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Topic 9 Options on Futures, Currency Derivatives and
Exotic Options
Pricing Options on Futures
Triangular Arbitrage and Interest Rate Parity
Pricing Currency Derivatives
Some Popular Exotic Options
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Introduction
Pricing more advanced futures and options are not necessarily hard to understand
Based on the same futures pricing model
Based on the same Black-Scholes model
If you know standard futures and option pricing models, you will automatically understand other exotic variations.
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Characteristics of Options on Futures
Options where exercise establishes either a long or
short position in a futures contract at the exercise
price
Exercise of long (short) call establishes a long (short)
futures.
Exercise of a long (short) put establishes a short (long)
futures.
Also called commodity options or futures options.
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Options on Futures
For example, a TCC ltd. futures call option has a
current price of $2.75
$1,000 multiplier / contract size
Therefore the cost is $2,750.00
If exercised when futures = 121, holder establishes long
futures position at 115, which is immediately marked to
market at 121 for a $6,000 credit to margin account {that is 121-
115 =6}.
Note: expiration can be the same month as futures or earlier,
depending on the contract.
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Pricing Options on Futures
The Intrinsic Value of an American Option on
Futures
Minimum value of American call on futures
Ca(f0,T,X) Max(0,f0 - X)
Minimum value of American put on futures
Pa(f0,T,X) Max(0,X - f0)
Difference between option price and intrinsic value is
time value.
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Pricing Options on Futures
The Lower Bound of a European Option on
Futures
Ce(f0,T,X) Max[0,(f0 - X)(1+r)-T]
How did we get this?
Note: f0 = S0(1+r)T
Pe(f0,T,X) Max[0,(X - f0)(1+r)-T]
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Pricing Options on Futures
Put-Call Parity of Options on Futures
Portfolio Current Value
A Long Futures 0
Long Put 0
B Long Call 0
Bonds
Payoffs from Portfolio
0ffT
0fX
0ffT
0ffT
TfX
XfT XfT
0fX 0ffT
),,( 0 XTfPe
),,( 0 XTfCe
TrfX )1)(( 0 0fX 0fX XfT
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Pricing Options on Futures
Put-Call Parity of Options on Futures
Pe(f0,T,X) = Ce(f0,T,X) + (X - f0)(1+r)-T.
Compare to put-call parity for options on spot:
Pe(S0,T,X) = Ce(S0,T,X) - S0 + X(1+r)-T.
If options on spot and options on futures expire at same
time, their values are equal;
implying f0 = S0(1+r)T.
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Pricing Options on Futures
Early Exercise of Call and Put Options on Futures
Deep in-the-money call may be exercised early because
behaves almost identically to futures
exercise frees up funds tied up in option but requires no funds to
establish futures
minimum value of European futures call is less than value if it
could be exercised
Similar arguments hold for puts
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Pricing Options on Futures
Options on Futures Pricing Models
Black model for pricing European options on
futures
Tdd
T
T/2/X)ln(fd
where
)]XN(d)N(d[feC
12
2
01
210
Trc
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Pricing Options on Futures
Options on Futures Pricing Models
Note that with the same expiration for options on spot as
options on futures, this formula gives the same price.
Black’s model ONLY works when futures and options have the
same termination date.
For puts
c cr T r T
2 0 1P Xe [1 N(d )] f e [1 N(d )]
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The Foreign Currency Market and Currency Derivatives
The Nature of Exchange Rates
Definition: the rate at which the currency of one
country can be translated into the currency of
another country.
Can be viewed as the price in one currency of
purchasing another currency. Similar concept to
price of any other asset.
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The Foreign Currency Market and Currency Derivatives
Foreign Currency Spot and Forward
Markets
Called the Interbank Market.
Notional value of currency forwards currently
exceed $10 trillion in any particular year.
Most heavily traded currencies are US dollar,
Euro, Yen, Swiss Franc and Pound Sterling.
