model stockpricesThe continuous time, continuousvariable process proves to be the mostuseful for the purposes of valuingderivatives
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Markov Processes (See pages 263-64)
In a Markov process future movementsin a variable depend only on where weare, not the history of how we gotwhere we areWe assume that stock prices followMarkov processes
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Weak-Form Market Efficiency
This asserts that it is impossible toproduce consistently superior returns witha trading rule based on the past history ofstock prices. In other words technicalanalysis does not work.A Markov process for stock prices isclearly consistent with weak-form marketefficiency
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Example of a Discrete TimeContinuous Variable Model
A stock price is currently at $40At the end of 1 year it is considered that itwill have a probability distribution ofφ(40,10) where φ(µ,σ) is a normaldistribution with mean µ and standarddeviation σ.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
QuestionsWhat is the probability distribution of thestock price at the end of 2 years?½ years?¼ years? ∆t years?
Taking limits we have defined acontinuous variable, continuous timeprocess
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Variances & StandardDeviations
In Markov processes changes insuccessive periods of time areindependentThis means that variances are additiveStandard deviations are not additive
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Variances & Standard Deviations(continued)
In our example it is correct to say thatthe variance is 100 per year.It is strictly speaking not correct to saythat the standard deviation is 10 peryear.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
A Wiener Process (See pages 265-67)
We consider a variable z whose value changescontinuouslyThe change in a small interval of time ∆t is ∆zThe variable follows a Wiener process if
1.
2. The values of ∆z for any 2 different (non-overlapping) periods of time are independent
(0,1) is where φε∆ε=∆ tz
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Properties of a Wiener Process
Mean of [z (T ) – z (0)] is 0Variance of [z (T ) – z (0)] is TStandard deviation of [z (T ) – z (0)] is
T
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Taking Limits . . .
What does an expression involving dz and dtmean?It should be interpreted as meaning that thecorresponding expression involving ∆z and ∆t istrue in the limit as ∆t tends to zeroIn this respect, stochastic calculus is analogous toordinary calculus
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Generalized Wiener Processes(See page 267-69)
A Wiener process has a drift rate (i.e.average change per unit time) of 0and a variance rate of 1In a generalized Wiener process thedrift rate and the variance rate can beset equal to any chosen constants
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Generalized Wiener Processes(continued)
The variable x follows a generalizedWiener process with a drift rate of aand a variance rate of b2 if
dx=a dt+b dz
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Generalized Wiener Processes(continued)
Mean change in x in time T is aTVariance of change in x in time T is b2TStandard deviation of change in x intime T is
tbtax ∆ε+∆=∆
b T
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
The Example Revisited
A stock price starts at 40 and has a probabilitydistribution of φ(40,10) at the end of the yearIf we assume the stochastic process is Markovwith no drift then the process is
dS = 10dzIf the stock price were expected to grow by $8on average during the year, so that the year-end distribution is φ(48,10), the process wouldbe
dS = 8dt + 10dz
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Itô Process (See pages 269)
In an Itô process the drift rate and thevariance rate are functions of time
dx=a(x,t) dt+b(x,t) dzThe discrete time equivalent
is only true in the limit as ∆t tends to zero
ttxbttxax ∆ε+∆=∆ ),(),(
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Why a Generalized Wiener Processis not Appropriate for Stocks
For a stock price we can conjecture that itsexpected percentage change in a short periodof time remains constant, not its expectedabsolute change in a short period of timeWe can also conjecture that our uncertainty asto the size of future stock price movements isproportional to the level of the stock price
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
An Ito Process for Stock Prices(See pages 269-71)
where µ is the expected return σ isthe volatility.
The discrete time equivalent is
dzSdtSdS σ+µ=
tStSS ∆εσ+∆µ=∆
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Monte Carlo Simulation
We can sample random paths for thestock price by sampling values for εSuppose µ= 0.14, σ= 0.20, and ∆t = 0.01,then
ε+=∆ SSS 02.00014.0
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Monte Carlo Simulation – One Path (See Table12.1, page 272)
Period
Stock Price at Start of Period
Random Sample for ε
Change in Stock Price, ∆S
0 20.000 0.52 0.236
1 20.236 1.44 0.611
2 20.847 -0.86 -0.329
3 20.518 1.46 0.628
4 21.146 -0.69 -0.262
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Itô’s Lemma (See pages 273-274)
If we know the stochastic processfollowed by x, Itô’s lemma tells us thestochastic process followed by somefunction G (x, t )Since a derivative security is a function ofthe price of the underlying and time, Itô’slemma plays an important part in theanalysis of derivative securities
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Taylor Series Expansion
A Taylor’s series expansion of G(x, t)gives
K+∆∂∂
+∆∆∂∂
∂+
∆∂∂
+∆∂∂
+∆∂∂
=∆
22
22
22
2
ttGtx
txG
xxGt
tGx
xGG
½
½
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Ignoring Terms of Higher OrderThan ∆t
t
x
xxGt
tGx
xGG
ttGx
xGG
∆
∆
∆+∆+∆=∆
∆+∆=∆
½
22
2
order of
is whichcomponent a has because
becomes this calculus stochastic In
havewecalculusordinary In
∂∂
∂∂
∂∂
∂∂
∂∂
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Substituting for ∆x
tbxGt
tGx
xGG
ttbtax
dztxbdttxadx
∆ε∂∂
+∆∂∂
+∆∂∂
=∆
∆∆ε∆∆
+=
222
2
½
order thanhigher of termsignoringThen + =
thatso),(),(
Suppose
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
The ε2∆t Term
tbxGt
tGx
xGG
ttttE
EEE
E
∆∂∂
+∆∂∂
+∆∂∂
=∆
∆∆
∆=∆ε
=ε
=ε−ε
=εφ≈ε
22
2
2
2
22
21
)(1)(
1)]([)(0)(,)1,0(
Hence ignored. be can and to alproportion is of variance The
that follows It
Since
2
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Taking Limits
Lemma sIto' is This
½ obtain We
ngSubstituti
½ limits Taking
dzbxGdtb
xG
tGa
xGdG
dzbdtadx
dtbxGdt
tGdx
xGdG
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
=
+=∂∂
+∂∂
+∂∂
=
22
2
22
2
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Application of Ito’s Lemmato a Stock Price Process
dzSSGdtS
SG
tGS
SGdG
tSGzdSdtSSd
½
and of function a For
is processprice stockThe
σ∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛σ
∂∂
+∂∂
+µ∂∂
=
σ+µ=
222
2
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Examples
dzdtdG
SG
dzGdtGrdGeSG
TtTr
2.
time at maturing contract a for stock a of price forward The 1.
σ+⎟⎟⎠
⎞⎜⎜⎝
⎛ σ−µ=
=
σ+−µ== −
2
ln
)(
2
)(
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.1
The Black-Scholes-Merton Model
Chapter 13
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.2
The Stock Price Assumption
Consider a stock whose price is SIn a short period of time of length ∆t, thereturn on the stock is normally distributed:
where µ is expected return and σ is volatility
( )ttSS
∆∆≈∆ σµφ ,
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.3
The Lognormal Property(Equations 13.2 and 13.3, page 282)
It follows from this assumption that
Since the logarithm of ST is normal, ST islognormally distributed
ln ln ,
ln ln ,
S S T T
S S T T
T
T
− ≈ −⎛⎝⎜
⎞⎠⎟
⎡
⎣⎢
⎤
⎦⎥
≈ + −⎛⎝⎜
⎞⎠⎟
⎡
⎣⎢
⎤
⎦⎥
0
2
0
2
2
2
φ µσ
σ
φ µσ
σ
or
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
The Lognormal Distribution
E S S e
S S e eT
T
TT T
( )
( ) ( )
=
= −0
02 2 2
1
var
µ
µ σ
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.5
Continuously Compounded Return, x(Equations 13.6 and 13.7), page 283)
,2
or
ln1=
or
2
0
0
⎟⎟⎠
⎞⎜⎜⎝
⎛−≈
=
Tx
SS
Tx
eSS
T
xTT
σσµφ
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.6
The Expected Return
The expected value of the stock price is S0eµT
The expected return on the stock is µ – σ2/2 not µ
This is because
are not the same
)]/[ln()]/(ln[ 00 SSESSE TT and
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.7
µ and µ- σ2/2
Suppose we have daily data for a period ofseveral monthsµ is the average of the returns in each day[=E(∆S/S)]µ- σ2/2 is the expected return over thewhole period covered by the datameasured with continuous compounding(or daily compounding, which is almost thesame)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.8
Mutual Fund Returns (See BusinessSnapshot 13.1 on page 285)
Suppose that returns in successive yearsare 15%, 20%, 30%, -20% and 25%The arithmetic mean of the returns is 14%The returned that would actually beearned over the five years (the geometricmean) is 12.4%
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.9
The Volatility
The volatility is the standard deviation of thecontinuously compounded rate of return in 1yearThe standard deviation of the return in time∆t isIf a stock price is $50 and its volatility is 25%per year what is the standard deviation ofthe price change in one day?
t∆σ
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.10
Estimating Volatility fromHistorical Data (page 286-88)
• Take observations S0, S1, . . . , Sn atintervals of τ years
• Calculate the continuously compoundedreturn in each interval as:
• Calculate the standard deviation, s , ofthe ui´s
• The historical volatility estimate is:
uS
Sii
i=
⎛⎝⎜
⎞⎠⎟
−
ln1
τ=σ
sˆ
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.11
Nature of Volatility
Volatility is usually much greater when themarket is open (i.e. the asset is trading)than when it is closedFor this reason time is usually measuredin “trading days” not calendar days whenoptions are valued
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.12
The Concepts Underlying Black-Scholes
The option price and the stock price dependon the same underlying source of uncertaintyWe can form a portfolio consisting of thestock and the option which eliminates thissource of uncertaintyThe portfolio is instantaneously riskless andmust instantaneously earn the risk-free rateThis leads to the Black-Scholes differentialequation
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.13
The Derivation of the Black-ScholesDifferential Equation
shares :ƒ+
derivative :1 of consisting portfolio a upset e W
ƒƒ½ƒƒƒ
222
2
S
zSS
tSSt
SS
zStSS
∂∂
−
∆σ∂∂
+∆⎟⎟⎠
⎞⎜⎜⎝
⎛σ
∂∂
+∂∂
+µ∂∂
=∆
∆σ+∆µ=∆
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.14
ƒƒ
bygiven is in time valueitsin change The
ƒƒ
bygiven is portfolio theof valueThe
SS
t
SS
∆∂∂
+∆−=∆Π
∆∂∂
+−=Π
Π
The Derivation of the Black-ScholesDifferential Equation continued
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.15
The Derivation of the Black-ScholesDifferential Equation continued
ƒƒ ½ƒƒ
:equation aldifferenti Scholes-Black get the toequations in these and ƒfor substitute We
Hence rate.
free-riskthebemust portfolioon thereturn The
2
222 r
SS
SrS
t
Str
=∂∂
σ+∂∂
+∂∂
∆∆Π∆=∆Π
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.16
The Differential Equation
Any security whose price is dependent on thestock price satisfies the differential equationThe particular security being valued is determinedby the boundary conditions of the differentialequationIn a forward contract the boundary condition is
ƒ = S – K when t =T The solution to the equation is
ƒ = S – K e–r (T – t )
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.17
The Black-Scholes Formulas(See pages 295-297)
TdT
TrKSd
TTrKSd
dNSdNeKp
dNeKdNScrT
rT
σ−=σ
σ−+=
σσ++
=
−−−=
−=−
−
10
2
01
102
210
)2/2()/ln(
)2/2()/ln(
)()(
)()(
where
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.18
The N(x) Function
N(x) is the probability that a normallydistributed variable with a mean of zeroand a standard deviation of 1 is less than xSee tables at the end of the book
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.19
Properties of Black-Scholes Formula
As S0 becomes very large c tends toS – Ke-rT and p tends to zero
As S0 becomes very small c tends to zeroand p tends to Ke-rT – S
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.20
Risk-Neutral Valuation
The variable µ does not appear in the Black-Scholes equationThe equation is independent of all variablesaffected by risk preferenceThe solution to the differential equation istherefore the same in a risk-free world as itis in the real worldThis leads to the principle of risk-neutralvaluation
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.21
Applying Risk-Neutral Valuation(See appendix at the end of Chapter 13)
1. Assume that the expectedreturn from the stock price isthe risk-free rate
2. Calculate the expected payofffrom the option
3. Discount at the risk-free rate
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.22
Valuing a Forward Contract withRisk-Neutral Valuation
Payoff is ST – KExpected payoff in a risk-neutral world isSerT – KPresent value of expected payoff is
e-rT[SerT – K]=S – Ke-rT
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.23
Implied Volatility
The implied volatility of an option is thevolatility for which the Black-Scholes priceequals the market priceThe is a one-to-one correspondencebetween prices and implied volatilitiesTraders and brokers often quote impliedvolatilities rather than dollar prices
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.24
An Issue of Warrants & ExecutiveStock Options
When a regular call option is exercised the stockthat is delivered must be purchased in the openmarketWhen a warrant or executive stock option isexercised new Treasury stock is issued by thecompanyIf little or no benefits are foreseen by the marketthe stock price will reduce at the time the issue ofis announced.There is no further dilution (See BusinessSnapshot 13.3.)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.25
The Impact of Dilution
After the options have been issued it is notnecessary to take account of dilution whenthey are valuedBefore they are issued we can calculatethe cost of each option as N/(N+M) timesthe price of a regular option with the sameterms where N is the number of existingshares and M is the number of new sharesthat will be created if exercise takes place
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.26
Dividends
European options on dividend-payingstocks are valued by substituting the stockprice less the present value of dividendsinto Black-ScholesOnly dividends with ex-dividend datesduring life of option should be includedThe “dividend” should be the expectedreduction in the stock price expected
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.27
American Calls
An American call on a non-dividend-payingstock should never be exercised earlyAn American call on a dividend-paying stockshould only ever be exercised immediatelyprior to an ex-dividend dateSuppose dividend dates are at times t1, t2,…tn. Early exercise is sometimes optimal attime ti if the dividend at that time is greaterthan
]1[ )( 1 ii ttreK −− +−
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 13.28
Black’s Approximation for Dealing withDividends in American Call Options
Set the American price equal to themaximum of two European prices:1. The 1st European price is for anoption maturing at the same time as theAmerican option2. The 2nd European price is for anoption maturing just before the final ex-dividend date
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.1
Options on Stock Indices,Currencies, and Futures
Chapter 14
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.2
European Options on StocksProviding a Dividend Yield
We get the same probabilitydistribution for the stock price at timeT in each of the following cases:1. The stock starts at price S0 andprovides a dividend yield = q2. The stock starts at price S0e–q T
and provides no income
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.3
European Options on StocksProviding Dividend Yieldcontinued
We can value European options byreducing the stock price to S0e–q T and thenbehaving as though there is no dividend
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.4
Extension of Chapter 9 Results(Equations 14.1 to 14.3)
rTqT KeeSc −− −≥ 0
Lower Bound for calls:
Lower Bound for puts
qTrT eSKep −− −≥ 0
Put Call Parity
qTrT eSpKec −− +=+ 0
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.5
Extension of Chapter 13 Results(Equations 14.4 and 14.5)
TTqrKSd
TTqrKSd
dNeSdNKep
dNKedNeScqTrT
rTqT
σσ−−+
=
σσ+−+
=
−−−=
−=−−
−−
)2/2()/ln(
)2/2()/ln(
)()(
)()(
02
01
102
210
where
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.6
The Binomial Model
S0u ƒu
S0d ƒd
S0 ƒ
p
(1 – p )
f=e-rT[pfu+(1-p)fd ]
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.7
The Binomial Modelcontinued
In a risk-neutral world the stock pricegrows at r-q rather than at r when thereis a dividend yield at rate qThe probability, p, of an up movementmust therefore satisfy
pS0u+(1-p)S0d=S0e (r-q)T
so that
p e du d
r q T
=−
−
−( )
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.8
Index Options (page 316-321)
The most popular underlying indices in theU.S. are
The Dow Jones Index times 0.01 (DJX)The Nasdaq 100 Index (NDX)The Russell 2000 Index (RUT)The S&P 100 Index (OEX)The S&P 500 Index (SPX)
Contracts are on 100 times index; they aresettled in cash; OEX is American and therest are European.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.9
LEAPS
Leaps are options on stock indices thatlast up to 3 yearsThey have December expiration datesThey are on 10 times the indexLeaps also trade on some individualstocks
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.10
Index Option Example
Consider a call option on an indexwith a strike price of 560Suppose 1 contract is exercisedwhen the index level is 580What is the payoff?
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.11
Using Index Options for PortfolioInsurance
Suppose the value of the index is S0 and the strikeprice is KIf a portfolio has a β of 1.0, the portfolio insuranceis obtained by buying 1 put option contract on theindex for each 100S0 dollars heldIf the β is not 1.0, the portfolio manager buys β putoptions for each 100S0 dollars heldIn both cases, K is chosen to give the appropriateinsurance level
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.12
Example 1
Portfolio has a beta of 1.0It is currently worth $500,000The index currently stands at 1000What trade is necessary to provideinsurance against the portfolio value fallingbelow $450,000?
