Calibration of Local Volatility Surfaces from Observed Market Call and Put Option Prices

Abstract

We present a novel, straightforward, robust, and precise calibration algorithm for local volatility surfaces based on observed market call and put option prices. The proposed local volatility reconstruction method is based on the widely recognized generalized Black–Scholes partial differential equation, which is numerically solved using a finite difference scheme. In the proposed method, sample points are strategically placed in the underlying and time domains. The unknown local volatility function is represented using the scattered interpolant function. The primary contribution of this study is that our proposed algorithm not only optimizes the volatility values at the sample points but also optimizes the positions of the sample positions using a least squares method. This optimization process improves the accuracy and robustness of our calibration method. Furthermore, we do not use the Tikhonov regularization technique, which was frequently used to obtain smooth solutions. To validate the practical efficiency and superior performance of the proposed reconstruction method for local volatility functions, we conduct a series of computational experiments using real-world market option prices such as the KOSPI 200, S &P 500, Hang Seng, and Euro Stoxx 50 indices. The proposed algorithm offers financial market practitioners a reliable tool for calibrating local volatility surfaces using only market option prices, enabling more accurate pricing and risk management of financial derivatives.