Chapter Nineteen

1. Exotic Options 19 Chapter Nineteen

2. 19.1 Packages 19.2 Nonstandard American Options 19.3 Forward Start Options 19.4 Compound Options 19.5 Chooser Options 19.6 Barrier Options 19.7 Binary Options Chapter Outline

3. 19.8 Lookback Options 19.9 Shout Options 19.10 Asian Options 19.11 Options to Exchange One Asset for Another 19.12 Basket Options 19.13 Hedging Issues 19.14 Static Options Replication Chapter Outline (Continued)

4. A package is a portfolio consisting of standard European calls, puts, forwards, cash, and the underlying asset itself. Often a package is structured so that it has no cost initially. For example, a Boston option has a deferred payment feature. The payoff at expiry is given by: Max(ST – K – cerT, – cerT) Recall that in Chapter 9 we discussed a number of different types of Packages: Bull spreads Bear spreads Straddles Strangles Et cetera 19.1 Packages

5. With a standard American option: Exercise price is nonvariant Exercise can take place at any time up to and including expiry. In practice, the American Options traded in the OTC market sometimes have non standard features. e.g.: Early exercise might be restricted to certain dates. This is called a Bermudan option. Early exercise might be allowed during only part of the life of the options. For example an early “lock out” period. The strike price might change over the life of the option. Valuation with binomial option pricing. 19.2 Nonstandard American Options

6. 19.3 Forward Start Options • Options that will start at some point in the future. • Consider a forward start at-the-money European call. • The option begins at T1 and matures at time T2 • The asset price is S0 now and S1 at T1 • To value the option, start at T1 and calculate the Black-Scholes value c of an option with life T1 – T2 • The value of the forward start option is found by taking an expectation in a risk-neutral world:

7. 19.3 Forward Start Options • If the option pays a known dividend yield of q: • If the option pays no dividends, q = 0 and:

8. 19.4 Compound Options • Options on Options. • Four main types: • Call on a call • Put on a call • Call on a put • Put on a put • Consider a call on a call: • On the first exercise date, T1 the holder has the option to pay the first strike price, K1 , and receive the call option, which then entitles him to the right to buy the asset for the second strike price, K2 , on the second expiry, T2 .

9. 19.5 Chooser Options • Also known as an as you like it option. • Entitles the holder choose whether the option is a call or a put. • Payoff at expiry (T1 )= max(c, p) • Readily valued with put-call parity: • This shows that the chooser option is a package consisting of: • A call option with strike K and maturity T2 • e-q(T2 - T1) put options with strike price Ke-(r-q)(T2 - T1) and maturity T1

10. 19.6 Barrier Options • Payoff depends on whether the underlying asset’s price reaches a certain level during a certain period of time • A number of flavors trade OTC: • Knock-out options • Knock-in options • One type of knock-out option is a down and out call • Plain vanilla call that disappears if the asset price reaches a certain barrier level, H. • Barrier options often have quite different properties from regular options. • e.g. increases in volatility can decrease option value!

11. 19.7 Binary Options • Options with discontinuous payoffs. • For example, a cash-or-nothing call • Pays a fixed amount of cash if the asset is above the strike price and nothing if it is below.

12. 19.8 Lookback Options • The payoff depends on the minimum or maximum asset price reached during the life of the option. • For example a call option with a reset provision is sometimes called a no regret option. • Can be valued as a Black-Scholes straddle or with binomial valuation.

13. Valuation of a Lookback Option • Notice that the exercise price of the call will be the smallest value of the stock price depending upon the path followed by the stock price to get there.

14. 38.02 33.06 28.10 28.75 28.10 24.44 20.77 $25 28.10 24.44 20.77 21.25 20.77 18.06 15.35 Three Period Binomial Process: Lookback Call Option Prices

15. Three Period Binomial Process: Lookback Call Option Prices $38.02 $13.02 33.06 $28.10 $3.10 28.75 28.10 $3.66 24.44 $20.77 $0 $25 $28.10 $6.85 24.44 $20.77 $0 21.25 $20.77 $2.71 18.06 $15.35 $0

16. Three Period Binomial Process: Lookback Call Option Prices $38.02 $13.02 33.06 $28.10 9.25 $3.10 28.75 28.10 $3.66 24.44 $20.77 2.33 $0 $25 $28.10 $6.85 24.44 $20.77 4.35 $0 21.25 $20.77 $2.71 18.06 $15.35 1.72 $0

17. Three Period Binomial Process: Lookback Call Option Prices $38.02 $13.02 33.06 $28.10 $3.10 9.25 28.75 $28.10 $3.66 6.61 24.44 $20.77 $0 2.33 $25 $28.10 $6.85 5.25 24.44 $20.77 $0 4.35 21.25 $20.77 $2.71 3.31 18.06 $15.35 $0 1.72

18. 19.9 Shout Options • European option where the holder can “shout” to the writer at one time during its life. • At the end of the life of the option, the option holder receives either the usual payoff or the intrinsic value at the time of the shout, whichever is greater. • Valuation by binomial or trinomial tree.

19. 19.10 Asian Options • Payoff depends on the average price of the underlying asset during at least some part of the life of the option. • The payoff from an average price call: Max(0,Save – K) • Average price options are less expensive than regular options and are arguably more appropriate for meeting some of the needs of corporate treasurers. E.g. hedging currency from continuous operations.

20. 19.11 Options to Exchange One Asset for Another • An option to buy yen with British pounds is, from the point of view of a U.S. investor, an exchange option.

21. 19.12 Basket Options • The payoff is dependent on the value of a portfolio of assets. • The assets are usually individual stocks or stock indices or currencies. • Valuation with Monte Carlo simulation.

22. 19.13 Hedging Issues • Exotics can sometimes be easier to hedge than the corresponding plain vanilla option: • E.g. Asian options as time passes, our uncertainty regarding the average price decreases. As a result, as expiry approaches, delta goes to zero. • Barrier options, however, can be much more difficult to hedge. • The delta of these options is discontinuous at the border.

23. 19.14 Static Options Replication • In Chapter 14 we covered dynamicoptions replication • Requires the position in the hedging assets to be rebalanced frequently • Can be very expensive due to transactions costs • An alternative approach that can sometimes be used to hedge an exotic option is static options replication • Find a portfolio of options that approximately replicate the exotic option. • If two portfolios are worth the same on a certain boundary, they are worth the same at all interior points of the boundary.

24. Summary • Exotic options are options with rules governing the payoff that are more complicated than standard options. • This chapter discussed 13 different types of exotic options. • Some can be valued analytically, others require numeric methods.