Uniform convergence of empirical characteristic functions in a complex domain with applications to option pricing

In Csörgo{double acute} and Totik (1983) and Csörgo{double acute} (1985) it has been shown that in the case of independent identically distributed (iid) random variables X₁, X₂, ..., X_n the empirical characteristic function (ecf) over(φ{symbol}, ̂)_n (u) converges uniformly, for | u | ≤ U_n to the characteristic function φ{symbol} (u) of X, on increasing intervals which union covers the whole real line. We show that if suitable moments exist then the uniform convergence is also valid for u in the complex domain x = u + i ν, | u | ≤ U_n, ν ∈ (a, b), where a < b depend on the cumulative distribution function F of X. This extension has an important application in Stochastic Finance in option pricing for Lévy processes. It allows us to prove the convergence of an empirical option pricing formula to the theoretical value of the option and opens a way towards option pricing based on empirical characteristic functions. Crown