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.13
Example 2
Portfolio has a beta of 2.0It is currently worth $500,000 and indexstands at 1000The risk-free rate is 12% per annumThe dividend yield on both the portfolioand the index is 4%How many put option contracts shouldbe purchased for portfolio insurance?
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.14
If index rises to 1040, it provides a40/1000 or 4% return in 3 monthsTotal return (incl dividends)=5%Excess return over risk-free rate=2%Excess return for portfolio=4%Increase in Portfolio Value=4+3-1=6%Portfolio value=$530,000
Calculating Relation Between Index Leveland Portfolio Value in 3 months
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.15
Determining the Strike Price (Table14.2, page 320)
Value of Index in 3 months
Expected Portfolio Value in 3 months ($)
1,080 570,000 1,040 530,000 1,000 490,000 960 450,000 920 410,000
An option with a strike price of 960 will provide protection
against a 10% decline in the portfolio value
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.16
Valuing European Index Options
We can use the formula for an optionon a stock paying a dividend yieldSet S0 = current index levelSet q = average dividend yieldexpected during the life of the option
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.17
Currency OptionsCurrency options trade on the PhiladelphiaExchange (PHLX)There also exists an active over-the-counter(OTC) marketCurrency options are used by corporationsto buy insurance when they have an FXexposure
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.18
The Foreign Interest Rate
We denote the foreign interest rate by rf
When a U.S. company buys one unit ofthe foreign currency it has aninvestment of S0 dollarsThe return from investing at the foreignrate is rf S0 dollarsThis shows that the foreign currencyprovides a “dividend yield” at rate rf
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.19
Valuing European CurrencyOptions
A foreign currency is an asset thatprovides a “dividend yield” equal to rf
We can use the formula for an optionon a stock paying a dividend yield :
Set S0 = current exchange rate Set q = rƒ
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.20
Formulas for European CurrencyOptions(Equations 14.7 and 14.8, page 322)
T
TfrrKSd
T
TfrrKSd
dNeSdNKep
dNKedNeScTrrT
rTTr
f
f
σ
σ−−+=
σ
σ+−+=
−−−=
−=−−
−−
)2/2()/ln(
)2/2()/ln(
)()(
)()(
0
2
0
1
102
210
where
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.21
Alternative Formulas(Equations 14.9 and 14.10, page 322)
F S e r r Tf0 0= −( )
Using
TddT
TKFd
dNFdKNep
dKNdNFecrT
rT
σ−=
σσ+
=
−−−=
−=−
−
12
20
1
102
210
2/)/ln(
)]()([
)]()([
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.22
Mechanics of Call Futures Options
When a call futures option is exercisedthe holder acquires
1. A long position in the futures2. A cash amount equal to the excess of
the futures price over the strike price
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.23
Mechanics of Put Futures Option
When a put futures option is exercised theholder acquires
1. A short position in the futures2. A cash amount equal to the excess of
the strike price over the futures price
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.24
The Payoffs
If the futures position is closed outimmediately:Payoff from call = F0 – KPayoff from put = K – F0
where F0 is futures price at time ofexercise
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.25
Put-Call Parity for FuturesOptions (Equation 14.11, page 329)
Consider the following two portfolios:1. European call plus Ke-rT of cash
2. European put plus long futures pluscash equal to F0e-rT
They must be worth the same at time T sothat
c+Ke-rT=p+F0 e-rT
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.26
Futures Price = $33Option Price = $4
Futures Price = $28Option Price = $0
Futures price = $30Option Price=?
Binomial Tree Example
A 1-month call option on futures has a strike price of29.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.27
Consider the Portfolio: long ∆ futuresshort 1 call option
Portfolio is riskless when 3∆ – 4 = -2∆ or∆ = 0.8
3∆ – 4
-2∆
Setting Up a Riskless Portfolio
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.28
Valuing the Portfolio( Risk-Free Rate is 6% )
The riskless portfolio is:long 0.8 futuresshort 1 call option
The value of the portfolio in 1 month is-1.6
The value of the portfolio today is -1.6e – 0.06/12 = -1.592
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.29
Valuing the Option
The portfolio that islong 0.8 futuresshort 1 option
is worth -1.592The value of the futures is zeroThe value of the option musttherefore be 1.592
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.30
Generalization of Binomial TreeExample (Figure 14.2, page 330)
A derivative lasts for time T and isdependent on a futures price
F0u ƒu
F0d ƒd
F0 ƒ
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.31
Generalization(continued)
Consider the portfolio that is long ∆ futures and short 1derivative
The portfolio is riskless when
∆ =−−
ƒ u dfF u F d0 0
F0u ∆ − F0 ∆ – ƒu
F0d ∆− F0∆ – ƒd
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.32
Generalization(continued)
Value of the portfolio at time Tis F0u ∆ –F0∆ – ƒu
Value of portfolio today is – ƒHence
ƒ = – [F0u ∆ –F0∆ – ƒu]e-rT
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.33
Generalization(continued)
Substituting for ∆ we obtainƒ = [ p ƒu + (1 – p )ƒd ]e–rT
where
p du d
=−−
1
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.34
Valuing European FuturesOptions
We can use the formula for an optionon a stock paying a dividend yield
Set S0 = current futures price (F0)Set q = domestic risk-free rate (r )
Setting q = r ensures that the expectedgrowth of F in a risk-neutral world iszero
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.35
Growth Rates For Futures Prices
A futures contract requires no initialinvestmentIn a risk-neutral world the expected returnshould be zeroThe expected growth rate of the futuresprice is therefore zeroThe futures price can therefore be treatedlike a stock paying a dividend yield of r
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.36
Black’s Formula(Equations 14.16 and 14.17, page 333)
The formulas for European options onfutures are known as Black’s formulas
[ ][ ]
TdT
TKFd
TTKFd
dNFdNKep
dNKdNFecrT
rT
σ−=σ
σ−=
σσ+
=
−−−=
−=−
−
10
2
01
102
210
2/2)/ln(
2/2)/ln(
)()(
)()(
where
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.37
Futures Option Prices vs SpotOption Prices
If futures prices are higher than spotprices (normal market), an Americancall on futures is worth more than asimilar American call on spot. AnAmerican put on futures is worth lessthan a similar American put on spotWhen futures prices are lower than spotprices (inverted market) the reverse istrue
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 14.38
Summary of Key Results
We can treat stock indices, currencies,and futures like a stock paying adividend yield of q
For stock indices, q = averagedividend yield on the index over theoption lifeFor currencies, q = rƒ
For futures, q = r
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 15.1
The Greek Letters
Chapter 15
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 15.2
Example
A bank has sold for $300,000 a European calloption on 100,000 shares of a nondividendpaying stock S0 = 49, K = 50, r = 5%, σ = 20%,T = 20 weeks, µ = 13%The Black-Scholes value of the option is$240,000How does the bank hedge its risk to lock in a$60,000 profit?
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 15.3
Naked & Covered Positions
Naked positionTake no action
Covered positionBuy 100,000 shares today
Both strategies leave the bankexposed to significant risk
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 15.4
Stop-Loss Strategy
This involves:Buying 100,000 shares as soon asprice reaches $50Selling 100,000 shares as soon asprice falls below $50This deceptively simple hedgingstrategy does not work well
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 15.5
Delta (See Figure 15.2, page 345)
Delta (∆) is the rate of change of theoption price with respect to the underlying
Optionprice
A
BSlope = ∆
Stock price
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 15.6
Delta Hedging
This involves maintaining a delta neutralportfolioThe delta of a European call on a stockpaying dividends at rate q is N (d 1)e– qT
The delta of a European put ise– qT [N (d 1) – 1]
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 15.7
Delta Hedgingcontinued
The hedge position must be frequentlyrebalancedDelta hedging a written option involves a“buy high, sell low” trading ruleSee Tables 15.2 (page 350) and 15.3(page 351) for examples of delta hedging
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 15.8
Using Futures for Delta Hedging
The delta of a futures contract is e(r-q)T
times the delta of a spot contractThe position required in futures for deltahedging is therefore e-(r-q)T times theposition required in the corresponding spotcontract
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 15.9
Theta
Theta (Θ) of a derivative (or portfolio ofderivatives) is the rate of change of the valuewith respect to the passage of timeThe theta of a call or put is usually negative.This means that, if time passes with the price ofthe underlying asset and its volatility remainingthe same, the value of the option declines
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 15.10
Gamma
Gamma (Γ) is the rate of change ofdelta (∆) with respect to the price of theunderlying assetGamma is greatest for options that areclose to the money (see Figure 15.9,page 358)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 15.11
Gamma Addresses Delta HedgingErrors Caused By Curvature(Figure 15.7, page 355)
S
CStock price
S'
Callprice
C''C'
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 15.12
Interpretation of GammaFor a delta neutral portfolio,
∆Π ≈ Θ ∆t + ½Γ∆S 2
∆Π
∆S
Negative Gamma
∆Π
∆S
Positive Gamma
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 15.13
Relationship Between Delta,Gamma, and Theta
For a portfolio of derivatives on a stockpaying a continuous dividend yield atrate q
Θ ∆ Γ Π+ − + =( )r q S S r12
2 2σ
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 15.14
Vega
Vega (ν) is the rate of change of thevalue of a derivatives portfolio withrespect to volatilityVega tends to be greatest for optionsthat are close to the money (See Figure15.11, page 361)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 15.15
Managing Delta, Gamma, & Vega
• ∆ can be changed by taking a position inthe underlying
To adjust Γ & ν it is necessary to take aposition in an option or other derivative
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 15.16
Rho
Rho is the rate of change of thevalue of a derivative with respectto the interest rate
For currency options there are 2rhos
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 15.17
Hedging in Practice
Traders usually ensure that their portfoliosare delta-neutral at least once a dayWhenever the opportunity arises, theyimprove gamma and vegaAs portfolio becomes larger hedgingbecomes less expensive
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 15.18
Scenario Analysis
A scenario analysis involves testing theeffect on the value of a portfolio of differentassumptions concerning asset prices andtheir volatilities
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 15.19
Hedging vs Creation of an OptionSynthetically
When we are hedging we takepositions that offset ∆, Γ, ν, etc.When we create an optionsynthetically we take positionsthat match ∆, Γ, & ν
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 15.20
Portfolio Insurance
In October of 1987 many portfoliomanagers attempted to create a putoption on a portfolio syntheticallyThis involves initially selling enough ofthe portfolio (or of index futures) tomatch the ∆ of the put option
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 15.21
Portfolio Insurancecontinued
As the value of the portfolio increases, the∆ of the put becomes less negative andsome of the original portfolio isrepurchasedAs the value of the portfolio decreases, the∆ of the put becomes more negative andmore of the portfolio must be sold
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 15.22
Portfolio Insurancecontinued
The strategy did not work well on October19, 1987...
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 16.1
Volatility Smiles
Chapter 16
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 16.2
Put-Call Parity Arguments
Put-call parity p +S0e-qT = c +K e–r T
holds regardless of the assumptionsmade about the stock price distributionIt follows thatpmkt-pbs=cmkt-cbs
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 16.3
Implied Volatilities
When pbs=pmkt, it must be true that cbs=cmkt
It follows that the implied volatilitycalculated from a European call optionshould be the same as that calculatedfrom a European put option when bothhave the same strike price and maturityThe same is approximately true ofAmerican options
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 16.4
Volatility Smile
A volatility smile shows the variation ofthe implied volatility with the strike priceThe volatility smile should be the samewhether calculated from call options orput options
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 16.5
The Volatility Smile for ForeignCurrency Options(Figure 16.1, page 377)
ImpliedVolatility
StrikePrice
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 16.6
Implied Distribution for ForeignCurrency Options (Figure 16.2, page 377)
Both tails are heavier than the lognormaldistributionIt is also “more peaked” than thelognormal distribution
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 16.7
The Volatility Smile for EquityOptions (Figure 16.3, page 380)
ImpliedVolatility
StrikePrice
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 16.8
Implied Distribution for EquityOptions (Figure 16.4, page 380)
The left tail is heavier and the right tail isless heavy than the lognormaldistribution
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 16.9
Other Volatility Smiles?
What is the volatility smile ifTrue distribution has a less heavy left tailand heavier right tailTrue distribution has both a less heavy lefttail and a less heavy right tail
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 16.10
Possible Causes of Volatility Smile
Asset price exhibiting jumps rather thancontinuous changeVolatility for asset price being stochastic
(One reason for a stochastic volatility inthe case of equities is the relationshipbetween volatility and leverage)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 16.11
Volatility Term Structure
In addition to calculating a volatility smile,traders also calculate a volatility termstructureThis shows the variation of impliedvolatility with the time to maturity of theoption
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 16.12
Volatility Term Structure
The volatility term structure tends to bedownward sloping when volatility is highand upward sloping when it is low
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 16.13
Example of a Volatility Surface(Table 16.2, page 382)
S tr ik e P r ic e
0 .9 0 0 .9 5 1 .0 0 1 .0 5 1 .1 0
1 m n th 1 4 .2 1 3 .0 1 2 .0 1 3 .1 1 4 .5
3 m n th 1 4 .0 1 3 .0 1 2 .0 1 3 .1 1 4 .2
6 m n th 1 4 .1 1 3 .3 1 2 .5 1 3 .4 1 4 .3
1 y e a r 1 4 .7 1 4 .0 1 3 .5 1 4 .0 1 4 .8
2 y e a r 1 5 .0 1 4 .4 1 4 .0 1 4 .5 1 5 .1
5 y e a r 1 4 .8 1 4 .6 1 4 .4 1 4 .7 1 5 .0
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Basic NumericalProcedures
Chapter 17
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Tree Approaches to DerivativesValuation
TreesMonte Carlo simulationFinite difference methods
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Binomial Trees
Binomial trees are frequently used toapproximate the movements in the price ofa stock or other assetIn each small interval of time the stockprice is assumed to move up by aproportional amount u or to move down bya proportional amount d
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Movements in Time ∆t(Figure 17.1, page 392)
Su
SdS
p
1 – p
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
1. Tree Parameters for assetpaying a dividend yield of q
Parameters p, u, and d are chosen so that the treegives correct values for the mean & variance of thestock price changes in a risk-neutral world
Mean: e(r-q)∆t = pu + (1– p )dVariance: σ2∆t = pu2 + (1– p )d 2 – e2(r-q)∆t
A further condition often imposed is u = 1/ d
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
2. Tree Parameters for assetpaying a dividend yield of q(Equations 17.4 to 17.7)
When ∆t is small a solution to the equations is
tqr
t
t
eadudap
ed
eu
∆−
∆σ−
∆σ
=−−
=
=
=
)(
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
The Complete Tree(Figure 17.2, page 394)
S0u 2
S0u 4
S0d 2
S0d 4
S0
S0u
S0d S0 S0
S0u 2
S0d 2
S0u 3
S0u
S0d
S0d 3
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Backwards Induction
We know the value of the option atthe final nodesWe work back through the treeusing risk-neutral valuation tocalculate the value of the option ateach node, testing for earlyexercise when appropriate
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Example: Put Option(Example 17.1, page 394)
S0 = 50; K = 50; r =10%; σ = 40%;T = 5 months = 0.4167;∆t = 1 month = 0.0833
The parameters implyu = 1.1224; d = 0.8909;a = 1.0084; p = 0.5073
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Example (continued)Figure 17.3, page 395
89.070.00
79.350.00
70.70 70.700.00 0.00
62.99 62.990.64 0.00
56.12 56.12 56.122.16 1.30 0.00
50.00 50.00 50.004.49 3.77 2.66
44.55 44.55 44.556.96 6.38 5.45
39.69 39.6910.36 10.31
35.36 35.3614.64 14.64
31.5018.50
28.0721.93
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Calculation of Delta
Delta is calculated from the nodes at time∆t
Delta =−−
= −2 16 6 96
56 12 44 550 41. .
. ..
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Calculation of Gamma
Gamma is calculated from the nodes attime 2∆t
∆ ∆
∆ ∆
1 2
2
064 37762 99 50
024 377 103650 39 69
064
1165003
=−
−= − =
−−
= −
−=
. ..
. ; . ..
.
..Gamma= 1
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Calculation of Theta
Theta is calculated from the central nodesat times 0 and 2∆t
Theta= per year
or - . per calendar day
377 4 4901667
4 3
0012
. ..
