Characteristic Functions in Option Pricing

Abstract

We consider in this chapter a general Itô process for stock prices. In a generalized valuation framework for options, the distribution function of stock price is analytically unknown. To express (quasi-) closed-form exercise probabilities and valuation formula, characteristic functions of the underlying stock returns (logarithms) are proven to be not only a powerful and convenient tool to achieve analytical tractability, but also a large accommodation for different stochastic processes and factors. In first section, we derive two important characteristic functions under Delta measure and forward measure respectively, under which two exercise probabilities can be calculated. The pricing formula for European-style options may be expressed in a form of inverse Fourier transform. As a result, we obtain a generalized principle for valuing options under the risk-neutral measure via characteristic functions. There is a corresponding extension for FX options.