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.
