Computation of the unknown volatility from integral option price observations in jump–diffusion models

Abstract

In this work we propose a simple and efficient algorithm to numerically approximate the time-dependent implied volatility for jump–diffusion models in option pricing that generalize the Black–Scholes equation. Here we use implicit–explicit difference schemes to compute the derivative part with fully implicit method and the integral term — in an explicit way. An average in time linearization of the diffusion term is applied, followed by a special decomposition of the unknown volatility function, which enables us to derive the implied volatility in an explicit form. Furthermore, the correctness of the algorithms is established. The presented numerical simulations demonstrate the capabilities of the current approach and confirm the robustness of the proposed methodology.