pricing derivatives & options
DESCRIPTION
Interest rate futures contracts Options on futures Mortgage-backed securities Interest rate caps- and floor Swap options Commodity linked bonds Zero-coupon treasury tripsTRANSCRIPT
Styring og Fondsmegling Dr.oecon Per B Solibakke
1
Pricing Derivatives and OptionsDerivatives is an investment whose value to day or at some future date is derived
entirely from the value of other assets, the underlying asset.Examples are:
Interest rate futures contractsOptions on futuresMortgage-backed securitiesInterest rate caps- and floorSwap optionsCommodity linked bondsZero-coupon treasury trips
Major Break-through in the valuation of derivatives came with two finance professors at MIT, Black and Scholes, came out with a formula that related the price of a call option to the price of the stock to which the option applies.
Styring og Fondsmegling Dr.oecon Per B Solibakke
2
Forwards and FuturesRepresents the obligation to buy (sell) a security or commodity at a pre-specified price, known as the forward price, at some future dateThe most important financial forward market is the inter-bank forward market for currencies, particularly dollars for yen and dollars for Euros.
Pricing Derivatives and Options: Examples
SwapsIs an agreement between two investors, or counterparties as they are sometimes called, to periodically exchange the cash flow of one security for the cash flow of another. The last date of exchange determines the swap maturity.Forwards, Futures and Swaps zero-cost instruments.
OptionsGives their buyers the right, but not the obligation, to buy (call option) or sell (sell option) an underlying security at a pre-specified price, known as the strike price.
Styring og Fondsmegling Dr.oecon Per B Solibakke
3
Pricing Derivatives and OptionsValues of Calls and Puts at Expiration:
Styring og Fondsmegling Dr.oecon Per B Solibakke
4
Pricing Derivatives and Options: Examples
Options cont.
WarrantsWarrants are options, usually calls, that companies issue on their own stock. In contrast too options, which are mere bets between investors on the value of the company’s underlying stock – in which the corporation never gets involved –warrants are contracts between a corporation and an investor.
Embedded OptionsA number of corporate securities have option like components. Ex. Convertible bonds, callable or refundable corporate bonds. Corporate equity and debt contain option like characteristics
Real AssetsMany real assets may be viewed as derivatives.Ex. Copper mine/Mortgage-back securities/Structured Notes
Styring og Fondsmegling Dr.oecon Per B Solibakke
5
Pricing Derivatives and Options: BasicsTwo basic components: 1. Concept of Perfect tracking
2. Principle of no arbitrage
A fair market price, is simply a no-arbitrage restriction between the tracking portfolio and the derivative.
It is always possible to develop a portfolio consisting of the underlying asset and a risk-free asset that perfectly tracks the future cash flows of a derivative. A perfect tracking portfolio is a combination of securities that perfectly replicates the future cash flows of another investment.
Note: In absence of tracking error, arbitrage exists if it costs more to buy the tracking portfolio than the derivative, or vice versa.
Styring og Fondsmegling Dr.oecon Per B Solibakke
6
Pricing Derivatives and Options: Forward ValueEx. Obligation (forward) to buy Ekornes stock one year from now for 100 kroner! The price
sells to day for 97 and Ekornes will not pay dividends in the period. One-year zero coupon bonds (par. 100) currently sell for 92.
97-92=+5Strategy 2 (tracking portfolio)
?Strategy 1(forward)
Cash Flow one Year from today
Cost Today
1 100S −
1 100S −
Strategy 2 costs 5 kroner. If strategy 1 cost > 5 kroner, arbitrage exists. Go short in strategy 1 and go long in strategy 2 (sell forward (short), buy the share and short 200 kroner par of zero coupon bonds). If Strategy 1 cost < 5 kroner, arbitrage exists. Go long in strategy 1 and go short in strategy 2 (buy forward (long), selling short the share and go long 200 kroner par of zero coupon bonds). kroner 5 is a fair value of the attractive obligation to buy Ekornesfor kroner 100 one year from now.
