1. Chapter 14 Exotic Options: I

2. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-2 Exotic Options Nonstandard options. Exotic options solve particular business problems that an ordinary option do not. They are often constructed by tweaking ordinary options in minor ways.

3. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-3 Exotic Options (cont’d) Relevant questions: –What is the rationale for the use of the exotic option? –How does the exotic payoff compare to that of a standard option? –Can the exotic option be approximated by a portfolio of other options? –Is the exotic option cheap or expensive relative to standard options? –How easily can the exotic option be hedged?

4. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-4 Asian Options The payoff of an Asian option is based on the average price over some period of time. An Asian options is an example of a path- dependent option. 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.

5. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-5 Asian Options (cont’d) Example –The exercise of the conversion option in convertible bonds is based on the average stock price over a 20-day period at the end of the bond’s life. Asian options are less valuable than otherwise equivalent ordinary options, since the averaged price of the underlying asset is less volatile than the asset price itself, and an option on a lower volatility asset is worth less.

6. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-6 Asian Options (cont’d) There are eight (2 3 ) basic kinds of Asian options: –Put or call. –Geometric or arithmetic average. –Average asset price is used in place of underlying. price or the strike price. Arithmetic versus geometric average: –Suppose we record the stock price every h periods from t = 0 to t = T. –Arithmetic average: Geometric average:

7. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-7 Asian Options (cont’d) 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, S T – G(T)]. –Geometric average strike put = max [0, G(T) – S T ].

8. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-8 Asian Options (cont’d) All four options above could also be computed using arithmetic average instead of geometric average. Relatively simple pricing formulas exist for pricing European options on the geometric average but not for arithmetic average options.

9. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-9 Asian Options (cont’d) Comparing Asian options

10. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-10 Asian Options (cont’d) XYZ’s hedging problem –XYZ has monthly revenue of 100m, and costs in dollars. –x t is the dollar price of a euro at time t. –In one year, the converted amount in dollars is –If we ignore interest, what we are trying to hedge is

11. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-11 Asian Options (cont’d) A solution for XYZ –An Asian put option that puts a floor K, on the average exchange rate received. The per euro payoff of this option would be

12. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-12 Asian Options (cont’d) Alternative solutions for XYZ’s hedging problem

13. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-13 Barrier Options The payoff depends on whether over the option life the underlying price reaches a specified level, called the barrier. –Path-dependent. –Since barrier puts and calls never pay more than standard puts and calls, they are no more expensive than standard puts and calls. –Widely used in practice.

14. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-14 Barrier Options (cont’d) Barrier puts and calls –Knock-out options: go out of existence (are “knocked-out”) down-and-out: if the asset price falls to reach the barrier. up-and-out: if the asset price rises to reach the barrier. –Knock-in options: come into existence (are “knocked-in”) down-and-in: if the asset price falls to reach the barrier. up-and-in: if the asset price rises to reach the barrier. –The important parity relation for barrier options is –Rebate options: make a fixed payment if the asset price reaches the barrier down rebates: if the asset price falls to reach the barrier. up rebates: if the asset price rises to reach the barrier.

15. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-15 Barrier Options (cont’d)

16. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-16 Barrier Options (cont’d)

17. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-17 Barrier Options (cont’d) See Section “Currency Hedging” for the use of the barrier options.

18. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-18 Compound Options An option to buy an option. Compound option has two strikes and two expirations, one each for underlying option and for the compound option. Suppose that the current time is t 0 and that we have a compound option which at time t 1 will give us the right to pay x to buy a European call option with strike K. This underlying call will expire at time T > t 1.

19. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-19 Compound Options (cont’d)

20. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-20 Compound Options (cont’d) At time t 1, the value of the compound option is The compound option is exercised for S t 1 > S * where S * satisfies

21. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-21 Compound Options (cont’d) In order for the compound call to be ultimately be valuable, there are two events that must take place: –At time t 1, S t 1 > S *. –At time T, S T > K.

22. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-22 Compound Options (cont’d) Because these two events must occur, the formula for a compound call contains a bivariate cumulative normal distribution, as opposed to the univariate distribution in the Black-Scholes formula (Details are not required, you may see Appendix A for your own interest).

23. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-23 Compound Options (cont’d) Four compound options –An option to buy a call (CallOnCall). –An option to sell a call (PutOnCall). –An option to buy a put (CallOnPut). –An option to sell a put (PutOnPut). Compound option parity

24. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-24 Gap Options A gap call option pays S – K 1 when S > K 2 The value of a gap call is where and

25. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-25 Gap Options (cont’d)

26. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-26 Gap Options (cont’d)

27. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-27 Gap Options (cont’d)

28. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-28 Exchange Options Pays off only if the underlying asset outperforms some other asset (benchmark)  out-performance option. Payoff: Max(0, S T  K T ) where K T is the price of a risky asset. The value of a European exchange call is where

29. Chapter 23 Exotic Options: II

30. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-30 Terminology

31. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-31 Definitions

32. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-32 All-or-Nothing Options Simple all-or-nothing options pay the holder a discrete amount of cash or a share if some particular event occurs. Cash-or-nothing –Call: pays $1 if S T >K and zero otherwise –Put: pays $1 if S T <K and zero otherwise Asset-or-nothing –Call: pays S (one unit share) if S T >K and zero otherwise –Put: pays S (one unit share) if S T <K and zero otherwise

33. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-33 All-or-Nothing Options (cont’d) + 1 asset-or-nothing call option with strike price K – K cash-or-nothing call option with strike price K = 1 ordinary call option with strike price K

34. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-34 All-or-Nothing Options (cont’d) Similarly, a put option can be created by buying K cash-or-nothing puts, and buying 1 asset-or- nothing put A gap option that pays S – K 1 if S > K 2 can be created by buying an asset call and selling K 1 cash calls, both with the strike price K 2

35. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-35 All-or-Nothing Options (cont’d) All-or-nothing options are easy to price but hard to hedge. Fig. 1 shows that a small swing in the stock price can determine whether the option is in- or out-of- the money, with the payoff changing discretely. Fig. 2 shows that hedging is straightforward and delta is well behaved when 3 months to expiration. However, with 2 minutes to expiration, the cash call delta at $40 is 15. For the at-the-money option, delta and gamma approach infinity at expiration because an arbitrarily small change in the price can result in a $1 change in the option’s value.

36. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-36 All-or-Nothing Options (cont’d)

37. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-37 All-or-Nothing Options (cont’d)