Derivatives

CHAPTER 23 BOPM: Extensions

Aims

• To show how the BOPM is used to price American options – these are path-dependent options and subject to early exercise.
• To adapt the BOPM to price options on stocks that pay a continuous dividend, options on foreign exchange, and options on futures contracts.
• To use the BOPM to price options on stocks that pay dividends at discrete intervals.
• To demonstrate how the binomial approach can be speeded up using control variate techniques and trinomial trees.
• To show how stock price movements in the binomial tree are determined by the ‘real world’ volatility of stock returns.

23.1 AMERICAN OPTIONS

So far we have used the BOPM to price European options (which can only be exercised at maturity). American options can be exercised at any time, over the life of the option. The question arises as to when it is optimal to exercise an American option and how this affects the price of American options. The following results hold for American options:

• For a call option on a non-dividend paying stock, early exercise is never optimal.
• For a call option on a dividend paying stock, early exercise is sometimes optimal.
• For a put option on a stock (with or without dividends), early exercise is sometimes optimal.

As we shall see American options on stock indices, commodities, currencies and futures contracts can be priced using results for options on stocks that pay continuous dividends.

23.1.1 European Put

We price a two-period European put with using RNV. ...