Styring og Fondsmegling Dr.oecon Per B Solibakke
7
Pricing Derivatives and Options: Binomial Model
The derivative valuation models induce that the current price of the underlying asset determines the price of the derivative to day. Hence, a call option has a known value at expiration date of the call when the stock price to day is known.
Linear functions of the underlying asset’s future payoffs:
Static investment strategies (buy and hold) ex. Forward contracts
Non-linear functions of the underlying asset’s future payoff:
Dynamic investment strategies (continuous rebalancing) ex. Option contracts
The ability to perfectly track a derivative’s payoff with a dynamic strategy requires that the following conditions are met:
1. The price of the underlying security must change smoothly; that is, it does not make large jumps.
2. It must be possible to trade both the derivative and the underlying security continuously.
3. Markets must be frictionless.
Styring og Fondsmegling Dr.oecon Per B Solibakke
8
If the price of the underlying security follows a binomial process, the investor can still perfectly track the derivative’s future cash flows.
UP
Down
Give high degree of accuracy, when the binomial periods are small and numerous.
Pricing Derivatives and Options: Binomial Model
Styring og Fondsmegling Dr.oecon Per B Solibakke
9
Binomial Model Tracking of a Structured Bond
OBX
1
1.500 (up state)
750 (down state)
1.275
1,10 (up state)
1,10 (down state)
331,75 (up state)
=100,00+6,75%(100,00) +225,00
106,75 (down state)
=100,00+6,75%(100,00) +0,00
Pricing Derivatives and Options: Binomial Model
Styring og Fondsmegling Dr.oecon Per B Solibakke
10
The binomial process allow perfect tracking of the value of the derivative applying a tracking portfolio consisting of the underlying asset and a risk-free bond.
Two steps: Identification of the tracking portfolio (TP) and valuation of the TP.Identification:Find the perfect tracking portfolio is the major task in valuing a derivative. With
binomial processes, the tracking portfolio is identified by solving two equations in two unknowns, where each equation corresponds to one of the two future nodes to which one can move (up or down).
Up node:
Down node:
(1 )u f uS B r V∆ + + =
(1 )d f dS B r V∆ + + =
Solved simultaneously yields a unique solution for ∆ and B.
Valuation:
The fair market value of the derivative equals the amount it costs to buy the tracking portfolio. Buying ∆ shares and B dollars of the risk free asset. Thus,
0V S B= ∆ +
Pricing Derivatives and Options: Binomial Model
Styring og Fondsmegling Dr.oecon Per B Solibakke
11
Pricing Derivatives and Options: Binomial Model
106,75331,75?Derivative0,3(750)+1,1B∆(1,500)+1,1B∆(1,275)+BTP portfolio
Down StateUp StateToday’s valueValueNext period
1.500 1,1 331,75750 1,1 106,75
BB
∆ ⋅ + ⋅ =∆ ⋅ + ⋅ =
Two equations in two unknownsThe Solution :
∆ = 0,3 and B = -107,50
0
0,3 1.275 107,50 275V S BV= ∆ += ⋅ − =
Valuation:
The value of the derivative equals the value of the tracking portfolio.
Example structured bond
Styring og Fondsmegling Dr.oecon Per B Solibakke
12
Pricing Derivatives and Options: Wall Street Approach
Based on our binomial results:
Important 1: The value of the derivative in relation to the value of the underlying asset does not depend on the probabilities of up and down.
Important 2: The grade of investor risk aversion was not necessary for calculation fair market values.
Why not?
This information is already captured by the price of the underlying asset on which we base out valuation of the derivative. Note: Once the stock price is known, risk aversion and mean return information are superfluous, not that they are irrelevant.
Styring og Fondsmegling Dr.oecon Per B Solibakke
13
Pricing Derivatives and Options: Wall Street Approach
The risk-neutral valuation method:
Step 1. Identify risk-neutral probabilities that are consistent with investors being risk neutral, given the current value of the underlying asset and its possible future values.
Step 2. Multiply each risk-neutral probability by the corresponding future value for the derivative and sum the products together.
Step 3. Discount the sum of the products in step 2 (the probability weighted average of the derivative’s possible future values) at the risk free rate.