.−= −
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Calculation of Vega
We can proceed as followsConstruct a new tree with a volatility of 41%instead of 40%.Value of option is 4.62Vega is
4 62 4 49 0 13. . .− =per 1% change in volatility
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Trees for Options on Indices,Currencies and Futures Contracts
As with Black-Scholes:For options on stock indices, q equals thedividend yield on the indexFor options on a foreign currency, q equals theforeign risk-free rateFor options on futures contracts q = r
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Binomial Tree for DividendPaying Stock
Procedure:Draw the tree for the stock price less thepresent value of the dividendsCreate a new tree by adding the presentvalue of the dividends at each node
This ensures that the tree recombines andmakes assumptions similar to those when theBlack-Scholes model is used
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Extensions of Tree Approach
Time dependent interest ratesThe control variate technique
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Alternative Binomial Tree(Section 17.4, page 406)
Instead of setting u = 1/d we can seteach of the 2 probabilities to 0.5 and
ttqr
ttqr
ed
eu∆σ−∆σ−−
∆σ+∆σ−−
=
=)2/(
)2/(
2
2
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Trinomial Tree (Page 409)
61
212
32
61
212
/1
2
2
2
2
3
+⎟⎟⎠
⎞⎜⎜⎝
⎛ σ−
σ∆
−=
=
+⎟⎟⎠
⎞⎜⎜⎝
⎛ σ−
σ∆
=
== ∆σ
rtp
p
rtp
udeu
d
m
u
t
S S
Sd
Su
pu
pm
pd
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Time Dependent Parameters in aBinomial Tree (page 409)
Making r or q a function of time does notaffect the geometry of the tree. Theprobabilities on the tree become functionsof time.We can make σ a function of time bymaking the lengths of the time stepsinversely proportional to the variance rate.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Monte Carlo Simulation and π
How could you calculate π by randomlysampling points in the square?
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Monte Carlo Simulation and Options When used to value European stock options, Monte
Carlo simulation involves the following steps:1. Simulate 1 path for the stock price in a risk neutral
world2. Calculate the payoff from the stock option3. Repeat steps 1 and 2 many times to get many sample
payoff4. Calculate mean payoff5. Discount mean payoff at risk free rate to get an
estimate of the value of the option
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Sampling Stock Price Movements(Equations 17.13 and 17.14, page 411)
In a risk neutral world the process for a stockprice is
We can simulate a path by choosing timesteps of length ∆t and using the discreteversion of this
where ε is a random sample from φ(0,1)tStSS ∆εσ+∆µ=∆ ˆ
dS S dt S dz= +$µ σ
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
A More Accurate Approach(Equation 17.15, page 412)
( )
( )
( ) ttetSttS
tttSttS
dzdtSd
∆εσ+∆σ−µ=∆+
∆σε+∆σ−µ=−∆+
σ+σ−µ=
or
is this of version discrete The Use
2/ˆ
2
2
2
)()(
2/ˆ)(ln)(ln
2/ˆln
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Extensions
When a derivative depends on severalunderlying variables we can simulatepaths for each of them in a risk-neutralworld to calculate the values for thederivative
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Sampling from NormalDistribution (Page 414)
One simple way to obtain a samplefrom φ(0,1) is to generate 12 randomnumbers between 0.0 & 1.0, take thesum, and subtract 6.0In Excel =NORMSINV(RAND()) givesa random sample from φ(0,1)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
To Obtain 2 Correlated NormalSamples
Obtain independent normal samples x1 andx2 and set
A procedure known as Cholesky’sdecomposition when samples are requiredfrom more than two normal variables
2212
11
1 ρ−+ρ=ε=ε
xxx
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Standard Errors in Monte CarloSimulation
The standard error of the estimate of theoption price is the standard deviation ofthe discounted payoffs given by thesimulation trials divided by the square rootof the number of observations.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Application of Monte CarloSimulation
Monte Carlo simulation can deal withpath dependent options, optionsdependent on several underlyingstate variables, and options withcomplex payoffsIt cannot easily deal with American-style options
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Determining Greek Letters
For ∆:1.Make a small change to asset price2.Carry out the simulation again using the same
random number streams3.Estimate ∆ as the change in the option price
divided by the change in the asset price
Proceed in a similar manner for other Greek letters
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Variance Reduction Techniques
Antithetic variable techniqueControl variate techniqueImportance samplingStratified samplingMoment matchingUsing quasi-random sequences
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Sampling Through the Tree
Instead of sampling from the stochasticprocess we can sample paths randomlythrough a binomial or trinomial tree tovalue a derivative
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Finite Difference Methods
Finite difference methods aim torepresent the differential equation inthe form of a difference equationWe form a grid by consideringequally spaced time values andstock price valuesDefine ƒi,j as the value of ƒ at timei∆t when the stock price is j∆S
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Finite Difference Methods(continued)
ƒ2ƒƒƒ
or ƒƒƒƒƒ
2
ƒƒƒset we
ƒƒ21ƒƒIn
2,1,1,
2
2
1,,,1,2
2
1,1,
2
222
SS
SSSS
SS
rS
SS
rSt
jijiji
jijijiji
jiji
∆
−+=
∂∂
∆⎟⎟⎠
⎞⎜⎜⎝
⎛∆−
−∆
−=
∂∂
∆
−=
∂∂
=∂∂
σ+∂∂
+∂∂
−+
−+
−+
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Implicit Finite Difference Method(Equation 17.25, page 420)
If we also set ƒ ƒ ƒ
we obtain the implicit finite difference method.This involves solving simultaneous equations of the form: ƒ ƒ ƒ ƒ
, ,
, , , ,
∂∂ t t
a b c
i j i j
j i j j i j j i j i j
=−
+ + =
+
− + +
1
1 1 1
∆
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Explicit Finite Difference Method(page 422-428)
ƒƒƒƒ :form the of equations solving involves This
method difference finite explicit the obtain wepoint )( the at arethey as point )( the at
same the be to assumed are and If
,,,,
2
11*
1*
11*
2
1
+++−+ ++=
+∂∂∂∂
jijjijjijji cba
i,j,jiSfSf
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Implicit vs Explicit FiniteDifference Method
The explicit finite difference method isequivalent to the trinomial tree approachThe implicit finite difference method isequivalent to a multinomial tree approach
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Implicit vs Explicit Finite Difference Methods(Figure 17.16, page 425)
ƒi , j ƒi +1, j
ƒi +1, j –1
ƒi +1, j +1
ƒi +1, jƒi , j
ƒi , j –1
ƒi , j +1
Implicit Method Explicit Method
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Other Points on Finite DifferenceMethods
It is better to have ln S rather than S as theunderlying variableImprovements over the basic implicit andexplicit methods:
Hopscotch methodCrank-Nicolson method
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.1
Chapter 18
Value at Risk
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.2
The Question Being Asked in VaR
“What loss level is such that we are X%confident it will not be exceeded in Nbusiness days?”
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.3
VaR and Regulatory Capital(Business Snapshot 18.1, page 436)
Regulators base the capital they requirebanks to keep on VaRThe market-risk capital is k times the 10-day 99% VaR where k is at least 3.0
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.4
VaR vs. C-VaR(See Figures 18.1 and 18.2)
VaR is the loss level that will not beexceeded with a specified probabilityC-VaR (or expected shortfall) is theexpected loss given that the loss is greaterthan the VaR levelAlthough C-VaR is theoretically moreappealing, it is not widely used
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.5
Advantages of VaR
It captures an important aspect of riskin a single numberIt is easy to understandIt asks the simple question: “How bad canthings get?”
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.6
Time Horizon
Instead of calculating the 10-day, 99% VaRdirectly analysts usually calculate a 1-day 99%VaR and assume
This is exactly true when portfolio changes onsuccessive days come from independentidentically distributed normal distributions
day VaR1-day VaR-10 ×= 10
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.7
Historical Simulation(See Tables 18.1 and 18.2, page 438-439))
Create a database of the daily movements in allmarket variables.The first simulation trial assumes that thepercentage changes in all market variables areas on the first dayThe second simulation trial assumes that thepercentage changes in all market variables areas on the second dayand so on
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.8
Historical Simulation continued
Suppose we use m days of historical dataLet vi be the value of a variable on day iThere are m-1 simulation trialsThe ith trial assumes that the value of themarket variable tomorrow (i.e., on day m+1) is
1−i
im v
vv
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.9
The Model-Building Approach
The main alternative to historical simulation is tomake assumptions about the probabilitydistributions of return on the market variablesand calculate the probability distribution of thechange in the value of the portfolio analyticallyThis is known as the model building approach orthe variance-covariance approach
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.10
Daily Volatilities
In option pricing we measure volatility “peryear”In VaR calculations we measure volatility“per day”
252year
day
σ=σ
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.11
Daily Volatility continued
Strictly speaking we should define σday asthe standard deviation of the continuouslycompounded return in one dayIn practice we assume that it is thestandard deviation of the percentagechange in one day
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.12
Microsoft Example (page 440)
We have a position worth $10 million inMicrosoft sharesThe volatility of Microsoft is 2% per day(about 32% per year)We use N=10 and X=99
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.13
Microsoft Example continued
The standard deviation of the change inthe portfolio in 1 day is $200,000The standard deviation of the change in 10days is
200 000 10 456, $632,=
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.14
Microsoft Example continued
We assume that the expected change inthe value of the portfolio is zero (This isOK for short time periods)We assume that the change in the value ofthe portfolio is normally distributedSince N(–2.33)=0.01, the VaR is
2 33 632 456 473 621. , $1, ,× =
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.15
AT&T Example (page 441)
Consider a position of $5 million in AT&TThe daily volatility of AT&T is 1% (approx16% per year)The S.D per 10 days is
The VaR is50 000 10 144, $158,=
158114 233 405, . $368,× =
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.16
Portfolio
Now consider a portfolio consisting of bothMicrosoft and AT&TSuppose that the correlation between thereturns is 0.3
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.17
S.D. of Portfolio
A standard result in statistics states that
In this case σX = 200,000 and σY = 50,000and ρ = 0.3. The standard deviation of thechange in the portfolio value in one day istherefore 220,227
YXYXYX σρσ+σ+σ=σ + 222
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.18
VaR for Portfolio
The 10-day 99% VaR for the portfolio is
The benefits of diversification are(1,473,621+368,405)–1,622,657=$219,369What is the incremental effect of the AT&Tholding on VaR?
657,622,1$33.210220,227 =××
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.19
The Linear Model
We assumeThe daily change in the value of a portfoliois linearly related to the daily returns frommarket variablesThe returns from the market variables arenormally distributed
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.20
The General Linear Modelcontinued (equations 18.1 and 18.2)
deviation standard sportfolio' theis and variableofy volatilit theis where
21
222
1 1
2
1
P
i
n
iijjiji
jiiiP
n
i
n
jijjijiP
n
iii
i
xP
σσ
ρσσαασασ
ρσσαασ
α
∑ ∑
∑∑
∑
= <
= =
=
+=
=
∆=∆
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.21
Handling Interest Rates: CashFlow Mapping
We choose as market variables bondprices with standard maturities (1mth,3mth, 6mth, 1yr, 2yr, 5yr, 7yr, 10yr, 30yr)Suppose that the 5yr rate is 6% and the7yr rate is 7% and we will receive a cashflow of $10,000 in 6.5 years.The volatilities per day of the 5yr and 7yrbonds are 0.50% and 0.58% respectively
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.22
Example continued
We interpolate between the 5yr rate of 6%and the 7yr rate of 7% to get a 6.5yr rateof 6.75%The PV of the $10,000 cash flow is
540,60675.1
000,105.6 =
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.23
Example continued
We interpolate between the 0.5% volatilityfor the 5yr bond price and the 0.58%volatility for the 7yr bond price to get 0.56%as the volatility for the 6.5yr bondWe allocate α of the PV to the 5yr bondand (1- α) of the PV to the 7yr bond
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.24
Example continued
Suppose that the correlation betweenmovement in the 5yr and 7yr bond pricesis 0.6To match variances
This gives α=0.074
)1(58.05.06.02)1(58.05.056.0 22222 α−α××××+α−+α=
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.25
Example continuedThe value of 6,540 received in 6.5 years
in 5 years and by
in 7 years.This cash flow mapping preserves valueand variance
484$074.0540,6 =×
056,6$926.0540,6 =×
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.26
When Linear Model Can be Used
Portfolio of stocksPortfolio of bondsForward contract on foreign currencyInterest-rate swap
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.27
The Linear Model and Options
Consider a portfolio of options dependenton a single stock price, S. Define
andSP
∆∆
=δ
SSx ∆
=∆
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.28
Linear Model and Optionscontinued (equations 18.3 and 18.4)
As an approximation
Similarly when there are many underlyingmarket variables
where δi is the delta of the portfolio withrespect to the ith asset
xSSP ∆δ=∆δ=∆
∑ ∆δ=∆i
iii xSP
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.29
Example
Consider an investment in options on Microsoftand AT&T. Suppose the stock prices are 120and 30 respectively and the deltas of theportfolio with respect to the two stock prices are1,000 and 20,000 respectivelyAs an approximation
where ∆x1 and ∆x2 are the percentage changesin the two stock prices
21 000,2030000,1120 xxP ∆×+∆×=∆
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.30
Skewness(See Figures 18.3, 18.4 , and 18.5)
The linear model fails to capture skewnessin the probability distribution of theportfolio value.
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.31
Quadratic Model
For a portfolio dependent on a single stockprice it is approximately true that
this becomes
2)(21 SSP ∆γ+∆δ=∆
22 )(21 xSxSP ∆γ+∆δ=∆
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.32
Quadratic Model continued
With many market variables we get anexpression of the form
where
This is not as easy to work with as the linearmodel
∑ ∑= =
∆∆γ+∆δ=∆n
i
n
ijiijjiiii xxSSxSP
1 1 21
jiij
ii SS
PSP
∂∂
=γ∂∂
=δ2
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.33
Monte Carlo Simulation (page 448-449)
To calculate VaR using M.C. simulation weValue portfolio todaySample once from the multivariatedistributions of the ∆xi
Use the ∆xi to determine market variablesat end of one dayRevalue the portfolio at the end of day
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.34
Monte Carlo Simulation
Calculate ∆PRepeat many times to build up aprobability distribution for ∆PVaR is the appropriate fractile of thedistribution times square root of NFor example, with 1,000 trial the 1percentile is the 10th worst case.
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.35
Speeding Up Monte Carlo
Use the quadratic approximation tocalculate ∆P
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.36
Comparison of Approaches
Model building approach assumes normaldistributions for market variables. It tendsto give poor results for low delta portfoliosHistorical simulation lets historical datadetermine distributions, but iscomputationally slower
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.37
Stress Testing
This involves testing how well a portfolioperforms under some of the most extrememarket moves seen in the last 10 to 20years
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.38
Back-Testing
Tests how well VaR estimates would haveperformed in the pastWe could ask the question: How often wasthe actual 10-day loss greater than the99%/10 day VaR?
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.39
Principal Components Analysis forInterest Rates (Tables 18.3 and 18.4 on page 451)
The first factor is a roughly parallel shift(83.1% of variation explained)The second factor is a twist (10% ofvariation explained)The third factor is a bowing (2.8% ofvariation explained)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.40
Using PCA to calculate VaR (page 453)
Example: Sensitivity of portfolio to rates ($m)
Sensitivity to first factor is from Table 18.3:10×0.32 + 4×0.35 – 8×0.36 – 7 ×0.36 +2 ×0.36 = – 0.08Similarly sensitivity to second factor = – 4.40
+2-7-8+4+105 yr4 yr3 yr2 yr1 yr
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18.41
Using PCA to calculate VaR continued
As an approximation
The f1 and f2 are independentThe standard deviation of ∆P (from Table18.4) is
The 1 day 99% VaR is 26.66 × 2.33 =62.12
21 40.408.0 ffP −−=∆
66.2605.640.449.1708.0 2222 =×+×
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.1
Estimating Volatilitiesand Correlations
Chapter 19
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.2
Standard Approach to EstimatingVolatility (page 461)
Define σn as the volatility per day betweenday n-1 and day n, as estimated at end of dayn-1Define Si as the value of market variable atend of day iDefine ui= ln(Si/Si-1)
σ n n ii
m
n ii
m
mu u
um
u
2 2
1
1
11
1
=−
−
=
−=
−=
∑
∑
( )
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.3
Simplifications Usually Made (page462)
Define ui as (Si-Si-1)/Si-1
Assume that the mean value of ui is zeroReplace m-1 by m
This gives
σn n ii
m
mu2 2
1
1= −=∑
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.4
Weighting Scheme
Instead of assigning equal weights to theobservations we can set
σ α
α
n i n ii
m
ii
m
u2 21
11
=
=
−=
=
∑
∑
where
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.5
ARCH(m) Model (page 463)
In an ARCH(m) model we also assignsome weight to the long-run variance rate,VL:
∑
∑
=
= −
=α+γ
α+γ=σ
m
ii
m
i iniLn uV
1
122
1
where
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.6
EWMA Model
In an exponentially weighted movingaverage model, the weights assigned tothe u2 decline exponentially as we moveback through timeThis leads to
21
21
2 )1( −− λ−+λσ=σ nnn u
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.7
Attractions of EWMA
Relatively little data needs to be storedWe need only remember the currentestimate of the variance rate and the mostrecent observation on the market variableTracks volatility changesRiskMetrics uses λ = 0.94 for dailyvolatility forecasting
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.8
GARCH (1,1) page 465
In GARCH (1,1) we assign some weight tothe long-run average variance rate
Since weights must sum to 1γ + α + β =1
21
21
2−− βσ+α+γ=σ nnLn uV
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.9
GARCH (1,1) continued
Setting ω = γV the GARCH (1,1) model is
and
β−α−ω
=1LV
21
21
2−− βσ+α+ω=σ nnn u
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.10
Example (Example 19.2, page 465)
Suppose
The long-run variance rate is 0.0002 sothat the long-run volatility per day is 1.4%
σ σn n nu21
21
20 000002 013 0 86= + +− −. . .
