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

1

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.

2

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.

3

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 121115 =6}.

Note: expiration can be the same month as futures or earlier,

depending on the contract.

4

Pricing Options on Futures

The Intrinsic Value of an American Option on

Futures

Minimum value of American call on futures

Minimum value of American put on futures

Ca(f0,T,X) Max(0,f0 - X)

Pa(f0,T,X) Max(0,X - f0)

Difference between option price and intrinsic value is

time value.

5

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]

6

Pricing Options on Futures

Put-Call Parity of Options on Futures

Payoffs from Portfolio

Portfolio

A

Current Value

Long Futures

Long Put

0

Pe ( f 0 , T , X )

fT X

fT X

fT f 0

X fT

fT f 0

X f0

B

Long Call

Bonds

Ce ( f 0 , T , X )

( X f 0 )(1 r ) T

0

X f0

X f0

0

fT f 0

fT X

X f0

fT f 0

7

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.

8

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

9

Pricing Options on Futures

Options on Futures Pricing Models

Black model for pricing European options on

futures

C e rc T [f 0 N(d1 ) XN(d2 )]

where

ln(f 0 /X) 2 /2 T

d1

T

d 2 d1

T

10

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

P XercT [1 N(d2 )] f0ercT [1 N(d1)]

11

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.

12

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.

13

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.

14

Pricing Foreign Currency

Derivatives

Cross-Rate Relationships

Lets say current spot rates are:

¥268/£

£0.3482/$

Then, using:

$ ¥ £

1

¥ £ $

The price of Yen in Australian dollars must equal:

$0.0107/ ¥ as (268)(0.3482)(0.0107)=1

15

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!

16

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.

17

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.

18

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

19

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

20

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.

21

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).

22

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.

23

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.

24

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.

25

Pricing Foreign Currency

Derivatives

The Garman-Kohlhagen Foreign Currency

Option Pricing Model

-r T

-r T

C=S0e c N(d1 ) Xe c N(d2 )

where

d1

ln(S0e

- rc T

d2 d1 T

/ X)+[rc ( 2 / 2)]T

T

Identical to Black-Scholes model for options on

stocks with dividend yields using r for dividend

yield.

26

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.

27

Some Exotic Options

Payoffs from Portfolio

Long asset-or-nothing option

Short cash-or-nothing option

ST X

ST X

0

0

ST

-X

0

ST X

This combination is equivalent to an ordinary European call.

28

Some Exotic Options

Recall that the Black-Scholes price is

S 0N (d 1 ) Xe rcT N (d 2 )

The first term is the price of the asset-or-nothing option.

The second term, ignoring the minus, is the price of a cashor-nothing option that pays off X if it expires in-the-money.

The prices of an asset-or-nothing option and cash-ornothing option are:

Oaon S0N(d1 )

Ocon Xe

rc T

N(d2 )

29

Some Exotic Options

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)

30

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.

31

Path-Dependent Options

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.

32

Path-Dependent Options

Asian options:

Average price options:

final payoff is determined by the average price of the

asset during the option’s life.

Average strike options

T

X

The average price substitutes for the exercise price at

expiration.

When are they used?:

Useful for hedging or speculating when the average is

acceptable.

33

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.

34

Path-Dependent Options

Lookback options

Four different types:

lookback call:

Min price for X

lookback put:

Xmax

Xmin

Max price for X

fixed-strike lookback call:

Max price for ST

fixed-strike lookback put:

Min price for ST

35

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?

36

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

37

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

1

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.

2

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.

3

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 121115 =6}.

Note: expiration can be the same month as futures or earlier,

depending on the contract.

4

Pricing Options on Futures

The Intrinsic Value of an American Option on

Futures

Minimum value of American call on futures

Minimum value of American put on futures

Ca(f0,T,X) Max(0,f0 - X)

Pa(f0,T,X) Max(0,X - f0)

Difference between option price and intrinsic value is

time value.

5

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]

6

Pricing Options on Futures

Put-Call Parity of Options on Futures

Payoffs from Portfolio

Portfolio

A

Current Value

Long Futures

Long Put

0

Pe ( f 0 , T , X )

fT X

fT X

fT f 0

X fT

fT f 0

X f0

B

Long Call

Bonds

Ce ( f 0 , T , X )

( X f 0 )(1 r ) T

0

X f0

X f0

0

fT f 0

fT X

X f0

fT f 0

7

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.

8

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

9

Pricing Options on Futures

Options on Futures Pricing Models

Black model for pricing European options on

futures

C e rc T [f 0 N(d1 ) XN(d2 )]

where

ln(f 0 /X) 2 /2 T

d1

T

d 2 d1

T

10

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

P XercT [1 N(d2 )] f0ercT [1 N(d1)]

11

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.

12

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.

13

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.

14

Pricing Foreign Currency

Derivatives

Cross-Rate Relationships

Lets say current spot rates are:

¥268/£

£0.3482/$

Then, using:

$ ¥ £

1

¥ £ $

The price of Yen in Australian dollars must equal:

$0.0107/ ¥ as (268)(0.3482)(0.0107)=1

15

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!

16

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.

17

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.

18

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

19

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

20

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.

21

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).

22

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.

23

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.

24

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.

25

Pricing Foreign Currency

Derivatives

The Garman-Kohlhagen Foreign Currency

Option Pricing Model

-r T

-r T

C=S0e c N(d1 ) Xe c N(d2 )

where

d1

ln(S0e

- rc T

d2 d1 T

/ X)+[rc ( 2 / 2)]T

T

Identical to Black-Scholes model for options on

stocks with dividend yields using r for dividend

yield.

26

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.

27

Some Exotic Options

Payoffs from Portfolio

Long asset-or-nothing option

Short cash-or-nothing option

ST X

ST X

0

0

ST

-X

0

ST X

This combination is equivalent to an ordinary European call.

28

Some Exotic Options

Recall that the Black-Scholes price is

S 0N (d 1 ) Xe rcT N (d 2 )

The first term is the price of the asset-or-nothing option.

The second term, ignoring the minus, is the price of a cashor-nothing option that pays off X if it expires in-the-money.

The prices of an asset-or-nothing option and cash-ornothing option are:

Oaon S0N(d1 )

Ocon Xe

rc T

N(d2 )

29

Some Exotic Options

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)

30

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.

31

Path-Dependent Options

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.

32

Path-Dependent Options

Asian options:

Average price options:

final payoff is determined by the average price of the

asset during the option’s life.

Average strike options

T

X

The average price substitutes for the exercise price at

expiration.

When are they used?:

Useful for hedging or speculating when the average is

acceptable.

33

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.

34

Path-Dependent Options

Lookback options

Four different types:

lookback call:

Min price for X

lookback put:

Xmax

Xmin

Max price for X

fixed-strike lookback call:

Max price for ST

fixed-strike lookback put:

Min price for ST

35

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?

36

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

37