Fewer steps than the tracking portfolio approach.
Styring og Fondsmegling Dr.oecon Per B Solibakke
14
Pricing Derivatives and Options: Wall Street Approach
A General formula for risk-neutral probabilities:
(1 ) 1
:1
f
f
f
u d r
wherer is the risk free rate
u rate of return at the up noded rate of return at the down noderearraging the Expression
r du d
π π
π
+ − = +
− −
−−
+ −=
−
10%1,1765
0,5882:
1 0,1 0,5882 0,871,1765 0,5882
fru up noded down noderearraging the Expression
π
=
− =− =
+ −= =
−
For our example:
331,75 0,87 106,75 (1 0,87) 302,5302,5 275
1.1
⋅ + ⋅ − =
=
Valuation of the derivative:
Styring og Fondsmegling Dr.oecon Per B Solibakke
15
Risk-Neutral Probabilities and Zero-cost Forward and Future Prices
( ) (1 )( ) /1 0u d fF F F F rπ π− + − − + =
(1 )u dF S Sπ π= ⋅ + − ⋅
The no-arbitrage futures price is the same as a weighted average of the expected futures prices at the end of the period, where the weights are the risk-neutral probabilities: (1 )u dF F Fπ π= ⋅ + − ⋅
At the end of the period (maturity): Fu = Su and Fd = Sd. Substitution gives us:
Pricing Derivatives and Options: Wall Street Approach
Because futures contracts are one class of popularly traded financial instruments with known terms (the future price) and known market values, it is often useful to infer risk-neutral probabilities from them.
To use the prices of zero-cost forwards and futures to obtain risk-neutral probabilities, it is necessary to slightly modify the risk-neutral valuation formulas.
For futures, the expected cash flows at the end of the period is
Styring og Fondsmegling Dr.oecon Per B Solibakke
16
1,10 (up state)
OBX
750 (down state)
1.500 (up state)
1,10 (down state)
1,275
1
1.500-F (up state)
750 -F (down state)
Using Risk-Neutral Probabilities to obtain Future Prices
Using 0,87 and 0,13 we can derive future prices:0.87(1.500) + 0,13(750) = 1402,50 consistent (1275 *1,1=1402,75)
Pricing Derivatives and Options: Wall Street Approach
Styring og Fondsmegling Dr.oecon Per B Solibakke
17
It is possible to rearrange to identify the risk neutral probabilities π and 1 –πfrom futures prices. This yields:
d
u d
d
u d
F FF F
and at maturityF SS S
π
π
−=
−
−=
−
If future prices can appreciate 10% (up state) or depreciate 10% (down state), we can calculate risk neutral probabilities.
In the up state, Fu=1.1F, in the down state Fd=0,9F. Applying the above equation:0.9 0,1 0,5
1,1 0,9 0, 2F F F
F F Fπ − ⋅= = =
⋅ − ⋅
Pricing Derivatives and Options: Wall Street Approach
Styring og Fondsmegling Dr.oecon Per B Solibakke
18
Pricing Derivatives and Options: Multiperiod Binomial
Risk neutral valuation can be applied whenever perfect tracking is possible. However, when large jumps in the value of the tracking portfolio or the derivative can occur, perfect tracking is in general not possible. The exception is a binomial price process.
Numerical example in a multiperiod setting:
100
110
90
121
38=121-83
99
38=121 - 83
16=99 - 83
8181
0
Styring og Fondsmegling Dr.oecon Per B Solibakke
19
Pricing Derivatives and Options: Multiperiod Binomial
Risk neutral valuation method(working backward through the tree diagram)121 π + 99 (1-π) = 110 π = 0,5
Value of the option at node U is:0,5 (38) + 0,5 (16) = 27
Value of the option at node D is (the same π)0,5 (16) + 0,5 (0) = 8
The value at the derivative is0,5 (27) + 0,5 (8) = 17,5
Styring og Fondsmegling Dr.oecon Per B Solibakke
20
Algebraic Representation of Two-Period Binomial Valuation
:(1 )
1(1 )
1
' :(1 )
1
uu udu
f
ud ddd
f
u d
f
Node uV VV
rV VV
r
To day s value VV VV
r
π π
π π
π π
+ −=
+
+ −=
+
+ −=
+
2 2
(1 ) (1 )(1 )1 1
2 (1 ) (1 )1
uu ud ud dd
f f
uu ud dd
f
V V V VVr r
V V VVr
π π π ππ π
π π π π
+ − + −= + −
+ +
+ − + −=
+
In one step:In two steps:
The discounted expected future value of the derivative with an expected value that is computed with risk-neutral probabilities rather than true probabilities.
