Orthogonal polynomials in pricing options by the PDE and Martingale approaches

This dissertation discusses two main applications of orthogonal polynomials to various option pricing problems. One application is for solving partial (integro) differential equations; the other is for estimating empirical implied volatility surfaces.; The first application, called pseudospectral methods, uses a linear combination of orthogonal polynomials as a trial solution form for the partial (integro) differential equation arising from various option pricing problems. This idea contrasts with finite difference methods, which is the standard numerical partial differential equation solving methods, where a piecewise linear function is assumed as a trial solution.; In chapter 2, pseudospectral methods are applied to standard option pricing under various model specifications. Particularly, a way of solving partial integro-differential equations by integrating quadrature method into pseudospectral methods is devised. In addition, forward equations are derived to deliver option prices with different strikes simultaneously.; In chapter 3, pseudospectral methods are applied to price several well-known exotic options to check whether the methods are robust in various situations.; Pseudospectral methods are found to outperform finite-difference methods when combined with the Broadie-Detemple technique.; Chapter 4 discusses the second application of orthogonal polynomials which are used to construct a trial surface for fitting volatilities implied from actual option data. Estimating the implied volatility surface and applying a semiparametric assumption of Black-Scholes formula yields an empirical risk-neutral density which is a key ingredient for pricing standard and path-independent exotic options. The implied volatility surface is also useful for inferring local volatility surface in order to calculate American and other path-dependent option prices.; The semiparametric polynomial method is different from the existing methods in at least one of the following aspects: It explicitly treats the maturity as a state variable; it has flexible functional form; and it assumes risk-neutral density conditional on information up to the current time. It is also discussed how to avoid the over-fitting problem which may arise when polynomials are used for interpolation.; From a simulation analysis and applications to actual option data, it is found that the semiparametric polynomial method performs well and is applicable in practice.