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.11
Example continued
Suppose that the current estimate of thevolatility is 1.6% per day and the mostrecent percentage change in the marketvariable is 1%.The new variance rate is
The new volatility is 1.53% per day0 000002 013 0 0001 086 0 000256 0 00023336. . . . . .+ × + × =
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.12
GARCH (p,q)
σ ω α β σn i n i jj
q
i
p
n ju2 2
11
2= + +−==
−∑∑
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.13
Maximum Likelihood Methods
In maximum likelihood methods wechoose parameters that maximize thelikelihood of the observations occurring
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.14
Example 1
We observe that a certain event happensone time in ten trials. What is our estimateof the proportion of the time, p, that ithappens?The probability of the event happening onone particular trial and not on the others is
We maximize this to obtain a maximumlikelihood estimate. Result: p=0.1
9)1( pp −
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.15
Example 2
Estimate the variance of observations from anormal distribution with mean zero
∑
∑
∏
=
=
=
=
⎥⎦
⎤⎢⎣
⎡−−
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛ −π
m
ii
m
i
i
m
i
i
um
v
vuv
vu
v
1
2
1
2
1
2
1
)ln(
2exp
21
:Result
:maximizing to equivalent is this logarithms Taking
:Maximize
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.16
Application to GARCH
We choose parameters that maximize
∑
∏
=
=
⎥⎦
⎤⎢⎣
⎡−−
⎟⎟⎠
⎞⎜⎜⎝
⎛−
π
m
i i
ii
i
im
i i
vuv
vu
v
1
2
2
1
)ln(
2exp
21
or
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.17
Excel Application (Table 19.1, page 469)
Start with trial values of ω, α, and βUpdate variancesCalculate
Use solver to search for values of ω, α, and βthat maximize this objective functionImportant note: set up spreadsheet so that youare searching for three numbers that are thesame order of magnitude (See page 470)
∑=
⎥⎦
⎤⎢⎣
⎡−−
m
i i
ii v
uv1
2
)ln(
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.18
Variance Targeting
One way of implementing GARCH(1,1)that increases stability is by using variancetargetingWe set the long-run average volatilityequal to the sample varianceOnly two other parameters then have to beestimated
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.19
How Good is the Model?
The Ljung-Box statistic tests forautocorrelationWe compare the autocorrelation of theui
2 with the autocorrelation of the ui2/σi
2
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.20
Forecasting Future Volatility(equation 19.3, page 472)
A few lines of algebra shows that
The variance rate for an option expiring onday m is
)()(][ 22Ln
kLkn VVE −σβ+α+=σ +
[ ]∑−
=+σ
1
0
21 m
kknE
m
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.21
Forecasting Future Volatilitycontinued (equation 19.4, page 473)
[ ]⎟⎟⎠
⎞⎜⎜⎝
⎛−
−+
β+α=
−
L
aT
L VVaTeV
T
a
)0(1
1ln
252
is optionday - a for annum per volatility The
Define
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.22
Volatility Term Structures (Table 19.4)
The GARCH (1,1) suggests that, when calculating vega,we should shift the long maturity volatilities less thanthe short maturity volatilitiesImpact of 1% change in instantaneous volatility forJapanese yen example:
0.060.270.460.610.84Volatilityincrease (%)
500100503010Option Life(days)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.23
Correlations and Covariances (page475-477)
Define xi=(Xi-Xi-1)/Xi-1 and yi=(Yi-Yi-1)/Yi-1
Alsoσx,n: daily vol of X calculated on day n-1σy,n: daily vol of Y calculated on day n-1covn: covariance calculated on day n-1The correlation is covn/(σu,n σv,n)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.24
Updating Correlations
We can use similar models to those forvolatilitiesUnder EWMA
covn = λ covn-1+(1-λ)xn-1yn-1
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.25
Positive Finite Definite Condition
A variance-covariance matrix, Ω, isinternally consistent if the positive semi-definite condition
for all vectors w
w wT Ω ≥ 0
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19.26
Example
The variance covariance matrix
is not internally consistent
1 0 0 90 1 0 9
0 9 0 9 1
.
.. .
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.1
Credit Risk
Chapter 20
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.2
Credit Ratings
In the S&P rating system, AAA is the bestrating. After that comes AA, A, BBB, BB,B, and CCCThe corresponding Moody’s ratings areAaa, Aa, A, Baa, Ba, B, and CaaBonds with ratings of BBB (or Baa) andabove are considered to be “investmentgrade”
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.3
Historical Data
Historical data provided by rating agenciesare also used to estimate the probability ofdefault
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.4
Cumulative Ave Default Rates (%)(1970-2003, Moody’s, Table 20.1, page 482)
1 2 3 4 5 7 10 Aaa 0.00 0.00 0.00 0.04 0.12 0.29 0.62
Aa 0.02 0.03 0.06 0.15 0.24 0.43 0.68
A 0.02 0.09 0.23 0.38 0.54 0.91 1.59
Baa 0.20 0.57 1.03 1.62 2.16 3.24 5.10
Ba 1.26 3.48 6.00 8.59 11.17 15.44 21.01
B 6.21 13.76 20.65 26.66 31.99 40.79 50.02
Caa 23.65 37.20 48.02 55.56 60.83 69.36 77.91
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.5
Interpretation
The table shows the probability ofdefault for companies starting with aparticular credit ratingA company with an initial credit rating ofBaa has a probability of 0.20% ofdefaulting by the end of the first year,0.57% by the end of the second year,and so on
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.6
Do Default Probabilities Increasewith Time?
For a company that starts with a goodcredit rating default probabilities tend toincrease with timeFor a company that starts with a poorcredit rating default probabilities tend todecrease with time
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.7
Default Intensities vs UnconditionalDefault Probabilities (page 482-483)
The default intensity (also called hazard rate) isthe probability of default for a certain time periodconditional on no earlier defaultThe unconditional default probability is theprobability of default for a certain time period asseen at time zeroWhat are the default intensities andunconditional default probabilities for a Caa ratecompany in the third year?
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.8
Recovery Rate
The recovery rate for a bond is usuallydefined as the price of the bondimmediately after default as a percent ofits face value
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.9
Recovery Rates(Moody’s: 1982 to 2003, Table 20.2, page 483)
Class Mean(%)
Senior Secured 51.6
Senior Unsecured 36.1
Senior Subordinated 32.5
Subordinated 31.1
Junior Subordinated 24.5
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.10
Estimating Default Probabilities
Alternatives:Use Bond PricesUse CDS spreadsUse Historical DataUse Merton’s Model
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.11
Using Bond Prices (Equation 20.2, page 484)
Average default intensity over life of bond is approximately
where s is the spread of the bond’s yieldover the risk-free rate and R is the recoveryrate
Rs
−1
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.12
More Exact CalculationAssume that a five year corporate bond pays a couponof 6% per annum (semiannually). The yield is 7% withcontinuous compounding and the yield on a similar risk-free bond is 5% (with continuous compounding)Price of risk-free bond is 104.09; price of corporate bondis 95.34; expected loss from defaults is 8.75Suppose that the probability of default is Q per year andthat defaults always happen half way through a year(immediately before a coupon payment.
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.13
Calculations (Table 20.3, page 485)
288.48QTotal
50.67Q0.798563.46103.4640Q4.5
54.01Q0.839564.34104.3440Q3.5
57.52Q0.882565.17105.1740Q2.5
61.20Q0.927765.97105.9740Q1.5
65.08Q0.975366.73106.7340Q0.5
PV of ExpLoss
DiscountFactor
LGDRisk-freeValue
RecoveryAmount
DefProb
Time(yrs)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.14
Calculations continued
We set 288.48Q = 8.75 to get Q = 3.03%This analysis can be extended to allowdefaults to take place more frequentlyWith several bonds we can use moreparameters to describe the defaultprobability distribution
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.15
The Risk-Free Rate
The risk-free rate when defaultprobabilities are estimated is usuallyassumed to be the LIBOR/swap zero rate(or sometimes 10 bps below theLIBOR/swap rate)To get direct estimates of the spread ofbond yields over swap rates we can lookat asset swaps
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.16
Real World vs Risk-NeutralDefault Probabilities
The default probabilities backed out ofbond prices or credit default swap spreadsare risk-neutral default probabilitiesThe default probabilities backed out ofhistorical data are real-world defaultprobabilities
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.17
A Comparison
Calculate 7-year default intensities fromthe Moody’s data (These are real worlddefault probabilities)Use Merrill Lynch data to estimateaverage 7-year default intensities frombond prices (these are risk-neutraldefault intensities)Assume a risk-free rate equal to the 7-year swap rate minus 10 basis point
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.18
Real World vs Risk NeutralDefault Probabilities, 7 yearaverages (Table 20.4, page 487)
Rating Real-world default probability per yr (bps)
Risk-neutral default probability per yr (bps)
Ratio Difference
Aaa 4 67 16.8 63 Aa 6 78 13.0 72 A 13 128 9.8 115 Baa 47 238 5.1 191 Ba 240 507 2.1 267 B 749 902 1.2 153 Caa 1690 2130 1.3 440
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.19
Risk Premiums Earned By BondTraders (Table 20.5, page 488)
Rating Bond Yield Spread over Treasuries
(bps)
Spread of risk-free rate used by market
over Treasuries (bps)
Spread to compensate for
default rate in the real world (bps)
Extra Risk Premium
(bps)
Aaa 83 43 2 38 Aa 90 43 4 43 A 120 43 8 69 Baa 186 43 28 115 Ba 347 43 144 160 B 585 43 449 93 Caa 1321 43 1014 264
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.20
Possible Reasons for These Results
Corporate bonds are relatively illiquidThe subjective default probabilities of bondtraders may be much higher than theestimates from Moody’s historical dataBonds do not default independently of eachother. This leads to systematic risk thatcannot be diversified away.Bond returns are highly skewed with limitedupside. The non-systematic risk is difficult todiversify away and may be priced by themarket
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.21
Which World Should We Use?
We should use risk-neutral estimates forvaluing credit derivatives and estimatingthe present value of the cost of defaultWe should use real world estimates forcalculating credit VaR and scenarioanalysis
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.22
Merton’s Model (page 489-491)
Merton’s model regards the equity as anoption on the assets of the firmIn a simple situation the equity value is
max(VT -D, 0)where VT is the value of the firm and D isthe debt repayment required
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.23
Equity vs. Assets
An option pricing model enables the valueof the firm’s equity today, E0, to be relatedto the value of its assets today, V0, and thevolatility of its assets, σV
E V N d De N d
dV D r T
Td d T
rT
V
V
V
0 0 1 2
10
2
2 12
= −
=+ +
= −
−( ) ( )
ln ( ) ( );
where
σ
σσ
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.24
Volatilities
σ∂∂
σ σE V VEEV
V N d V0 0 1 0= = ( )
This equation together with the option pricingrelationship enables V0 and σV to bedetermined from E0 and σE
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.25
Example
A company’s equity is $3 million and thevolatility of the equity is 80%The risk-free rate is 5%, the debt is $10million and time to debt maturity is 1 yearSolving the two equations yields V0=12.40and σv=21.23%
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.26
Example continued
The probability of default is N(-d2) or12.7%The market value of the debt is 9.40The present value of the promisedpayment is 9.51The expected loss is about 1.2%The recovery rate is 91%
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.27
The Implementation of Merton’sModel (e.g. Moody’s KMV)
Choose time horizonCalculate cumulative obligations to time horizon.This is termed by KMV the “default point”. Wedenote it by DUse Merton’s model to calculate a theoreticalprobability of defaultUse historical data or bond data to develop aone-to-one mapping of theoretical probabilityinto either real-world or risk-neutral probability ofdefault.
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.28
Credit Risk in DerivativesTransactions (page 491-493)
Three casesContract always an assetContract always a liabilityContract can be an asset or a liability
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.29
General Result
Assume that default probability is independent of thevalue of the derivativeConsider times t1, t2,…tn and default probability is qiat time ti. The value of the contract at time ti is fi andthe recovery rate is RThe loss from defaults at time ti is qi(1-R)E[max(fi,0)].Defining ui=qi(1-R) and vi as the value of a derivativethat provides a payoff of max(fi,0) at time ti, the costof defaults is
∑=
n
iiivu
1
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.30
Credit Risk Mitigation
NettingCollateralizationDowngrade triggers
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.31
Default Correlation
The credit default correlation between twocompanies is a measure of their tendency todefault at about the same timeDefault correlation is important in riskmanagement when analyzing the benefits ofcredit risk diversificationIt is also important in the valuation of somecredit derivatives, eg a first-to-default CDS andCDO tranches.
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.32
Measurement
There is no generally accepted measure ofdefault correlationDefault correlation is a more complexphenomenon than the correlation betweentwo random variables
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.33
Binomial Correlation Measure(page 499)
One common default correlation measure,between companies i and j is the correlationbetween
A variable that equals 1 if company i defaultsbetween time 0 and time T and zerootherwiseA variable that equals 1 if company j defaultsbetween time 0 and time T and zerootherwise
The value of this measure depends on T.Usually it increases at T increases.
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.34
Binomial Correlation continued
Denote Qi(T) as the probability that companyA will default between time zero and time T,and Pij(T) as the probability that both i and jwill default. The default correlation measureis
])()(][)()([
)()()()(
22 TQTQTQTQ
TQTQTPT
jjii
jiijij
−−
−=β
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.35
Survival Time Correlation
Define ti as the time to default for companyi and Qi(ti) as the probability distribution fortiThe default correlation betweencompanies i and j can be defined as thecorrelation between ti and tjBut this does not uniquely define the jointprobability distribution of default times
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.36
Gaussian Copula Model (page 496-499)
Define a one-to-one correspondence between the timeto default, ti, of company i and a variable xi by
Qi(ti ) = N(xi ) or xi = N-1[Q(ti)]where N is the cumulative normal distribution function.This is a “percentile to percentile” transformation. The ppercentile point of the Qi distribution is transformed to thep percentile point of the xi distribution. xi has a standardnormal distributionWe assume that the xi are multivariate normal. Thedefault correlation measure, ρij between companies i andj is the correlation between xi and xj
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.37
Binomial vs Gaussian CopulaMeasures (Equation 20.10, page 499)
The measures can be calculated fromeach other
function ondistributiy probabilitnormal bivariate cumulative the is where
that so
M
TQTQTQTQ
TQTQxxMT
xxMTP
jjii
jiijjiij
ijjiij
])()(][)()([
)()(];,[)(
];,[)(
22 −−
−ρ=β
ρ=
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.38
Comparison (Example 20.4, page 499)
The correlation number depends on thecorrelation metric usedSuppose T = 1, Qi(T) = Qj(T) = 0.01, avalue of ρij equal to 0.2 corresponds to avalue of βij(T) equal to 0.024.In general βij(T) < ρij and βij(T) is anincreasing function of T
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.39
Example of Use of Gaussian Copula(Example 20.3, page 498)
Suppose that we wish to simulate thedefaults for n companies . For eachcompany the cumulative probabilities ofdefault during the next 1, 2, 3, 4, and 5years are 1%, 3%, 6%, 10%, and 15%,respectively
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.40
Use of Gaussian Copula continued
We sample from a multivariate normaldistribution to get the xi
Critical values of xi areN -1(0.01) = -2.33, N -1(0.03) = -1.88,N -1(0.06) = -1.55, N -1(0.10) = -1.28,N -1(0.15) = -1.04
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.41
Use of Gaussian Copula continued
When sample for a company is less than-2.33, the company defaults in the first yearWhen sample is between -2.33 and -1.88, the companydefaults in the second yearWhen sample is between -1.88 and -1.55, the companydefaults in the third yearWhen sample is between -1,55 and -1.28, the companydefaults in the fourth yearWhen sample is between -1.28 and -1.04, the companydefaults during the fifth yearWhen sample is greater than -1.04, there is no defaultduring the first five years
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.42
A One-Factor Model for theCorrelation Structure (Equation 20.7, page 498)
The correlation between xi and xj is aiaj
The ith company defaults by time T when xi < N-1[Qi(T)]or
The probability of this is
2
1
1])([
i
iii
aMaTQNZ
−
−<
−
iiii ZaMax 21−+=
[ ]⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−
−=
−
2
1
1)()(
i
iii
aMaTQNNMTQ
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.43
Credit VaR (page 499-502)
Can be defined analogously to MarketRisk VaRA T-year credit VaR with an X%confidence is the loss level that we are X%confident will not be exceeded over Tyears
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.44
Calculation from a Factor-BasedGaussian Copula Model (equation 20.11, page500)
Consider a large portfolio of loans, each of whichhas a probability of Q(T) of defaulting by time T.Suppose that all pairwise copula correlations areρ so that all ai’s areWe are X% certain that M is less than N-1(1- X) =- N-1(X)It follows that the VaR is
[ ]⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
ρ−ρ+
=−−
1)()(
),(11 XNTQN
NTXV
ρ
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 20.45
CreditMetrics (page 500-502)
Calculates credit VaR by consideringpossible rating transitionsA Gaussian copula model is used to definethe correlation between the ratingstransitions of different companies
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.1
Credit Derivatives
Chapter 21
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.2
Credit Derivatives
Derivatives where the payoff depends onthe credit quality of a company or countryThe market started to grow fast in the late1990sBy 2003 notional principal totaled $3trillion
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.3
Credit Default Swaps
Buyer of the instrument acquires protection from theseller against a default by a particular company orcountry (the reference entity)Example: Buyer pays a premium of 90 bps per yearfor $100 million of 5-year protection against companyXPremium is known as the credit default spread. It ispaid for life of contract or until defaultIf there is a default, the buyer has the right to sellbonds with a face value of $100 million issued bycompany X for $100 million (Several bonds aretypically deliverable)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.4
CDS Structure (Figure 21.1, page 508)
Default Protection Buyer, A
Default Protection Seller, B
90 bps per year
Payoff if there is a default byreference entity=100(1-R)
Recovery rate, R, is the ratio of the value of the bond issuedby reference entity immediately after default to the face valueof the bond
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.5
Other Details
Payments are usually made quarterly orsemiannually in arrearsIn the event of default there is a final accrualpayment by the buyerSettlement can be specified as delivery of thebonds or in cashSuppose payments are made quarterly in theexample just considered. What are the cashflows if there is a default after 3 years and 1month and recovery rate is 40%?
