Fast reconstruction of time-dependent market volatility for European options

Abstract

This paper presents a robust and fast numerical algorithm to reconstruct the implied volatility as a piecewise linear function of time. This is done from a set of market observations in the Black–Scholes world. We use a fully implicit finite difference scheme to solve the partial differential equations. To find the time-dependent volatility function, we minimize the cost function defined as the sum of the squared errors between the theoretical prices and the prices observed on the market. On the last time step, right before each maturity, we apply a decomposition of the numerical option value with respect to the volatility which increases the stability and the solvability of the problem considered. We employ a predictor–corrector technique due to the non-uniqueness of the volatility function minimizer. The paper is concluded with profound numerical experiments with synthetic and real market data.