Note: π can vary along the nodes of the three diagram.
Pricing Derivatives and Options: Multiperiod Binomial
Styring og Fondsmegling Dr.oecon Per B Solibakke
21
Numerical Techniques:
The techniques for valuing virtually all the new financial instruments developed by Wall Street firms consists of Numerical Methods; that is, no algebraic formula is used to compute the value of the derivative as a function of the value of the underlying security. Instead, a computer is fed a number corresponding to the price of the underlying security along with some important parameter values. The computer derives the numerical value of the derivative + sometimes, the number of shares of the underlying share in the tracking portfolio.
Simulation:
Generate random numbers to generate outcomes and then averages the outcomes of some variable to obtain values (together with the standard deviation). Exclude mortgages and American put options.
Pricing Derivatives and Options: Valuation Techniques
Styring og Fondsmegling Dr.oecon Per B Solibakke
22
Binomial-Like Numerical Methods:
Simplification. First, the binomial method can be used to approximate many kinds of continuous distributions if the time periods are cut to extremely small intervals. One popular continuous distribution is the lognormal distribution. The natural logarithm of the return of a security is normally distributed when the price movements of the security are determined with by the lognormal distribution. Once the annualized standard deviation, σ, of the normal distribution is known, u and dare estimated as follows:
/ 1
(2.718281828)
/
T Nu e and du
whereT number of years to ExpirationN number of binomial periodse Exponential ConsTantThus
T N square root of the number of years per binomioal period
σ= =
===
=
Pricing Derivatives and Options: Valuation Techniques
Styring og Fondsmegling Dr.oecon Per B Solibakke
23
Pricing Derivatives and Options: Valuation Techniques
All derivative valuation procedures make use of a short term risk-free return. The most common used input for the risk free rate is LIBOR.
Styring og Fondsmegling Dr.oecon Per B Solibakke
24
Pricing Derivatives and Options
Summary and ConclusionsThe price movements of a derivative are perfectly correlated over short time intervals with the price movements of the underlying asset on which it is based.
Hence, a portfolio of the underlying asset and a riskless security can be formed that perfectly tracks the future cash flows of the derivative. To prevent arbitrage, the tracking portfolio and the derivative must have the same value.
Styring og Fondsmegling Dr.oecon Per B Solibakke
25
Pricing Derivatives and Options: Put-Call Parity
Profiles: Buy a call and sell a put
Styring og Fondsmegling Dr.oecon Per B Solibakke
26
Pricing Derivatives and Options: Put-Call Parity
Styring og Fondsmegling Dr.oecon Per B Solibakke
27
Pricing Derivatives and Options: Corporate Securities
It is possible to view equity as a call option on the assets of the firm and to view risky corporate debt as riskless debt worth PV(K) plus a short position in a put option on the asset of the firm(-p0) with a strike price of K.
The call option characteristic of equity arises because of the limited liability of corporate equity holders.
E0 = Max(Vo – Debt(K), 0)K V0
1. Equity
2. Debt
K S0
The put option characteristic of debt arises because of the put-call formula.
Debt are assets less a call option:D0 = S0 – c0
The put-call: c0-p0=S0 – PV(Debt), substitutionD0 = PV(Debt(K)) – p0
Styring og Fondsmegling Dr.oecon Per B Solibakke
28
Pricing Derivatives and Options: Corporate Securities
Hence, any characteristics of the assets of the firm that affect option values will alter the values of debt and equity (for example the variance of the asset return).