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.6
Attractions of the CDS Market
Allows credit risks to be traded in thesame way as market risksCan be used to transfer credit risks to athird partyCan be used to diversify credit risks
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.7
Using a CDS to Hedge a Bond
Portfolio consisting of a 5-year par yieldcorporate bond that provides a yield of 6% and along position in a 5-year CDS costing 100 basispoints per year is (approximately) a long positionin a riskless instrument paying 5% per year
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.8
Valuation Example (page 510-512)
Conditional on no earlier default a referenceentity has a (risk-neutral) probability of default of2% in each of the next 5 years. (This is a defaultintensity)Assume payments are made annually in arrears,that defaults always happen half way through ayear, and that the expected recovery rate is 40%Suppose that the breakeven CDS rate is s perdollar of notional principal
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.9
Unconditional Default andSurvival Probabilities (Table 21.1)
0.90390.01845
0.92240.01884
0.94120.01923
0.96040.01962
0.98000.02001
SurvivalProbability
DefaultProbability
Time(years)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.10
Calculation of PV of PaymentsTable 21.2 (Principal=$1)
4.0704sTotal
0.7040s0.77880.9039s0.90395
0.7552s0.81870.9224s0.92244
0.8101s0.86070.9412s0.94123
0.8690s0.90480.9604s0.96042
0.9322s0.95120.9800s0.98001
PV of ExpPmt
DiscountFactor
ExpectedPaymt
SurvivalProb
Time (yrs)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.11
Present Value of Expected Payoff(Table 21.3; Principal = $1)
0.0511Total
0.00880.79850.01110.40.01844.5
0.00950.83950.01130.40.01883.5
0.01020.88250.01150.40.01922.5
0.01090.92770.01180.40.01961.5
0.01170.97530.01200.40.02000.5
PV of Exp.Payoff
DiscountFactor
ExpectedPayoff
Rec.Rate
DefaultProbab.
Time(yrs)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.12
PV of Accrual Payment Made in Event of aDefault. (Table 21.4; Principal=$1)
0.0426sTotal
0.0074s0.79850.0092s0.01844.5
0.0079s0.83950.0094s0.01883.5
0.0085s0.88250.0096s0.01922.5
0.0091s0.92770.0098s0.01961.5
0.0097s0.97530.0100s0.02000.5
PV of PmtDiscFactor
ExpectedAccr Pmt
DefaultProb
Time
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.13
Putting it all together
PV of expected payments is4.0704s+0.0426s=4.1130sThe breakeven CDS spread is given by4.1130s = 0.0511 or s = 0.0124 (124 bps)The value of a swap negotiated sometime ago with a CDS spread of 150bpswould be 4.1130×0.0150-0.0511 or0.0106 times the principal.
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.14
Implying Default Probabilitiesfrom CDS spreads
Suppose that the mid market spread for a 5 yearnewly issued CDS is 100bps per yearWe can reverse engineer our calculations toconclude that the default intensity is 1.61% peryear.If probabilities are implied from CDS spreadsand then used to value another CDS the result isnot sensitive to the recovery rate providing thesame recovery rate is used throughout
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.15
Other Credit Derivatives
Binary CDSFirst-to-default Basket CDSTotal return swapCredit default optionCollateralized debt obligation
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.16
Binary CDS (page 513)
The payoff in the event of default is a fixedcash amountIn our example the PV of the expectedpayoff for a binary swap is 0.0852 and thebreakeven binary CDS spread is 207 bps
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.17
CDS Forwards and Options (page 514-515)
Example: European option to buy 5 yearprotection on Ford for 280 bps starting in oneyear. If Ford defaults during the one-year life ofthe option, the option is knocked outDepends on the volatility of CDS spreads
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.18
Total Return Swap (page 515-516)
Agreement to exchange total return on acorporate bond for LIBOR plus a spreadAt the end there is a payment reflecting thechange in value of the bondUsually used as financing tools bycompanies that want an investment in thecorporate bond
Total ReturnPayer
Total Return Receiver
Total Return on Bond
LIBOR plus 25bps
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.19
First to Default Basket CDS(page 516)
Similar to a regular CDS except that severalreference entities are specified and there is apayoff when the first one defaultsThis depends on “default correlation”Second, third, and nth to default deals aredefined similarly
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.20
Collateralized Debt Obligation(Figure 21.3, page 517)
A pool of debt issues are put into a specialpurpose trustTrust issues claims against the debt in anumber of tranches
First tranche covers x% of notional and absorbs firstx% of default lossesSecond tranche covers y% of notional and absorbsnext y% of default lossesetc
A tranche earn a promised yield on remainingprincipal in the tranche
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.21
Bond 1Bond 2Bond 3
6
Bond n
Average Yield8.5%
Trust
Tranche 11st 5% of lossYield = 35%
Tranche 22nd 10% of loss
Yield = 15%
Tranche 33rd 10% of lossYield = 7.5%
Tranche 4Residual lossYield = 6%
CDO Structure
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.22
Synthetic CDO
Instead of buying the bonds thearranger of the CDO sells credit defaultswaps.
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.23
Single Tranche Trading (Table 21.6, page518)
This involves trading tranches of standard portfolios that are notfundedCDX IG (Aug 4, 2004):
iTraxx IG (Aug 4, 2004)
14.5bps47.5bps135.5bps
347bps41.8%Quote
15-30%10-15%7-10%3-7%0-3%Tranche
20bps43bps70bps168bps27.6%Quote
12-22%9-12%6-9%3-6%0-3%Tranche
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.24
Valuation of Correlation DependentCredit Derivatives (page 519-520)
A popular approach is to use a factor-based Gaussian copula model to definecorrelations between times to default thetime to defaultOften all pairwise correlations and all theunconditional default distributions areassumed to be the sameMarket likes to imply a pairwise correlationfrom market quotes.
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.25
Valuation of Correlation DependentCredit Derivatives continued
The probability of k defaults by time T conditional on Mis
This enables cash flows conditional on M to becalculated. By integrating over M the unconditionaldistributions are obtained
⎟⎟⎠
⎞⎜⎜⎝
⎛
ρ−ρ−
=−
1)]([
)(1 MTQN
NMTQ
( ) kNk MTQMTQkkN
N −−−
]1[)(!)!(
!
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.26
Convertible Bonds
Often valued with a tree where during a timeinterval ∆t there is
a probability pu of an up movementA probability pd of a down movementA probability 1-exp(-λt) that there will be a default
In the event of a default the stock price falls tozero and there is a recovery on the bond
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.27
The Probabilities
ud
eu
duauep
dudeap
t
t
d
t
u
1
)( 2
=
=
−−
=
−−
=
∆λ−σ
∆λ−
∆λ−
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.28
Node Calculations
Define:Q1: value of bond if neither converted nor
calledQ2: value of bond if calledQ3: value of bond if convertedValue at a node =max[min(Q1,Q2),Q3]
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.29
Example 21.1 (page 522)
9-month zero-coupon bond with face value of $100Convertible into 2 sharesCallable for $113 at any timeInitial stock price = $50,volatility = 30%,no dividendsRisk-free rates all 5%Default intensity, λ, is 1%Recovery rate=40%
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 21.30
The Tree (Figure 21.4, page 522)
G76.42
D 152.8566.34
B 132.69 H57.60 57.60
A 115.19 E 115.1950.00 50.00
106.93 C 106.36 I43.41 43.41
101.20 F 100.0037.6898.61 J
32.71100.00
Default Default Default0.00 0.00 0.00
40.00 40.00 40.00
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.1
Exotic Options
Chapter 22
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.2
Types of Exotics
PackageNonstandard AmericanoptionsForward start optionsCompound optionsChooser optionsBarrier options
Binary optionsLookback optionsShout optionsAsian optionsOptions to exchangeone asset for anotherOptions involvingseveral assets
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.3
Packages (page 529)
Portfolios of standard optionsExamples from Chapter 10: bull spreads,bear spreads, straddles, etcOften structured to have zero costOne popular package is a range forwardcontract
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.4
Non-Standard American Options(page 530)
Exercisable only on specific dates(Bermudans)Early exercise allowed during onlypart of life (initial “lock out” period)Strike price changes over the life(warrants, convertibles)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.5
Forward Start Options (page 531)
Option starts at a future time, T1
Most common in employee stock optionplansOften structured so that strike priceequals asset price at time T1
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.6
Compound Option (page 531)
Option to buy or sell an optionCall on callPut on callCall on putPut on put
Can be valued analyticallyPrice is quite low compared with aregular option
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.7
Chooser Option “As You Like It”(page 532)
Option starts at time 0, matures at T2
At T1 (0 < T1 < T2) buyer chooses whether itis a put or callThis is a package!
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.8
Chooser Option as a Package
1
2
1))(()(
1
)(1
)(
1
),0max(
),max(
1212
1212
TT
SKeec
TeSKecp
pcT
TTqrTTq
TTqTTr
time at maturing put aplus time at maturing call a is This
therefore is time at value The
parity call-put From is value the time At
−+
−+=
−−−−−
−−−−
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.9
Barrier Options (page 535)
Option comes into existence only if stockprice hits barrier before option maturity
‘In’ optionsOption dies if stock price hits barrierbefore option maturity
‘Out’ options
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.10
Barrier Options (continued)
Stock price must hit barrier from below‘Up’ options
Stock price must hit barrier from above‘Down’ options
Option may be a put or a callEight possible combinations
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.11
Parity Relations
c = cui + cuo
c = cdi + cdo
p = pui + puo
p = pdi + pdo
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.12
Binary Options (page 535)
Cash-or-nothing: pays Q if ST > K,otherwise pays nothing.
Value = e–rT Q N(d2)Asset-or-nothing: pays ST if ST > K,otherwise pays nothing.
Value = S0 N(d1)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.13
Decomposition of a Call Option
Long Asset-or-Nothing optionShort Cash-or-Nothing option where payoff
is K
Value = S0 N(d1) – e–rT KN(d2)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.14
Lookback Options (page 536)
Lookback call pays ST – Smin at time TAllows buyer to buy stock at lowestobserved price in some interval of timeLookback put pays Smax– ST at time TAllows buyer to sell stock at highestobserved price in some interval of timeAnalytic solution
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.15
Shout Options (page 537)
Buyer can ‘shout’ once during option lifeFinal payoff is either
Usual option payoff, max(ST – K, 0), orIntrinsic value at time of shout, Sτ – K
Payoff: max(ST – Sτ , 0) + Sτ – KSimilar to lookback option but cheaperHow can a binomial tree be used tovalue a shout option?
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.16
Asian Options (page 538)
Payoff related to average stock priceAverage Price options pay:
Call: max(Save – K, 0)Put: max(K – Save , 0)
Average Strike options pay:Call: max(ST – Save , 0)Put: max(Save – ST , 0)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.17
Asian Options
No analytic solutionCan be valued by assuming (as anapproximation) that the average stockprice is lognormally distributed
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.18
Exchange Options (page 540)
Option to exchange one asset foranotherFor example, an option to exchangeone unit of U for one unit of VPayoff is max(VT – UT, 0)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.19
Basket Options (page 541)
A basket option is an option to buy or sella portfolio of assetsThis can be valued by calculating the firsttwo moments of the value of the basketand then assuming it is lognormal
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.20
How Difficult is it toHedge Exotic Options?
In some cases exotic options areeasier to hedge than thecorresponding vanilla options.(e.g., Asian options)In other cases they are more difficult tohedge (e.g., barrier options)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.21
Static Options Replication(Section 22.13, page 541)
This involves approximately replicating an exoticoption with a portfolio of vanilla optionsUnderlying principle: if we match the value of anexotic option on some boundary , we havematched it at all interior points of the boundaryStatic options replication can be contrasted withdynamic options replication where we have totrade continuously to match the option
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.22
Example
A 9-month up-and-out call option an a non-dividend paying stock where S0 = 50, K = 50,the barrier is 60, r = 10%, and σ = 30%Any boundary can be chosen but the naturalone isc (S, 0.75) = MAX(S – 50, 0) when S < 60c (60, t ) = 0 when 0 ≤ t ≤ 0.75
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.23
Example (continued)
We might try to match the followingpoints on the boundaryc(S , 0.75) = MAX(S – 50, 0) for S < 60c(60, 0.50) = 0
c(60, 0.25) = 0 c(60, 0.00) = 0
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.24
Example continued(See Table 22.1, page 543)
We can do this as follows:
+1.00 call with maturity 0.75 & strike 50
–2.66 call with maturity 0.75 & strike 60
+0.97 call with maturity 0.50 & strike 60
+0.28 call with maturity 0.25 & strike 60
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.25
Example (continued)
This portfolio is worth 0.73 at time zerocompared with 0.31 for the up-and out optionAs we use more options the value of thereplicating portfolio converges to the value ofthe exotic optionFor example, with 18 points matched on thehorizontal boundary the value of the replicatingportfolio reduces to 0.38; with 100 points beingmatched it reduces to 0.32
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 22.26
Using Static OptionsReplication
To hedge an exotic option we shortthe portfolio that replicates theboundary conditionsThe portfolio must be unwound whenany part of the boundary is reached
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 23.1
Weather, Energy, andInsurance Derivatives
Chapter 23
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 23.2
Pricing Issues (page 551)
To a good approximation many underlyingvariables in insurance, weather, andenergy derivatives contracts can beassumed to have zero systematic risk.This means that we can calculateexpected payoff in the real world anddiscount at the risk-free rate
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 23.3
Weather Derivatives: Definitions(page 552)
Heating degree days (HDD): For each daythis is max(0, 65 – A) where A is theaverage of the highest and lowesttemperature in ºF.Cooling Degree Days (CDD): For eachday this is max(0, A – 65)Contracts specify the weather station to beused
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 23.4
Weather Derivatives: Products
A typical product is a forward contract oran option on the cumulative CDD or HDDduring a monthWeather derivatives are often used byenergy companies to hedge the volume ofenergy required for heating or coolingduring a particular monthHow would you value an option on AugustCDD at a particular weather station?
