Robust and accurate construction of the local volatility surface using the Black–Scholes equation

In this study, we develop a numerical method for the robust and accurate construction of a local volatility (LV) surface using the generalized Black–Scholes (BS) equation from the given option price data. The BS equation is a partial differential equation and has been used to model financial option pricing. Constant volatility was used in the classical BS model. However, it is well known that the constant volatility BS model is practically unsuitable because real financial market data demonstrate non-constant volatility behavior. The LV function is dependent on the asset prices and time. One of the difficulties in reconstructing an unknown LV surface is uniqueness. We extend a previous study of reconstructing time-dependent volatility, which is unique, to time- and space-dependent volatility surfaces. We propose an algorithm comprising four steps: the first step is estimating constant implied volatility; the second step is finding the influential region using the probability density function of a log-normal distribution; the third step is calculating the time-dependent volatility function; and the final step is reconstructing the LV surface. We use a finite difference method to numerically solve the BS model and a nonlinear fitting function to compute the LV surface. We perform computational experiments using synthetic and real market data. The numerical results demonstrate the robust and accurate construction of an unknown LV surface using the proposed method.