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Pricing Foreign Currency Derivatives
Cross-Rate Relationships must hold
Else, arbitrage opportunities exist
Example:
$/£ = price of British pound in Australian dollars
¥ /$ = price of Australian dollars in yen
£/ ¥ = price of yen in pounds
Therefore: ($/ ¥)(¥/£)(£/$) = 1
If not, then (triangular) arbitrage can be executed.
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Pricing Foreign Currency Derivatives
Cross-Rate Relationships
Lets say current spot rates are:
¥268/£
£0.3482/$
Then, using:
The price of Yen in Australian dollars must equal:
$0.0107/ ¥ as (268)(0.3482)(0.0107)=1
1$
£
£
¥
¥
$
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Pricing Foreign Currency Derivatives
Cross-Rate Relationships
What happens if $0.0134/ ¥?
Yen is overvalued relative to the dollar
Take 1 pound and convert to ¥268.
Take ¥268 and convert to (268)(0.0134)=$3.591
Then convert back:
Take $3.591 and convert to (3.591)(0.3482)=£1.25
This is a 25% return, with no risk!
Such arbitrage no longer exists
Go to the nearest FX changer in a Bank – check to see if money can be made from the quoted cross-rates!
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Pricing Foreign Currency Derivatives
Interest Rate Parity
The relationship between spot and forward / futures prices of a
currency. Same as cost of carry model in other forward and
futures markets.
Proves that one cannot borrow and convert a domestic currency to
another foreign currency, sell a futures, earn the foreign risk-free
rate and convert back risklessly, earning a rate higher than the
domestic rate.
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Pricing Foreign Currency Derivatives
Interest Rate Parity – an example:
S0 = spot rate in domestic currency per foreign currency.
Let’s say Yen per Singapore Dollar.
Foreign interest rate is r.
In this case it is the Singapore risk-free rate.
Holding period is T.
Domestic rate is r.
This would be the risk-free rate in Japan.
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Pricing Foreign Currency Derivatives
Interest Rate Parity
Take S0(1+ r)-T Yen and buy (1+ r)-T Singapore Dollars.
Place your Singapore dollars in the bank earning rreturn.
Simultaneously, you sell one forward contract to deliver 1 Singapore
Dollar at T at price F0.
At time T:
Your Singapore dollars will be worth 1 dollar.
Your (1+ r)-T will have grown by (1+ r)-T x (1+ r)T = 1.
Your forward contract obliges you to deliver the Singapore dollar and receive
F0 Yen
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Pricing Foreign Currency Derivatives
Interest Rate Parity
What has this led to?
You invested S0(1+ r)-T Yen and received F0 Yen.
You earn the Japanese risk free rate as the above transaction is riskless
That is:
F0 = S0(1+ r)-T(1 + r)T
This is called interest rate parity.
Sometimes written as
F0 = S0(1 + r)T/(1 + r)T
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Pricing Foreign Currency Derivatives
Difference between domestic and foreign rate is
analogous to difference between risk-free rate
and dividend yield on stock index futures.
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Pricing Foreign Currency Derivatives
Currency Options
Minimum value of American foreign currency call is
Ca(S0,T,X) Max(0,S0 - X).
Minimum value of American foreign currency put is
Pa(S0,T,X) Max(0,X - S0).
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Pricing Foreign Currency Derivatives
The Lower Bound of European Foreign
Currency Options
Calls:
Ce(S0,T,X) Max[0,S0(1+ r)-T - X(1+r)-T].
This must hold for American options too.
Note similarity to case of call on stock with dividend yield.
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Pricing Foreign Currency Derivatives
Puts:
Pe(S0,T,X) Max[0,X(1+r)-T - S0(1+ r)-T].
This must hold for American options too.
Note similarity to case of put on stock with dividend yield.
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Pricing Foreign Currency Derivatives
Put-Call Parity
S0(1+ r)-T + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T
Note similarity to put-call parity for options on stocks.
