Actuarial Finance

12 Further topics in the binomial tree model

In Chapter 11, we presented the general binomial tree model. In that market model, we determined replicating portfolios and computed no-arbitrage prices for vanilla European options, from which we derived risk-neutral pricing formulas. As mentioned in Section 6.5, in practice, most traded options are American while many underlying assets pay a regular stream of income such as dividends.

The main objective of this chapter is to build upon the knowledge gained in Chapter 11 to extend the binomial tree to more realistic situations. The specific objectives are to:

• determine replicating portfolios;
• compute no-arbitrage prices;
• derive risk-neutral formulas,

for

• American-style options (especially American put options);
• options on stocks paying continuous dividends;
• currency options;
• futures options.

12.1 American options

Until now, in the binomial model, we have only considered European-style options. We will now analyze American-style options, which can be exercised at any time prior to maturity, and determine how an issuer can manage this additional risk when the dynamics of the underlying asset comes from a multi-period binomial tree.

12.1.1 American put options

Recall from Chapter 6 that it is never optimal to early-exercise an American call whereas it could be optimal to early-exercise an American put option, when both derivatives are written on a non-dividend-paying stock. These two conclusions being model-free, they hold in ...