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The Fundamental Transform for Pricing Options

Abstract

In his excellent book, Option Valuation Under Stochastic Volatility: With Mathematica Code, Alan Lewis (2000) offers a powerful valuation approach that is applicable to a wide range of European options. His method requires that the fundamental transform, a generalization of the characteristic function that allows complex arguments, be available. It also requires the Fourier transform of the option payoff. Since payoff transforms are available for a wide set of options, however, Lewis' approach is readily applicable to path independent European options of various sorts. The advantage of this approach over other approaches that use Fourier transforms is that, once the fundamental transform of a given model is obtained, it can be used repeatedly to price European options for which the payoff transform is known. This greatly simplifies pricing, since payoff transforms are much easier to obtain than the Fourier transform of the option price itself.

In this chapter, we explain Lewis' fundamental transform approach for option valuation, and we also present his subsequent paper (Lewis, 2001) that uses Parseval's identity to obtain option prices. In that paper, simple variations in the contours of integrations give rise to different forms of the call price encountered in the literature. Finally, we present Lewis' (2000) volatility of ...