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Chapter 14Chapter 14Exotic Options: IExotic Options: I

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Exotic (nonstandard) options

Exotic options solve particular business problems that an ordinary option cannot

They are constructed by tweaking ordinary options in minor ways

Relevant questions: How does the exotic payoff compare to ordinary option

payoff? Can the exotic option be approximated by a portfolio of

other options? Is the exotic option cheap or expensive relative to

standard options? What is the rationale for the use of the exotic option? How easily can the exotic option be hedged?

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Asian options

The payoff of an Asian option is based on the average price over some period of time path-dependent

Situations when Asian options are useful: When a business cares about the average exchange rate

over time When a single price at a point in time might be subject to

manipulation When price swings are frequent due to thin markets

Example: The exercise of the conversion option in convertible

bonds is based on the stock price over a 20-day period at the end of the bond’s life

Asian options are less valuable than otherwise identical ordinary options

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Asian options (cont.)

There are eight (23) basic kinds of Asian options: Put or call Geometric or arithmetic average Average asset price is used in place of underlying

price or strike

Arithmetic versus geometric average: Suppose we record the stock price every h periods

from t=0 to t=T Arithmetic average: Geometric average:

A TN

Sihi

N( )

1

1G T S S Sh h Nh

N( ) ( ) / 21

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Asian options (cont.)

Average used as the asset price: Average price option Geometric average price call = max [0, G(T) – K] Geometric average price put = max [0, K – G(T)]

Average used as the strike price: Average strike option Geometric average strike call = max [0, ST – G(T)] Geometric average strike put = max [0, G(T) – ST]

All four options above could also be computed using arithmetic average instead of geometric average

Simple pricing formulas exist for geometric average options but not for arithmetic average options

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Asian options (cont.)

Example of use of average price option: Receiving funds in a foreign currency every month

for a year, with translation of the cash flow into US Dollars taking place every month. At the end of the year, the firm is concerned with the average exchange rate obtained, i.e. does not want to have received funds at a rate below a specific exchange rate.

Example of use of average strike option: Acquiring shares of a security S gradually by

purchasing a fixed amount every month. Wanting to unload the full position at the end of the year, the firm needs an option that yields a payoff proportional to the difference between average security value and terminal value.

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Asian options (cont.)

Comparing Asian options:

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Asian options (cont.)

For Asian options that average the stock price, the averaging reduces the volatility of the value of the stock entered in the payoff.

Hence the price of the option is less than an otherwise similar traditional option.

For Asian options that average the strike price, more averaging reduces the correlation between the terminal value of the stock and the strike price (computed as average of past stock prices), hence increasing the value of the option.

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Asian options (cont.)

XYZ’s hedging problem XYZ has monthly revenue of 100m, and costs in

dollars x is the dollar price of a euro In one year, the converted amount in dollars is

Ignoring interest what needs to be hedged is

A solution for XYZ An Asian put option that puts a floor K, on the

average rate received

100 12 12

1

12m

x ei

r i

i

( ) /

x =x

ii

ii12121

121

12

max K xii

01

12 1

12

,

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Asian options (cont.)

Alternative solutions for XYZ’s hedging problem

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Barrier options

The payoff depends on whether over the option life the underlying price reaches the barrier path dependent

Barrier puts and calls: Knock-out options: go out of existence if the asset price

down-and-out: has to fall to reach the barrier up-and-out: has to rise to reach the barrier

Knock-in options: come into existence if the asset price down-and-in: has to fall to reach the barrier up-and-in: has to rise to reach the barrier

Rebate options: make a fixed payment if the asset price down rebates: has to fall to reach the barrier up rebates: has to rise to reach the barrier

Barrier options are less valuable than otherwise identical ordinary options

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Barrier options (cont.)

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Barrier options (cont.)

“Knock-In” option + “Knock-out” option = Ordinary option

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Barrier options (cont.)

Overall use of Barrier options:

They come into or go out of existence depending on whether a barrier has been reached, hence they are cheaper than standard options. One could prefer to pay less and have an option that is alive only “if needed”, or have the option disappear if not needed after all.

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Compound options

An option to buy an option

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Gap options

A gap call option pays S – K1 when S > K2

The value of a gap call is

where

and

C S,K ,K , ,r,T, Se N d K e N d- T -rT( ) = ( ) ( )1 2 1 1 2

dln S / K r T

T1

221

2 ( ) ( )

d d T2 1

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Gap Call, paying S-90 when S>100

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Gap Put, paying 90-S when S<100

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Gap options (cont.)

Advantage of Gap options: the payout can be anything, and is not tied to the exercise price the way traditional options are.

From an insurance point of view, one can see this as the assurance that one will be 100% covered if the loss exceed a certain amount. Or can create a cheaper Gap option by choosing a higher “deductible”.

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Exchange options

Pays off only if the underlying asset outperforms some other asset (benchmark) outperformance option

The value of a European exchange call is

where

,

and

C S,K, ,r,T, Se N d Ke N d-

ST -

KT

( ) = ( ) ( )

1 2

dln Se / Ke t

T

St

Kt

1

21

2

( )

d d T2 1

S K S K2 2 2