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Pricing Foreign Currency Derivatives
The Garman-Kohlhagen Foreign Currency Option Pricing Model
Identical to Black-Scholes model for options on stocks with dividend yields using r for dividend yield.
c c
c
- T -r T
0 1 2
- T 2
0 c1
2 1
C=S e N(d ) Xe N(d )
where
ln(S e / X)+[r ( /2)]Td
Td d T
r
r
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Some Exotic Options
Digital Options
Digital options, sometimes called binary options,
are of two types.
Asset-or-nothing options pay the holder the asset if the
option expires in the money and nothing otherwise.
Cash-or-nothing options pay the holder a fixed
amount of cash if the option expires in the money and
nothing otherwise.
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This combination is equivalent to an ordinary European call.
Long asset-or-nothing option 0
Short cash-or-nothing option 0 -X
0
Payoffs from Portfolio
XST XST
TS
XST
Some Exotic Options
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Recall that the Black-Scholes price is
The first term is the price of the asset-or-nothing option.
The second term, ignoring the minus, is the price of a cash-
or-nothing option that pays off X if it expires in-the-money.
The prices of an asset-or-nothing option and cash-or-
nothing option are:
c
aon 0 1
r T
con 2
O S N(d )
O Xe N(d )
0 1 2( ) ( )cr TS N d Xe N d
Some Exotic Options
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Chooser Options
Enable the investor to decide at a specific time after
purchasing the option but before expiration that the
option will be a call or a put.
Assume that decision must be made at time t < T
The chooser option is identical to
an ordinary call expiring at T with exercise price X plus
an ordinary put expiring at t with exercise price X(1+r)-(T-t)
Some Exotic Options
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Some Exotic Options
Chooser Options
Example: Comparison with a Straddle
Coca Cola Amatil U.S. chooser option in which choice must be made in 25 days.
Call/put expires in 47 days. S0 = 9.32, X = 9,
= .76, rc = .0512.
T = 47/365 = .1288, t = 25/365 = .0685
so T - t = .1288 - .0685 = .0603.
Exercise price on put used to price the chooser is 9(1.0512)-.0603 = 8.9729.
Using Black-Scholes model, put is worth $0.5457. Thus the call is worth $1.1914, the put is equal to $0.5457, for a total of $1.7371.
A straddle would cost $1.19 + $0.81 = $2.
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Payoff determined by the sequence of prices
followed by the asset and not just by the price
of the asset at expiration.
Priced using the binomial model
In practice the binomial model is difficult to use
for path-dependent options.
Path-Dependent Options
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Asian options:
Average price options:
final payoff is determined by the average price of the
asset during the option’s life.
Average strike options
The average price substitutes for the exercise price at
expiration.
When are they used?:
Useful for hedging or speculating when the average is
acceptable.
Path-Dependent Options
X
T
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Path-Dependent Options
Lookback Options
Also called a no-regrets option, it permits
purchase of the asset at its lowest price during
the option’s life or sale of the asset at its highest
price during the option’s life.
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Lookback options
Four different types: lookback call:
Min price for X
lookback put:
Max price for X
fixed-strike lookback call:
Max price for ST
fixed-strike lookback put:
Min price for ST
Path-Dependent Options
Xmin
Xmax
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Path-Dependent Options
Barrier Options
Terminate /Activate if the asset price hits a certain level, called the
barrier.
The former is called a knock-out option (or simply out-options)
The latter is called a knock-in options (or simply in-options).
If the barrier is above the current price, it is called an up-option.
If the barrier is below the current price, it is called a down-option.
Will Barrier Options be more or less expensive than Standard
European Options?
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Path-Dependent Options
Other Exotic Options:
compound and installment options
multi-asset options, exchange options, min-max
options (rainbow options), alternative options,
outperformance options
shout, cliquet and lock-in options
contingent premium, pay-later and deferred strike
options
forward-start and tandem options