A method of inversion of Fourier transforms and its applications

Abstract

The problem of inversion of Fourier transforms is a frequently discussed topic in the theory of PDEs, Stochastic Processes and many other branches of Analysis. We consider here in more details an application of a method proposed in Financial Modeling. As a motivating example consider a frictionless market with no arbitrage opportunities and a constant riskless interest rate r > 0. Assuming the existence of a risk-neutral equivalent martingale measure Q, we get the option value V = e −rTE Q[ϕ] at time 0 and maturity T > 0, where ϕ is a reward function and the expectation E Q is taken with respect to the equivalent martingale measure Q. Usually, the reward function ϕ has a simple structure. Hence, the main problem is to approximate properly the respective density function and then to approximate E Q [ϕ]. Here we offer an approximant for the density function without proof of any convergence results. These problems will be considered in details in our future publications.