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 23.5
Energy Derivatives (page 553-556)
Main energy sources:OilGasElectricity
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 23.6
Oil Derivatives (page 553)
Virtually all derivatives available on stocks andstock indices are also available in the OTCmarket with oil as the underlying assetFutures and futures options traded on the NewYork Mercantile Exchange (NYMEX) and theInternational Petroleum Exchange (IPE) are alsopopular
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 23.7
Natural Gas Derivatives (page 554)
A typical OTC contract is for the delivery ofa specified amount of natural gas at aroughly uniform rate to specified locationduring a month.NYMEX and IPE trade contracts thatrequire delivery of 10,000 million Britishthermal units of natural gas to a specifiedlocation
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 23.8
Electricity Derivatives (page 555)
Electricity is an unusual commodity in thatit cannot be storedThe U.S is divided into about 140 controlareas and a market for electricity iscreated by trading between control areas.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 23.9
Electricity Derivatives continued
A typical contract allows one side toreceive a specified number of megawatthours for a specified price at a specifiedlocation during a particular monthTypes of contracts:
5x8, 5x16, 7x24, daily or monthly exercise, swing options
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 23.10
How an Energy Producer HedgesRisks
Estimate a relationship of the formY=a+bP+cT+ε
where Y is the monthly profit, P is theaverage energy prices, T is temperature,and ε is an error termTake a position of –b in energy forwardsand –c in weather forwards.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 23.11
Modeling Energy Prices (equation 23.1,page 555)
For oil a is about 0.5 and σ is about 20%;for natural gas these parameters are about1.0 and 40%; for electricity they are about15 and 150%.
dzdtSatSd σ+−θ= ]ln)([ln
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 23.12
Insurance Derivatives (page 556-557)
CAT bonds are an alternative to traditionalreinsuranceThis is a bond issued by a subsidiary of aninsurance company that pays a higher-than-normal interest rate.If claims of a certain type are above a certainlevel the interest and possibly the principal onthe bond are used to meet claims
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.1
More on Models andNumerical Procedures
Chapter 24
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.2
Three Alternatives to GeometricBrownian Motion
Constant elasticity of variance(CEV)Mixed Jump diffusionVariance Gamma
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.3
CEV Model (page 562 to 563))
When α = 1 the model is Black-ScholescaseWhen α > 1 volatility rises as stock pricerisesWhen α < 1 volatility falls as stock pricerisesEuropean option can be value analyticallyin terms of the cumulative non-central chisquare distribution
dzSSdtqrdS ασ+−= )(
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.4
CEV Models Implied Volatilities
σimp
K
α < 1
α > 1
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.5
Mixed Jump Diffusion Model (page563 to 564)
Merton produced a pricing formula when the stockprice follows a diffusion process overlaid with randomjumps
dp is the random jump k is the expected size of the jumpλ dt is the probability that a jump occurs in the nextinterval of length dt
dpdzdtkSdS +σ+λ−µ= )(/
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.6
Jumps and the Smile
Jumps have a big effect on the impliedvolatility of short term optionsThey have a much smaller effect on theimplied volatility of long term options
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.7
The Variance-Gamma Model (page564 to 566)
g is change over time T in a variable that followsa gamma process. This is a process where smalljumps occur frequently and there are occasionallarge jumpsConditional on g, ln ST is normal. Its varianceproportional to gThere are 3 parameters
v, the variance rate of the gamma processσ2, the average variance rate of ln S per unit timeθ, a parameter defining skewness
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.8
Understanding the Variance-Gamma Model
g defines the rate at which informationarrives during time T (g is sometimesreferred to as measuring economic time)If g is large the the change in ln S has arelatively large mean and varianceIf g is small relatively little informationarrives and the change in ln S has arelatively small mean and variance
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.9
Time Varying Volatility
Suppose the volatility is σ1 for the first yearand σ2 for the second and thirdTotal accumulated variance at the end ofthree years is σ1
2 + 2σ22
The 3-year average volatility is
2 22 2 2 1 2
1 223 2 ;
3σ + σ
σ = σ + σ σ =
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.10
Stochastic Volatility Models (equations24.2 and 24.3, page 567)
When V and S are uncorrelated aEuropean option price is the Black-Scholes price integrated over thedistribution of the average variance
VL
S
dzVdtVVadV
dzVdtqrS
dS
αξ+−=
+−=
)(
)(
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.11
Stochastic Volatility Models continued
When V and S are negatively correlatedwe obtain a downward sloping volatilityskew similar to that observed in the marketfor equitiesWhen V and S are positively correlated theskew is upward sloping
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.12
The IVF Model (page 568)
SdztSdttqtrdS
SdzSdtqrdS
),()]()([by replaced is)(
modelmotion Brownian geomeric usual The prices.
option observed matchesexactly that priceasset for the process a create todesigned
is modelfunction y volatilitimplied The
σ+−=
σ+−=
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.13
The Volatility Function (equation 24.4)
The volatility function that leads to themodel matching all European option pricesis
)()]()([)(2)],([ 222
2
KcKKctqtrKctqtctK
mkt
mktmktmkt
∂∂∂∂−++∂∂
=σ
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.14
Strengths and Weaknesses of theIVF Model
The model matches the probabilitydistribution of stock prices assumed by themarket at each future timeThe models does not necessarily get thejoint probability distribution of stock pricesat two or more times correct
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.15
Numerical Procedures
Topics:Path dependent options using treeBarrier optionsOptions where there are two stochasticvariablesAmerican options using Monte Carlo
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.16
Path Dependence:The Traditional View
Backwards induction works well forAmerican options. It cannot be used forpath-dependent optionsMonte Carlo simulation works well forpath-dependent options; it cannot be usedfor American options
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.17
Extension of BackwardsInductionBackwards induction can be used for some path-dependent optionsWe will first illustrate the methodology usinglookback options and then show how it can beused for Asian options
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.18
Lookback Example (Page 570)
Consider an American lookback put on a stock whereS = 50, σ = 40%, r = 10%, ∆t = 1 month & the life ofthe option is 3 monthsPayoff is Smax-ST
We can value the deal by considering all possiblevalues of the maximum stock price at each node
(This example is presented to illustrate the methodology. It is not the mostefficient way of handling American lookbacks (See Technical Note 13)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.19
Example: An American Lookback PutOption (Figure 24.2, page 570)
S0 = 50, σ = 40%, r = 10%, ∆t = 1 month,
56.12
56.124.68
44.55
50.00
6.38
62.99
62.993.36
50.00
56.12 50.006.12 2.66
36.69
50.00
10.31
70.7070.70
0.00
62.99 56.126.87 0.00
56.12
56.12 50.0011.57 5.45
44.55
35.3650.0014.64
50.005.47 A
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.20
Why the Approach Works
This approach works for lookback options becauseThe payoff depends on just 1 function of the path followedby the stock price. (We will refer to this as a “pathfunction”)The value of the path function at a node can be calculatedfrom the stock price at the node & from the value of thefunction at the immediately preceding nodeThe number of different values of the path function at anode does not grow too fast as we increase the number oftime steps on the tree
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.21
Extensions of the Approach
The approach can be extended so that thereare no limits on the number of alternativevalues of the path function at a nodeThe basic idea is that it is not necessary toconsider every possible value of the pathfunctionIt is sufficient to consider a relatively smallnumber of representative values of thefunction at each node
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.22
Working ForwardFirst work forward through the treecalculating the max and min values ofthe “path function” at each nodeNext choose representative values ofthe path function that span the rangebetween the min and the max
Simplest approach: choose the min, themax, and N equally spaced values betweenthe min and max
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.23
Backwards Induction
We work backwards through the tree in theusual way carrying out calculations for each ofthe alternative values of the path function thatare considered at a nodeWhen we require the value of the derivative ata node for a value of the path function that isnot explicitly considered at that node, we uselinear or quadratic interpolation
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.24
Part of Tree to Calculate Valueof an Option on the ArithmeticAverage(Figure 24.3, page 572)
S = 50.00
Average S46.6549.0451.4453.83
Option Price5.6425.9236.2066.492
S = 45.72
Average S43.8846.7549.6152.48
Option Price 3.430 3.750 4.079 4.416
S = 54.68
Average S47.9951.1254.2657.39
Option Price 7.575 8.101 8.635 9.178
X
Y
Z
0.5056
0.4944
S=50, X=50, σ=40%, r=10%, T=1yr,∆t=0.05yr. We are at time 4∆t
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.25
Part of Tree to Calculate Value of anOption on the Arithmetic Average(continued)
Consider Node X when the average of 5observations is 51.44
Node Y: If this is reached, the average becomes51.98. The option price is interpolated as 8.247
Node Z: If this is reached, the average becomes50.49. The option price is interpolated as 4.182
Node X: value is(0.5056×8.247 + 0.4944×4.182)e–0.1×0.05 = 6.206
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.26
Using Trees with Barriers(Section 24.5, page 573)
When trees are used to valueoptions with barriers, convergencetends to be slowThe slow convergence arises fromthe fact that the barrier isinaccurately specified by the tree
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.27
True Barrier vs Tree Barrier for aKnockout Option: The Binomial Tree Case
Barrier assumed by treeTrue barrier
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.28
Inner and Outer Barriers for Trinomial Trees(Figure 24.4, page 574)
Outer barrierTrue barrier
Inner Barrier
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.29
Alternative Solutionsto Valuing Barrier Options
Interpolate between value when innerbarrier is assumed and value whenouter barrier is assumedEnsure that nodes always lie on thebarriersUse adaptive mesh methodology
In all cases a trinomial tree ispreferable to a binomial tree
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.30
Modeling Two CorrelatedVariables (Section 24.6, page 576)
APPROACHES:1.Transform variables so that they are not
correlated & build the tree in the transformedvariables
2.Take the correlation into account by adjustingthe position of the nodes
3.Take the correlation into account by adjustingthe probabilities
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.31
Monte Carlo Simulation andAmerican Options
Two approaches:The least squares approachThe exercise boundary parameterizationapproach
Consider a 3-year put option where theinitial asset price is 1.00, the strike price is1.10, the risk-free rate is 6%, and there isno income
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.32
Sampled Paths
1.341.220.881.0081.010.840.921.0070.900.770.761.0061.521.561.111.0050.920.970.931.0041.031.071.221.0031.541.261.161.0021.341.081.091.001t=3t=2t=1t=0Path
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.33
The Least Squares Approach (page579)
We work back from the end using a leastsquares approach to calculate thecontinuation value at each timeConsider year 2. The option is in themoney for five paths. These giveobservations on S of 1.08, 1.07, 0.97, 0.77,and 0.84. The continuation values are 0.00,0.07e-0.06, 0.18e-0.06, 0.20e-0.06, and 0.09e-0.06
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.34
The Least Squares Approachcontinued
Fitting a model of the form V=a+bS+cS2 weget a best fit relation
V=-1.070+2.983S-1.813S2
for the continuation value VThis defines the early exercise decision att=2. We carry out a similar analysis at t=1
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.35
The Least Squares Approachcontinued
In practice more complex functional formscan be used for the continuation value andmany more paths are sampled
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.36
The Early Exercise BoundaryParametrization Approach (page 582)
We assume that the early exercise boundarycan be parameterized in some wayWe carry out a first Monte Carlo simulation andwork back from the end calculating the optimalparameter valuesWe then discard the paths from the first MonteCarlo simulation and carry out a new MonteCarlo simulation using the early exerciseboundary defined by the parameter values.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 24.37
Application to Example
We parameterize the early exerciseboundary by specifying a critical assetprice, S*, below which the option isexercised.At t=1 the optimal S* for the eight paths is0.88. At t=2 the optimal S* is 0.84In practice we would use many more pathsto calculate the S*
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 25.1
Martingales andMeasures
Chapter 25
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 25.2
Derivatives Dependent on a SingleUnderlying Variable
dzσ dtµƒ
dƒ
dzσ dtµƒ
d ƒ
ƒƒ
dzsdtmd
222
2
111
1
.
+=
+=
θ
+=θθ
θ
Suppose and prices withon dependent sderivative two Imagine
process the follows that security) traded a of
price they necessaril (not , variable, a Consider
21
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 25.3
Forming a Riskless Portfolio
tƒƒσµƒƒσµ=
ƒƒσƒƒσ
ƒσƒσ
∆−∆Π
−=Π
−
Π
)(
)()(
21122121
211122
11
22
derivative 2nd the of and derivative 1st the of +
of consisting , portfolio riskless a up set can We
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 25.4
or
:gives This
:riskless is portfolio the Since
2
2
1
1
121221
σrµ
σrµ
r σr σσµσµ
t=r
−=
−
−=−
Π∆∆Π
Market Price of Risk (Page 590)
This shows that (µ – r )/σ is the same for allderivatives dependent on the sameunderlying variable, θWe refer to (µ – r )/σ as the market price ofrisk for θ and denote it by λ
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 25.5
Extension of the Analysisto Several Underlying Variables(Equations 25.12 and 25.13, page 593)
then
withvariables underlying several on depends If
σλrµ
dzσµ dtƒ
dƒ
f
n
iii
n
iii
∑
∑
=
=
=−
+=
1
1
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 25.6
Martingales (Page 594)
A martingale is a stochastic process withzero driftA variable following a martingale has theproperty that its expected future valueequals its value today
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 25.7
Alternative Worlds
dzfdtfrdf
dzσfdtrfdf
σ+λσ+=
λ
+=
)(
is risk of price market the where worlda In
worldneutral-risk ltraditiona the In
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 25.8
The Equivalent MartingaleMeasure Result (Page 595)
ion)considerat under period the during income no
provide to assumed are and pricessecurity derivative all
for martingale a is that shows lemma sIto' then ,security a
of volatility the to equal set weIf
gff
gfg
(
λ
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 25.9
Forward Risk Neutrality
We refer to a world where the market priceof risk is the volatility of g as a world that isforward risk neutral with respect to g.If Eg denotes a world that is FRN wrt g
fg
E fgg
T
T
0
0
=⎛⎝⎜
⎞⎠⎟
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 25.10
Alternative Choices for theNumeraire Security g
Money Market AccountZero-coupon bond priceAnnuity factor
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 25.11
Money Market Accountas the Numeraire
The money market account is an accountthat starts at $1 and is always invested atthe short-term risk-free interest rateThe process for the value of the account is
dg=rg dtThis has zero volatility. Using the moneymarket account as the numeraire leads tothe traditional risk-neutral world
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 25.12
Money Market Accountcontinued
worldneutral-risk ltraditiona the in nsexpectatio denotes where
becomes
equation the ,= and 1= Since
E
feEf
gfE
gf
egg
T
rdt
T
Tg
rdt
T
T
T
ˆ
ˆ 0
0
0
0
0
0
⎥⎦
⎤⎢⎣
⎡ ∫=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
∫
−
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 25.13
Zero-Coupon Bond Maturing attime T as Numeraire
price bond the wrtFRN is that worlda in nsexpectatio denotes and price bond coupon-zero the is ),( where
becomes
equation The
T
TT
T
Tg
ETP
fETPf
gfE
gf
0
][),0(0
0
0
=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 25.14
Forward Prices
In a world that is FRN wrt P(0,T), theexpected value of a security at time T is itsforward price
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 25.15
Interest Rates
In a world that is FRN wrt P(0,T2) theexpected value of an interest ratelasting between times T1 and T2 is theforward interest rate
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 25.16
Annuity Factor as the Numeraire
⎥⎦
⎤⎢⎣
⎡=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
)()0(0
0
0
TAfEAf
gfE
gf
TA
T
Tg
becomes
equation The
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 25.17
Annuity Factors and Swap Rates
Suppose that s(t) is the swap ratecorresponding to the annuity factor A.Then:
s(t)=EA[s(T)]
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 25.18
Extension to Several IndependentFactors (Page 599)
∑∑
∑∑
∑
∑
==
==
=
=
σ+⎥⎦
⎤⎢⎣
⎡σλ+=
σ+⎥⎦
⎤⎢⎣
⎡σλ+=
σ+=
σ+=
m
iiig
m
iigi
m
iiif
m
iifi
m
iiig
m
iiif
dztgtdttgttrtdg
dztftdttfttrtdf
dztgtdttgtrtdg
dztftdttftrtdf
1,
1,
1,
1,
1,
1,
)()()()()()(
)()()()()()(
)()()()()(
)()()()()(
consistent internally are that worldsother For
worldneutral-riskltraditiona the In
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 25.19
Extension to Several IndependentFactors continued
hold. results the of rest the and martingalea is case, factor-one the in As
where worldas wrtFRN is that worlda define We
gf
σλg
igi ,=
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 25.20
Applications(Section 25.6, page 600)
Valuation of a European call option wheninterest rates are stochasticValuation of an option to exchange oneasset for another
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 25.21
Change of Numeraire(Section 25.7, page 602)
wvwghwv
vhg
q
vqv
and between ncorrelatio the is and of volatility the is of
volatility the is whereby increases variable a of drift the , to from
security numeraire the change weWhen
ρσ=σσρσ
,,,
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 26.1
Interest Rate Derivatives:The Standard Market Models
Chapter 26
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 26.2
The Complications in ValuingInterest Rate Derivatives (page 611)
We need a whole term structure to define thelevel of interest rates at any timeThe stochastic process for an interest rate ismore complicated than that for a stock priceVolatilities of different points on the termstructure are differentInterest rates are used for discounting the payoffas well as for defining the payoff
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 26.3
Approaches to PricingInterest Rate Options
Use a variant of Black’smodelUse a no-arbitrage (yieldcurve based) model
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 26.4
Black’s Model
Similar to the model proposed byFischer Black for valuing options onfuturesAssumes that the value of an interestrate, a bond price, or some othervariable at a particular time T in thefuture has a lognormal distribution
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 26.5
Black’s Model (continued)
The mean of the probability distribution isthe forward value of the variableThe standard deviation of the probabilitydistribution of the log of the variable is
where σ is the volatilityThe expected payoff is discounted at theT-maturity rate observed today
σ T
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 26.6
Black’s Model (Eqn 26.1 and 26.2, page 611-612)
TddT
TKFd
dNFdKNTPpdKNdNFTPc
σ−=σ
σ+=
−−−=−=
12
20
1
102
210
;2/)/ln(
)]()()[,0()]()()[,0(
• K : strike price• F0 : forward value of
variable today
• T : option maturity• σ : volatility of F
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 26.7
Black’s Model: Delayed Payoff
• K : strike price• F0 : forward value of
variable• σ : volatility of F
• T : time whenvariable is observed
• T * : time of payoff
TddT
TKFd
dNFdKNTPp
dKNdNFTPc
σ−=σ
σ+=
−−−=
−=
12
20
1
102*
210*
;2/)/ln(
)]()()[,0(
)]()()[,0(
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 26.8
Validity of Black’s Model
Two assumptions: 1. The expected value of the underlying
variable is its forward price 2. We can discount expected payoffs at
rate observed in the market todayIt turns out that these assumptions offseteach other in the applications of Black’smodel that we will consider
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 26.9
Black’s Model for European BondOptions
Assume that the future bond price is lognormal
Both the bond price and the strike price shouldbe cash prices not quoted prices
TddT
TKFd
dNFdKNTPpdKNdNFTPc
BB
BB
B
B
σ−=σ
σ+=
−−−=−=
12
2
1
12
21
;2/)/ln(
)]()()[,0()]()()[,0(
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 26.10
Forward Bond and Forward Yield
Approximate duration relation between forwardbond price, FB, and forward bond yield, yF
where D is the (modified) duration of theforward bond at option maturity
F
FF
B
BF
B
B
yyDy
FFyD
FF ∆
−≈∆
∆−≈∆ or
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 26.11
Yield Vols vs Price Vols (Equation 26.8,page 617)
This relationship implies the followingapproximation
where σy is the forward yield volatility, σB isthe forward price volatility, and y0 istoday’s forward yield Often σy is quoted with the understandingthat this relationship will be used tocalculate σB
yB Dy σ=σ 0
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 26.12
Theoretical Justification for BondOption Model
model sBlack' to leads This
Also
is price option the time at maturing bond coupon-zero a wrtFRN is that worlda in Working
0][
)]0,[max(),0(,
FBE
KBETPT
TT
TT
=
−
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 26.13
Caps and Floors
A cap is a portfolio of call options on LIBOR. Ithas the effect of guaranteeing that the interestrate in each of a number of future periods willnot rise above a certain levelPayoff at time tk+1 is Lδk max(Rk-RK, 0) where L isthe principal, δk =tk+1-tk , RK is the cap rate, and Rkis the rate at time tk for the period between tk andtk+1A floor is similarly a portfolio of put options onLIBOR. Payoff at time tk+1 is Lδk max(RK -Rk , 0)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 26.14
Caplets
A cap is a portfolio of “caplets”Each caplet is a call option on a futureLIBOR rate with the payoff occurring inarrearsWhen using Black’s model we assumethat the interest rate underlying eachcaplet is lognormal
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 26.15
Black’s Model for Caps(Equation 26.13, p. 621)
The value of a caplet, for period (tk, tk+1) is
• Fk : forward interest rate for (tk, tk+1)
• σk : forward rate volatility
• L: principal• RK : cap rate• δk=tk+1-tk
-= and where
kkk
kkKk
Kkkk
tddt
tRFd
dNRdNFtPL
σσ
σ+=
−δ +
12
2
1
211
2/)/ln(
)]()()[,0(
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 26.16
When Applying Black’s ModelTo Caps We Must ...
