Regularization for the Inverse Problem of Finding the Purely Time-Dependent Volatility

Abstract

We consider the inverse problem of finding the volatility σ∈Lρ(0, T) such that \(U_{BS}(X,K,r,t,{{\int }_{0}^{t}}\sigma ^{2}(\tau )d\tau )=u(t)\), 0≤t≤T, where UBS is the Black–Scholes formula and u(t) is the observable fair price of an European call option. The problem is ill-posed. Using the residual method, we shall regularize the problem. An explicit error estimate is given.