The options result implies that the more debt a firm has, the less in the money is the implicit option in equity. Thus, knowing how option risk is affected by the degree to which options is in or out of money may shed light on how the mix of debt and equity affects the risk of the firm’s debt and equity securities.
Finally, because stock is an option on the assets of the firm, a call option on the stock of a firm is really an option on an option, or a compound option (Geske,79).
Styring og Fondsmegling Dr.oecon Per B Solibakke
29
Assume that the one-period risk-free rate is constant, and that the ratio of price in the next period to price in this period is always u or d.
1
11
f
f
r dand
u du r
u d
π
π
+ −=
−− −
− =−
The hypothetical probabilities that would exist in a risk-neutral world must make the expected return on the stock equal the risk free rate. The risk neutral probabilities satisfy: pu +(1-p)d=1+rf , giving the relationship
Pricing Derivatives and Options: Binomial Valuation
Styring og Fondsmegling Dr.oecon Per B Solibakke
30
The proper no-arbitrage call value c0, as a function of the stock price S0 becomes:
Pricing Derivatives and Options: Binomial Valuation
[ ] [ ]0 00
0 00
max ,0 (1 ) max ,01
:1 ! (1 ) max 0,
(1 ) !( 1)!
f
Nj N j j N j
Njf
uS K dS Kc
rand the GENERAL binomial formula
Nc u d S Kr j N
π π
π π π− −
=
− + − −=
+
= − − + −∑
Ex. Find the value of a three month at-the-money call option on DNB, trading at 32 kroner. Assume rf =0, u=2, d=0.5 and the number of years 3.
1 0 0,5 0,5 1 1 2, 1 12 0,5 1,5 3 3 3
andπ π+ −= = = − = − =
−
2 2 2 31 1 2 1 2 2(256 32) 3 (64 32) 3 (0) (0) 15, 413 3 3 3 3 3
− + − + + =
Since the discount rate is 0, 15,41 is the value of the call option.
Styring og Fondsmegling Dr.oecon Per B Solibakke
31
Pricing Derivatives and Options: Binomial Valuation
32
64
16
128
32
8
256
224
64
32
16
0
0
4
Styring og Fondsmegling Dr.oecon Per B Solibakke
32
Pricing Derivatives and Options: Black&Scholes Valuation
Black-Scholes FormulaIf a stock that pays no dividends before expiration of an option has return that is
lognormally distributed, can be continuously traded in frictionless market, and has a constant variance, then, for a constant risk-free rate, the value of a European call option on that stock with a strike price of K and T years to expiration is given by
0 0 1 1
01
( ) ( )
ln( / ( ))2
r Tc S N d Ke N d Twhere
S PV K TdT
σ
σσ
− ⋅= − −
= +
The Greek letter σ is the annualized standard deviation of the natural logarithm of the stock return, ln() represents the natural logarithm, and N(z) is the probability that a normally distributed variable with a mean of zero and variance of 1 is less than z.
Styring og Fondsmegling Dr.oecon Per B Solibakke
33
Pricing Derivatives and Options: Black&Scholes Valuation
Ex. Black-Scholes FormulaThe non-dividend paying stock Ekornes has a current price of 150 kroner and a
volatility of 20 percent per year. What is the price of a 3 month European option with strike price of 150 when the risk-free rate is 5%:
30,0512
1ln(30 / 28 ) 0,2 3 /12 0,0815 0,2 0,5 0,865
2 0,2 0,5 20,2 3 /12ed− ⋅
⋅= + = + =
⋅
0
0
0
30 (0,74) 27.652 (0,74 0,2 0,5)30 (0,74) 27.652 (0,74 0,2 0,5)23.1105 20, 4326 2,678
c N Nc N Nc
= ⋅ − ⋅ − ⋅= ⋅ − ⋅ − ⋅= − =
Styring og Fondsmegling Dr.oecon Per B Solibakke
34
Pricing Derivatives and Options: Estimating Volatility
Using Historical DataNote that B&S is based on the volatility of instantaneous volatility:Procedure for calculation of historical volatility:
1. Obtain historical returns for the stock the option is written on.2. Covert returns to gross returns (1+return in decimal form).3. Take the natural logarithm of the decimal version of the gross return.4. Compute the unbiased sample variance of the logged return series and
annualize it by multiplying it by the square root of ratio of 365 to the number of days in the return interval.
