Derivative Pricing, Numerical Methods

Numerical methods are needed for derivatives pricing in cases where analytic solutions are either unavailable or not easily computable. Examples of the former case include American-style options and most discretely observed path-dependent options. Examples of the latter type include the analytic formula for valuing continuously observed Asian options in [28], which is very hard to calculate for a wide range of parameter values often encountered in practice, and the closed form solution for the price of a discretely observed partial barrier option in [32], which requires high dimensional numerical integration. The subject of numerical methods in the area of derivatives valuation and hedging is very broad. A wide range of different types of contracts are available, and in many cases there are several candidate models for the stochastic evolution of the underlying state variables. Many subtle numerical issues can arise in various contexts. A complete description of these would be very lengthy, so here we will only give a sample of the issues involved. Our plan is to first present a detailed description of the different methods available in the context of the Black–Scholes–Merton [6, 43] model for simple European and American-style equity options. There are basically two types of numerical methods: techniques used to solve partial differential equations (PDEs) (these include the popular binomial and trinomial tree approaches), and Monte Carlo methods. We will then describe how the methods can be adapted to more general contexts, such as derivatives dependent on more than one underlying factor, path-dependent derivatives, and derivatives with discontinuous payoff functions. Familiarity with the basic theory of derivative pricing will generally be assumed, though some of the basic concepts will be reviewed when we discuss the binomial method. Readers seeking further information about these topics should consult texts such as [33, 56], which are also useful basic references for numerical methods. To set the stage, consider the case of a nondividend paying stock with a price S, which evolves according to geometric Brownian motion