EITHERUse spot volatilitiesVolatility different for each caplet
ORUse flat volatilitiesVolatility same for each caplet within aparticular cap but varies according tolife of cap
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 26.17
Theoretical Justification for CapModel
modelsBlack'toleads This][
Also)]0,[max(),0(
is priceoption the at time maturing bondcoupon -zero
a wrt FRN is that worldain Working
1
11
1
kkk
Kkkk
k
FRE
RREtPt
=
−
+
++
+
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 26.18
Swaptions
A swaption or swap option gives theholder the right to enter into an interestrate swap in the futureTwo kinds
The right to pay a specified fixed rate andreceive LIBORThe right to receive a specified fixed rate andpay LIBOR
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 26.19
Black’s Model for EuropeanSwaptionsWhen valuing European swap options it isusual to assume that the swap rate islognormalConsider a swaption which gives the right topay sK on an n -year swap starting at time T .The payoff on each swap payment date is
where L is principal, m is payment frequencyand sT is market swap rate at time T
max )0,( KT ssmL
−
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 26.20
Black’s Model for EuropeanSwaptions continued (Equation 26.15, page 627)
The value of the swaption is
s0 is the forward swap rate; σ is the swap ratevolatility; ti is the time from today until the i th swappayment; and
)]()( [ 210 dNsdNsLA K−
Am
P tii
m n
==
∑1 01
( , )
TddT
Tssd K σ−=σ
σ+= 12
20
1 ;2/)/ln(where
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 26.21
Theoretical Justification for SwapOption Model
modelsBlack'toleads This][
Also)]0,[max(
is priceoption the swap, theunderlyingannuity the
wrt FRN is that worldain Working
0ssE
ssLAE
TA
KTA
=
−
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 26.22
Relationship Between Swaptionsand Bond Options
An interest rate swap can be regarded asthe exchange of a fixed-rate bond for afloating-rate bondA swaption or swap option is therefore anoption to exchange a fixed-rate bond for afloating-rate bond
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 26.23
Relationship Between Swaptionsand Bond Options (continued)
At the start of the swap the floating-ratebond is worth par so that the swaption canbe viewed as an option to exchange a fixed-rate bond for parAn option on a swap where fixed is paid andfloating is received is a put option on thebond with a strike price of parWhen floating is paid and fixed is received, itis a call option on the bond with a strikeprice of par
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 26.24
Deltas of Interest RateDerivatives
Alternatives:Calculate a DV01 (the impact of a 1bps parallelshift in the zero curve)Calculate impact of small change in the quote foreach instrument used to calculate the zero curveDivide zero curve (or forward curve) into bucketsand calculate the impact of a shift in each bucketCarry out a principal components analysis forchanges in the zero curve. Calculate delta withrespect to each of the first two or three factors
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 27.1
Convexity, Timing, andQuanto Adjustments
Chapter 27
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 27.2
Forward Yields and ForwardPricesWe define the forward yield on a bond as the yieldcalculated from the forward bond priceThere is a non-linear relation between bond yields and bondpricesIt follows that when the forward bond price equals theexpected future bond price, the forward yield does notnecessarily equal the expected future yield
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 27.3
Relationship Between Bond Yieldsand Prices (Figure 27.1, page 636)
BondPrice
YieldY3
B 1
Y1Y2
B 3B 2
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 27.4
Convexity Adjustment for BondYields (Eqn 27.1, p. 637)
Suppose a derivative provides a payoff at time Tdependent on a bond yield, yT observed at time T .Define:G(yT) : price of the bond as a function of its yield
y0 : forward bond yield at time zeroσy : forward yield volatilityThe expected bond price in a world that is FRN wrtP(0,T) is the forward bond priceThe expected bond yield in a world that is FRN wrtP(0,T) is
)()(
21Yield Bond Forward
0
0220 yG
yGTy y ′′′
σ−
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 27.5
Convexity Adjustment for SwapRate
The expected value of the swap rate for the period T toT+τ in a world that is FRN wrt P(0,T) is
where G(y) defines the relationship between price andyield for a bond lasting between T and T+τ that pays acoupon equal to the forward swap rate
)()(
21Rate Swap Forward
0
0220 yG
yGTy y ′′′
σ−
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 27.6
Example 27.1 (page 638)
An instrument provides a payoff in 3 years equalto the 1-year zero-coupon rate multiplied by$1000Volatility is 20%Yield curve is flat at 10% (with annualcompounding)The convexity adjustment is 10.9 bps so that thevalue of the instrument is 101.09/1.13 = 75.95
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 27.7
Example 27.2 (Page 638-639)
An instrument provides a payoff in 3 years =to the 3-year swap rate multiplied by $100Payments are made annually on the swapVolatility is 22%Yield curve is flat at 12% (with annualcompounding)The convexity adjustment is 36 bps so thatthe value of the instrument is 12.36/1.123 =8.80
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 27.8
Timing Adjustments (Equation 27.4, page640)
The expected value of a variable, V, in a world that isFRN wrt P(0,T*) is the expected value of the variablein a world that is FRN wrt P(0,T) multiplied by
where R is the forward interest rate between T and T*expressed with a compounding frequency of m, σR isthe volatility of R, R0 is the value of R today, σV is thevolatility of F, and ρ is the correlation between R andV
⎥⎦
⎤⎢⎣
⎡+
−σσρ− T
mRTTRRVVR
/1)(exp
0
*0
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 27.9
Example 27.3 (page 640)
A derivative provides a payoff 6 years equal to the valueof a stock index in 5 years. The interest rate is 8% withannual compounding1200 is the 5-year forward value of the stock indexThis is the expected value in a world that is FRN wrtP(0,5)To get the value in a world that is FRN wrt P(0,6) wemultiply by 1.00535The value of the derivative is 1200×1.00535/(1.086)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 27.10
Quantos(Section 27.3, page 641)
Quantos are derivatives where thepayoff is defined using variablesmeasured in one currency and paid inanother currencyExample: contract providing a payoff ofST – K dollars ($) where S is the Nikkeistock index (a yen number)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 27.11
Diff Swap
Diff swaps are a type of quantoA floating rate is observed in one currencyand applied to a principal in anothercurrency
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 27.12
Quanto Adjustment (page 642)
The expected value of a variable, V, in aworld that is FRN wrt PX(0,T) is itsexpected value in a world that is FRN wrtPY(0,T) multiplied by exp(ρVWσVσWT) whereW is the forward exchange rate (units of Yper unit of X) and ρVW is the correlationbetween V and W.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 27.13
Example 27.4 (page 642)
Current value of Nikkei index is 15,000This gives one-year forward as 15,150.75Suppose the volatility of the Nikkei is 20%,the volatility of the dollar-yen exchangerate is 12% and the correlation betweenthe two is 0.3The one-year forward value of the Nikkeifor a contract settled in dollars is15,150.75e0.3 ×0.2×0.12×1 or 15,260.23
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 27.14
Quantos continued
twothe between ncorrelatio of tcoefficien the is
and ), of unit per of (units rate exchange the of volatility the is , of volatility the is where
by increases variable a of rate growth the ,currency in worldneutral risk ltraditiona the to currency in worldneutral
risk ltraditiona the from move weWhen
ρ
σσσρσ
XYV
VX
Y
SV
SV
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 27.15
Siegel’s Paradox
is?explain thyou Can . ofdrift a have to1
for process expect the we, of ratedrift a has for process theGiven that
)/1()/1]([)/1( thatlemma sIto' from implies This
][process neutral-risk thefollows )currency
ofuntit per currency of (units rate exchangeAn
2
YX
XY
SSYX
SXY
rrSrr
SdzSdtSrrSd
SdzSdtrrdSX
YS
−−
σ−σ+−=
σ+−=
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 27.16
When is a Convexity, Timing, orQuanto Adjustment Necessary
A convexity or timing adjustment isnecessary when interest rates are used ina nonstandard way for the purposes ofdefining a payoffNo adjustment is necessary for a vanillaswap, a cap, or a swap option
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.1
Interest Rate Derivatives:Models of the Short Rate
Chapter 28
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.2
Term Structure Models
Black’s model is concerned withdescribing the probability distribution ofa single variable at a single point intimeA term structure model describes theevolution of the whole yield curve
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.3
The Zero Curve
The process for the instantaneous short rate, r,in the traditional risk-neutral world defines theprocess for the whole zero curve in this worldIf P(t, T ) is the price at time t of a zero-couponbond maturing at time T
where is the average r between times t and T[ ] P t T E e r T t( , ) $ ( )= − −
r
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.4
Equilibrium Models
Rendleman & Bartter: Vasicek: Cox, Ingersoll, & Ross (CIR):
dr r dt r dz
dr a b r dt dz
dr a b r dt r dz
= +
= − +
= − +
µ σ
σ
σ
( )
( )
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.5
Mean Reversion(Figure 28.1, page 651)
Interestrate
HIGH interest rate has negative trend
LOW interest rate has positive trend
ReversionLevel
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.6
Alternative Term Structuresin Vasicek & CIR(Figure 28.2, page 652)
Zero Rate
Maturity
Zero Rate
Maturity
Zero Rate
Maturity
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.7
Equilibrium vs No-ArbitrageModels
In an equilibrium model today’sterm structure is an outputIn a no-arbitrage model today’sterm structure is an input
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.8
Developing No-ArbitrageModel for r
A model for r can be made to fitthe initial term structure byincluding a function of time inthe drift
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.9
Ho and Lee
dr = θ(t )dt + σdzMany analytic results for bond pricesand option pricesInterest rates normally distributedOne volatility parameter, σAll forward rates have the samestandard deviation
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.10
Initial ForwardCurve
ShortRate
r
r
r
rTime
Diagrammatic Representation ofHo and Lee (Figure 28.3, page 655)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.11
Hull and White Model
dr = [θ(t ) – ar ]dt + σdzMany analytic results for bond prices andoption pricesTwo volatility parameters, a and σInterest rates normally distributedStandard deviation of a forward rate is adeclining function of its maturity
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.12
Diagrammatic Representation of Hulland White (Figure 28.4, page 656)
ShortRate
r
r
r
rTime
Forward RateCurve
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.13
Black-Karasinski Model (equation 28.18)
Future value of r is lognormalVery little analytic tractability
[ ] dztdtrrtatrd )()ln()()()ln( σ+−θ=
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.14
Options on Zero-Coupon Bonds(equation 28.20, page 658)
In Vasicek and Hull-White model, price of call maturingat T on a bond lasting to s is
LP(0,s)N(h)-KP(0,T)N(h-σP)Price of put is
KP(0,T)N(-h+σP)-LP(0,s)N(h)where
[ ]
TTsσ
KLa
eeaKTP
sLPh
P
aTTsa
PP
P
)( Lee-HoFor
price. strike theis and principal theis 2
112),0(
),0(ln1 2)(
−σ=
−−
σ=σ
σ+
σ=
−−−
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.15
Options on Coupon BearingBonds
In a one-factor model a European option on acoupon-bearing bond can be expressed as aportfolio of options on zero-coupon bonds.We first calculate the critical interest rate at theoption maturity for which the coupon-bearingbond price equals the strike price at maturityThe strike price for each zero-coupon bond is setequal to its value when the interest rate equalsthis critical value
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.16
Interest Rate Trees vs Stock PriceTrees
The variable at each node in aninterest rate tree is the ∆t-periodrateInterest rate trees work similarly tostock price trees except that thediscount rate used varies fromnode to node
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.17
Two-Step Tree Example(Figure 28.6, page 660))
Payoff after 2 years is MAX[100(r – 0.11), 0]pu=0.25; pm=0.5; pd=0.25; Time step=1yr
10%0.35**
12%1.11*
10%0.23
8%0.00
14%3
12%1
10%0
8%0
6%0 *: (0.25×3 + 0.50×1 + 0.25×0)e–0.12×1
**: (0.25×1.11 + 0.50×0.23 +0.25×0)e–0.10×1
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.18
Alternative Branching Processesin a Trinomial Tree(Figure 28.7, page 661)
(a) (b) (c)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.19
Procedure for Building Tree
dr = [θ(t ) – ar ]dt + σdz
1.Assume θ(t ) = 0 and r (0) = 02.Draw a trinomial tree for r to match the mean
and standard deviation of the process for r3.Determine θ(t ) one step at a time so that the
tree matches the initial term structure
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.20
Example (page 662 to 667)
σ = 0.01 a = 0.1 ∆t = 1 year The zero curve is as shown in
Table 28.1 on page 665
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.21
Building the First Tree for the ∆trate R
Set vertical spacingChange branching when jmax nodes frommiddle where jmax is smallest integergreater than 0.184/(a∆t)Choose probabilities on branches so thatmean change in R is -aR∆t and S.D. ofchange is
tR ∆σ=∆ 3
t∆σ
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.22
The Initial Tree(Figure 28.8, page 663)
A
B
C
D
E
F
G
H
I
Node A B C D E F G H I
R 0.000% 1.732% 0.000% -1.732% 3.464% 1.732% 0.000% -1.732% -3.464%
p u 0.1667 0.1217 0.1667 0.2217 0.8867 0.1217 0.1667 0.2217 0.0867
p m 0.6666 0.6566 0.6666 0.6566 0.0266 0.6566 0.6666 0.6566 0.0266
p d 0.1667 0.2217 0.1667 0.1217 0.0867 0.2217 0.1667 0.1217 0.8867
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.23
Shifting Nodes
Work forward through treeRemember Qij the value of a derivativeproviding a $1 payoff at node j at time i∆tShift nodes at time i∆t by αi so that the(i+1)∆t bond is correctly priced
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.24
The Final Tree(Figure 28.9, Page 665)
A
B
C
D
E
F
G
H
I
Node A B C D E F G H I
R 3.824% 6.937% 5.205% 3.473% 9.716% 7.984% 6.252% 4.520% 2.788%
p u 0.1667 0.1217 0.1667 0.2217 0.8867 0.1217 0.1667 0.2217 0.0867
p m 0.6666 0.6566 0.6666 0.6566 0.0266 0.6566 0.6666 0.6566 0.0266
p d 0.1667 0.2217 0.1667 0.1217 0.0867 0.2217 0.1667 0.1217 0.8867
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.25
Extensions
The tree building procedure can beextended to cover more general models ofthe form:
dƒ(r ) = [θ(t ) – a ƒ(r )]dt + σdzWe set x=f(r) and proceed similarly tobefore
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 28.26
Calibration to determine a and σ
The volatility parameters a and σ are chosen so that themodel fits the prices of actively traded instruments suchas caps and European swap options as closely aspossibleWe minimize a function of the form
where Ui is the market price of the ith calibratinginstrument, Vi is the model price of the ith calibratinginstrument and P is a function that penalizes big changesor curvature in a and σ
∑=
+−n
iii PVU
1
2)(
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 29.1
Interest Rate Derivatives:HJM and LMM
Chapter 29
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 29.2
HJM Model: Notation
P(t,T ): price at time t of a discount bondwith principal of $1 maturing at T
Ωt : vector of past and present values ofinterest rates and bond prices at time tthat are relevant for determining bondprice volatilities at that time
v(t,T,Ωt ): volatility of P(t,T)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 29.3
Notation continued
ƒ(t,T1,T2): forward rate as seen at t for theperiod between T1 and T2
F(t,T): instantaneous forward rate as seen att for a contract maturing at T
r(t): short-term risk-free interest rate at tdz(t): Wiener process driving term structure
movements
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 29.4
Modeling Bond Prices (Equation 29.1,page 680)
) for process a get we approach Letting for
process the determine to lemma sIto' use can we
Because all for
providing function any choose can We
t,TFTT),Tf(t,T
TTTtPTtP),Tf(t,T
tttvv
tdzTtPTtvdtTtPtrTtdP
t
t
(.