Styring og Fondsmegling Dr.oecon Per B Solibakke
35
Pricing Derivatives and Options: Estimating Volatility
The implied volatility approachLook at other options on the same security. If market values of the options exist,
there is a unique implied volatility that maked the B&S model consistent with the market price of a particular option.
Method 2: Newton Raphson technique.
S 100 PriceK 125 Strike Markør 0r 12 % Interest rate Initial 1.24304tau 0.25 Maturitysigma 0.471234 Volatility Derivative 25.5193d(1) -0.70193 N'(d1) 0.510386d(2) -0.93754
C (call price) 3 Mål kjøp 3
The EXCEL-book: NewtRaphImp-vol.xls shows a particular form for implementation.
Styring og Fondsmegling Dr.oecon Per B Solibakke
36
Pricing Derivatives and Options: B&S Greeks
The Greeks of B&S Formula
Delta: The sensitivity to Stock price Changes (Gamma measures Delta changes)
Vega: The sensitivity to Volatility Changes
Theta: The sensitivity to Expiration Changes
Rho: The sensitivity to Risk-free Interest rate Changes
Styring og Fondsmegling Dr.oecon Per B Solibakke
37
Pricing Derivatives and Options: Complex Assets
The Forward Price Version of the Black-Scholes Model0 0 1 1
01
( ) ( )
ln( / )2
r Tc e F N d KN d T
where
F K TdT
σ
σσ
− ⋅ = − −
= +
Forward Prices$.40 1,06) $0,39261,08
) $102 1,06 $4 1,06 $4 $100,0017) ($800 $20) 1,06 $826.80) ($18 $1 $1) (1,06 $19,08
USi US
iiiiiiv
⋅=
⋅ − ⋅ − =− ⋅ =
+ − ⋅ =
1 10 1 0
1 10 1 0
10
0,3926 ( ) $0,5 ( 0,25) ln(0,3926 / 0,5) 0,125 0,84; $0,011,06 0, 25
$100,0017 ( ) $100 ( 0, 25) ln(100,0017 /100) 0,125 0,13; $9,391,06 0, 25
$826,80 ( ) $850 ()
N d N di c where d c
N d N dii c where d c
N d N diii c
⋅ − ⋅ −= = + = − =
⋅ − ⋅ −= = + = =
⋅ − ⋅= 1
1 0
1 10 1 0
0, 25) ln(826,80 / 850 0,125 0,13; $68,221,06 0, 25
$19,08 ( ) $20 ( 0, 25) ln($19,08 / 20) 0,125 0,06; $1, 431,06 0,25
where d c
N d N div c where d c
−= + = =
⋅ − ⋅ −= = + = − =
Call Option Prices
Styring og Fondsmegling Dr.oecon Per B Solibakke
38
Pricing Derivatives and Options
Summary and ConclusionsDespite a few biases in the Black & Scholes option pricing formula, it appears
that the formula work reasonably well when properly implemented.
Black-Scholes Option-Pricing Formula
S 25 Current stock priceX 25 Exercise pricer 6.00 % Risk-free rate of interestT 0.5 Time to maturity of option (in years)Sigma 30 % Stock volatility
d1 0.2475 <-- (LN(S/X)+(r+0.5*sigma^2)*T)/(sigma*SQRT(T))d2 0.0354 <-- d1-sigma*SQRT(T)
N(d1) 0.5977 <-- Uses formula NormSDist(d1)N(d2) 0.5141 <-- Uses formula NormSDist(d2)
Call price 2.47 <-- S*N(d1)-X*exp(-r*T)*N(d2)Put price 1.73 <-- call price - S + X*Exp(-r*T): by Put-Call parity
1.73 <-- X*exp(-r*T)*N(-d2) - S*N(-d1): direct formula
The Excel-book: Black&ScholesImpl.xls shows various implementations.