Fourier-Based Valuation Methods in Mathematical Finance

Abstract

In this survey the current state of Fourier-based methods to compute prices in advanced financial models is discussed. The key point of the Fourier-based approach is the separation of the payoff function from the distribution of the underlying process. These two ingredients enter into the integral representation of the pricing formula in the form of its Fourier transform and its characteristic function or equivalently its moment-generating function, respectively. To price derivatives which have a financial asset such as a stock, an index or an FX rate as underlying, exponential Levy models are considered. The approach is able to handle path-dependent options as well. For this purpose the characteristic functions of the running supremum and the running infimum of the driving Levy process are investigated in detail. Pricing of interest rate derivatives is considered in the second part. In this context it is natural to use time-inhomogeneous Levy processes, also called processes with independent increments and absolutely continuous characteristics (PIIAC), as drivers. We discuss various possibilities to model the basic quantities in interest rate theory, which are instantaneous forward rates, bond prices, Libor rates, and forward processes. Valuation formulas for caps, floors, and swaptions are derived.