Reconstruction of the local volatility function using the Black–Scholes model

Abstract

In this paper, we propose a robust and accurate numerical algorithm to reconstruct a local volatility function using the Black–Scholes (BS) partial differential equation (PDE). Using the BS PDE and given market data, option prices at strike prices and expiry times, a time-dependent local volatility function is computed. The proposed algorithm consists of the following steps: (1) The time-dependent volatility function is computed using a recently developed method; (2) A Monte Carlo simulation technique is used to find the effective region which has a strong influence on option prices; and we partition the effective domain into several sub-regions and define a local volatility function based on the time-dependent volatility function on the sub-regions; and (3) Finally, we calibrate the local volatility function using the fully implicit finite difference method and the conjugate gradient optimization algorithm. We demonstrate the robustness and accuracy of the proposed local volatility reconstruction algorithm using manufactured volatility surface and real market data.