)],(ln[)],(ln[
0),,(
)(),(),,(),()(),(
1221
12
2121 −
−=
=Ω
Ω+=
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 29.5
The process for F(t,T)Equation 29.4 and 29.5, page 681)
factor one than more is there whenhold results Similar
have must we(),(
write weif that means result This
dτs(t,)ΩT,s(t,)ΩT, m(t,
)dzΩT,s(t,dtΩT,t,mTtdF
tdzTtvdtTtvTtvTtdF
T
t ttt
tt
tTtTt
∫ Ωτ=
+=
Ω−ΩΩ=
),
)
)(),,(),,(),,(),(
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 29.6
Tree Evolution of Term Structureis Non-Recombining
Tree for the short rate r is non-Markov(see Figure 29.1, page 682)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 29.7
The LIBOR Market Model
The LIBOR market model is a modelconstructed in terms of the forward ratesunderlying caplet prices
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 29.8
Notation
t kF t t tm t t
t F t tv t P t t t
t t
k
k k k
k k
k k
k k k
: th reset date forward rate between times and
: index for next reset date at time volatility of at time volatility of ( , at time
( ):( )( ): ( )( ): )
:
+
+ −
1
1
ς
δ
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 29.9
Volatility Structure
We assume a stationary volatility structurewhere the volatility of depends only on the number of accrual periods between the next reset date and [i.e., it is a function onlyof ]
F t
tk m t
k
k
( )
( )−
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 29.10
In Theory the Λ’s can bedetermined from Cap Prices
yinductivel determined be to s the allows This
have must weprices cap to fit perfect a provides
model the If caplet. the for volatility the is If when of volatility the as Define i
'
),()()(
11
22
1
Λ
δΛ=σ
σ=−Λ
∑=
−−
+
k
iiikkk
kkk
k
t
ttitmktF
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 29.11
Example 29.1 (Page 684)
If Black volatilities for the first threecaplets are 24%, 22%, and 20%, then
Λ0=24.00%Λ1=19.80%Λ2=15.23%
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 29.12
Example 29.2 (Page 684)
n 1 2 3 4 5
σn(%) 15.50 18.25 17.91 17.74 17.27
Λn-1(%) 15.50 20.64 17.21 17.22 15.25
n 6 7 8 9 10
σn(%) 16.79 16.30 16.01 15.76 15.54
Λn-1(%) 14.15 12.98 13.81 13.60 13.40
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 29.13
The Process for Fk in a One-Factor LIBOR Market Model
dF F dz
P t t
k k m t k
i
= + −
+
K Λ ( )
( , ),
The drift depends on the world chosenIn a world that is forward risk - neutralwith respect to the drift is zero1
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 29.14
Rolling Forward Risk-Neutrality(Equation 29.12, page 685)
It is often convenient to choose a worldthat is always FRN wrt a bond maturing atthe next reset date. In this case, we candiscount from ti+1 to ti at the δi rateobserved at time ti. The process for Fk is
dFF
FF
dt dzk
k
i i i m t k m t
i ij m t
i
k m t=+
+− −
=−∑
δδ
Λ ΛΛ( ) ( )
( )( )1
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 29.15
The LIBOR Market Model andHJM
In the limit as the time between resetstends to zero, the LIBOR market modelwith rolling forward risk neutrality becomesthe HJM model in the traditional risk-neutral world
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 29.16
Monte Carlo Implementation ofLMM Model(Equation 29.14, page 685)
We assume no change to the drift betweenreset dates so that
F t F tF t
Lk j k ji i j i j k j
j j
k j
j k
i
k k j j( ) ( )exp( )
+− − −
=−=
+−
⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎡
⎣⎢⎢
⎤
⎦⎥⎥
∑1
2
1 2δ
δδ ε δ
Λ Λ ΛΛ
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 29.17
Multifactor Versions of LMM
LMM can be extended so that there areseveral components to the volatilityA factor analysis can be used to determinehow the volatility of Fk is split intocomponents
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 29.18
Ratchet Caps, Sticky Caps, andFlexi Caps
A plain vanilla cap depends only on oneforward rate. Its price is not dependent onthe number of factors.Ratchet caps, sticky caps, and flexi capsdepend on the joint distribution of two ormore forward rates. Their prices tend toincrease with the number of factors
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 29.19
Valuing European Options in theLIBOR Market Model
There is an analytic approximation thatcan be used to value European swapoptions in the LIBOR market model. Seeequations 29.18 and 29.19 on page 689
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 29.20
Calibrating the LIBOR MarketModel
In theory the LMM can be exactly calibrated tocap prices as described earlierIn practice we proceed as for short rate modelsto minimize a function of the form
where Ui is the market price of the ith calibratinginstrument, Vi is the model price of the ithcalibrating instrument and P is a function thatpenalizes big changes or curvature in a and σ
∑=
+−n
iii PVU
1
2)(
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 29.21
Types of Mortgage-BackedSecurities (MBSs)
Pass-ThroughCollateralized MortgageObligation (CMO)Interest Only (IO)Principal Only (PO)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 29.22
Option-Adjusted Spread(OAS)
To calculate the OAS for aninterest rate derivative we value itassuming that the initial yield curveis the Treasury curve + a spreadWe use an iterative procedure tocalculate the spread that makesthe derivative’s model price =market price. This is the OAS.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 30.1
Chapter 30
Swaps Revisited
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 30.2
Valuation of Swaps
The standard approach is to assume thatforward rates will be realizedThis works for plain vanilla interest rateand plain vanilla currency swaps, but doesnot necessarily work for non-standardswaps
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 30.3
Variations on Vanilla InterestRate Swaps
Principal different on two sidesPayment frequency different on two sidesCan be floating-for-floating instead of floating-for-fixedIt is still correct to assume that forward rates arerealizedHow should a swap exchanging the 3-monthLIBOR for 3-month CP rate be valued?
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 30.4
Compounding Swaps (Business Snapshot30.2, page 699)
Interest is compounded instead of being paidExample: the fixed side is 6% compoundedforward at 6.3% while the floating side isLIBOR plus 20 bps compounded forward atLIBOR plus 10 bps.This type of compounding swap can bevalued using the “assume forward rates arerealized” rule. This is because we can enterinto a series of forward contracts that havethe effect of exchanging cash flows for theirvalues when forward rates are realized.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 30.5
Currency Swaps
Standard currency swaps can be valuedusing the “assume forward LIBOR rate arerealized” rule.Sometimes banks make a smalladjustment because LIBOR in currency Ais exchanged for LIBOR plus a spread incurrency B
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 30.6
More Complex Swaps
LIBOR-in-arrears swapsCMS and CMT swapsDifferential swaps
These cannot be accurately valued byassuming that forward rates will berealized
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 30.7
LIBOR-in Arrears Swap (Equation 30.1,page 701)
Rate is observed at time ti and paid at time ti rather thantime ti+1
It is necessary to make a convexity adjustment to eachforward rate underlying the swapSuppose that Fi is the forward rate between time ti andti+1 and σi is its volatilityWe should increase Fi by
when valuing a LIBOR-in-arrears swap
ii
iiiii
FtttF
τ+−σ +
1)( 1
22
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 30.8
CMS swaps
Swap rate observed at time ti is paid attime ti+1We must
make a convexity adjustment becausepayments are swap rates (= yield on a paryield bond)Make a timing adjustment because paymentsare made at time ti+1 not ti
See equation 30.2 on page 702
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 30.9
Differential Swaps
Rate is observed in currency Y andapplied to a principal in currency XWe must make a quanto adjustment to therateSee equation 30.3 on page 704.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 30.10
Equity Swaps (page 704-705)
Total return on an equity index isexchanged periodically for a fixed orfloating returnWhen the return on an equity index isexchanged for LIBOR the value of theswap is always zero immediately after apayment. This can be used to value theswap at other times.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 30.11
Swaps with Embedded Options(page 705-708)
Accrual swapsCancelable swapsCancelable compounding swaps
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 30.12
Other Swaps (page 708-709)
Indexed principal swapCommodity swapVolatility swapBizarre deals (for example, the P&G 5/30swap in Business Snapshot 30.4 on page709)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 31.1
Real Options
Chapter 31
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 31.2
An Alternative to the NPV Rulefor Capital Investments
Define stochastic processes for the keyunderlying variables and use risk-neutralvaluationThis approach (known as the real optionsapproach) is likely to do a better job atvaluing growth options, abandonmentoptions, etc than NPV
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 31.3
The Problem with using NPV toValue Options
Consider the example from Chapter 11: risk-free rate=12%; strike price = $21
Suppose that the expected return required by investorsin the real world on the stock is 16%. What discount rateshould we use to value an option with strike price $21?
Stock Price = $22
Stock price = $20
Stock Price=$18
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 31.4
Correct Discount Rates areCounter-Intuitive
Correct discount rate for a call option is42.6%Correct discount rate for a put option is–52.5%
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 31.5
General Approach to Valuation
We can value any asset dependent on avariable θ by
Reducing the expected growth rate of θ by λswhere λ is the market price of θ-risk and s isthe volatility of θAssuming that all investors are risk-neutral
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 31.6
Extension to Many UnderlyingVariables
When there are several underlyingvariables θi we reduce the growth rate ofeach one by its market price of risk timesits volatility and then behave as though theworld is risk-neutralNote that the variables do not have to beprices of traded securities
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 31.7
Estimating the Market Price ofRisk Using CAPM (equation 31.2, page 716)
rate free-riskterm-short the is and market; the on return
expected the is return; smarket' the of volatility the is market; the on returns and variable the in changes percentage between
ncorrelatio ousinstantane the is where
by given is variable a of risk of price market The
r
r
m
m
mm
µσ
ρ
−µσρ
=λ )(
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 31.8
Example of Application of Real OptionsApproach to Valuing Amazon.com (BusinessSnapshot 31.1; Schwartz and Moon)
Estimate stochastic processes for the company’s salesrevenue and its average growth rate.Estimated the market price of risk and other keyparameters (cost of goods sold as a percent of sales,variable expenses as a percent of sales, fixed expenses,etc.)Use Monte Carlo simulation to generate differentscenarios in a risk-neutral world.The stock price is the average of the present values ofthe net cash flows discounted at the risk-free rate.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 31.9
Commodity Prices
Futures prices can be used to define theprocess followed by a commodity price ina risk-neutral world.We can build in mean reversion and use aprocess for constructing trinomial treesthat is analogous to that used for interestrates in Chapter 28
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 31.10
Example (page 671)
A company has to decide whether to invest $15million to obtain 6 million barrels of oil at the rateof 2 million barrels per year for three years. Thefixed operating costs are $6 million per year andthe variable costs are $17 per barrel. The spotprice of oil $20 per barrel and 1, 2, and 3-yearfutures prices are $22, $23, and $24,respectively. The risk-free rate is 10% perannum for all maturities.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 31.11
The Process for Oil
We assume that this is d ln(S)=[θ(t)-aln(S)] dt+σ dzWhere a=0.1 and σ=0.2
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 31.12
Tree Assuming θ(t)=0; Fig 31.1
E J0.6928 0.6928
B F K0.3464 0.3464 0.3464
A C G L0.0000 0.0000 0.0000 0.0000
D H M-0.3464 -0.3464 -0.3464
I N-0.6928 -0.6928
0.88670.12170.16670.22170.08670.12170.16670.22170.1667pd
0.02660.65660.66660.65660.02660.65660.66660.65660.6666pm
0.08670.22170.16670.12170.88670.22170.16670.12170.1667pu
IHGFEDCBANode
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 31.13
Final Tree for Oil Prices; Fig 31.2
E J44.35 45.68
B F K30.49 31.37 32.30
A C G L20.00 21.56 22.18 22.85
D H M15.25 15.69 16.16
I N11.10 11.43
0.88670.12170.16670.22170.08670.12170.16670.22170.1667pd
0.02660.65660.66660.65660.02660.65660.66660.65660.6666pm
0.08670.22170.16670.12170.88670.22170.16670.12170.1667pu
IHGFEDCBANode
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 31.14
Valuation of Base Project; Fig 31.3
E J42.24 0.00
B F K38.32 21.42 0.00
A C G L14.46 10.80 5.99 0.00
D H M-9.65 -5.31 0.00
I N-13.49 0.00
0.88670.12170.16670.22170.08670.12170.16670.22170.1667pd
0.02660.65660.66660.65660.02660.65660.66660.65660.6666pm
0.08670.22170.16670.12170.88670.22170.16670.12170.1667pu
IHGFEDCBANode
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 31.15
Valuation of Option to Abandon; Fig 31.4
(No Salvage Value; No Further Payments)
E J0.00 0.00
B F K0.00 0.00 0.00
A C G L1.94 0.80 0.00 0.00
D H M9.65 5.31 0.00
I N13.49 0.00
0.88670.12170.16670.22170.08670.12170.16670.22170.1667pd
0.02660.65660.66660.65660.02660.65660.66660.65660.6666pm
0.08670.22170.16670.12170.88670.22170.16670.12170.1667pu
IHGFEDCBANode
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 31.16
Value of Expansion Option; Fig 31.5 (CompanyCan Increase Scale of Project by 20% for $2million)
E J6.45 0.00
B F K5.66 2.28 0.00
A C G L1.06 0.34 0.00 0.00
D H M0.00 0.00 0.00
I N0.00 0.00
0.88670.12170.16670.22170.08670.12170.16670.22170.1667pd
0.02660.65660.66660.65660.02660.65660.66660.65660.6666pm
0.08670.22170.16670.12170.88670.22170.16670.12170.1667pu
IHGFEDCBANode
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 32.1
Derivatives Mishaps andWhat We Can Learn
from Them
Chapter 32
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 32.2
Big Losses by FinancialInstitutions
Allied Irish Bank ($700 million)Barings ($1 billion)Daiwa ($1 billion)Kidder Peabody ($350 million)LTCM ($4 billion)Midland Bank ($500 million)National Westminster Bank ($130 million)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 32.3
Big Losses by Non-FinancialCorporations
Allied Lyons ($150 million)Gibsons Greetings ($20 million)Hammersmith and Fulham ($600 million)Metallgesellschaft ($1.8 billion)Orange County ($2 billion)Procter and Gamble ($90 million)Shell ($1 billion)Sumitomo ($2 billion)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 32.4
Lessons for All Users ofDerivatives
Risk must be quantified and risk limits setExceeding risk limits not acceptable evenwhen profits resultDo not assume assume that a trader witha good track record will always be rightBe diversifiedScenario analysis and stress testing isimportant
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 32.5
Lessons for Financial Institutions
Do not give too much independence to startradersSeparate the front middle and back officeModels can be wrongBe conservative in recognizing inceptionprofitsDo not sell clients inappropriate productsLiquidity risk is importantThere are dangers when many are followingthe same strategy