Derivatives: Markets, Valuation, and Risk Management

CHAPTER 9

Valuing Options Numerically

The last two chapters focused on valuing options analytically. Analytical valuation equations were possible because, in general, the options were European-style with only one exercise opportunity. For other types of options, the valuation problem is not so simple. With American-style options, for example, there are an infinite number of early exercise opportunities between now and the expiration date, and the decision to exercise early depends on a number of factors including all subsequent exercise opportunities. An analytical solution for the American-style option valuation problem (i.e., a valuation equation) has not been found.¹ The same is true for many Asian-style options (e.g., options written on an arithmetic average) and many European-style options with multiple sources of underlying price risk (e.g., spread options). In such cases, options must be valued numerically. Moreover, even in instances where analytical solutions to option contract values are possible (e.g., accrual options), numerical methods are often easier to apply.

The purpose of this chapter is to discuss numerical methods for valuing options. All of them are developed within the Black-Scholes/Merton (BSM) option valuation framework. The underlying asset's price is assumed to follow a geometric Brownian motion (i.e., to be log-normally distributed at any future instant in time), and a risk-free hedge between the option and its underlying asset(s) is possible. Three of ...