1. Chapter 14
Exotic Options: I

2. 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?

3. 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

4. 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:

5. 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

6. 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.

7. Asian options (cont.)
Comparing Asian options:

8. 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.

9. Asian options (cont.) XYZ’s hedging problem A solution for XYZ
XYZ has monthly revenue of m, 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

10. Asian options (cont.)
Alternative solutions for XYZ’s hedging problem

11. 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

12. Barrier options (cont.)

13. Barrier options (cont.)
“Knock-In” option + “Knock-out” option = Ordinary option

14. 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.

15. Compound options
An option to buy an option

16. Gap options A gap call option pays S – K1 when S > K2
The value of a gap call is
where
and

17. Gap Call, paying S-90 when S>100

18. Gap Put, paying 90-S when S<100

19. 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”.

20. 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