Calibrating with a smile: A Mellin transform approach to volatility surface calibration

Abstract

The implied volatility in the Black-Scholes framework is not a constant but a function of both the strike price (“smile/skew”) and the time to expiry. A popular approach to recovering the volatility surface is through the use of deterministic volatility function models via Dupire’s equation. A new method for volatility surface calibration based on the Mellin transform is proposed. An explicit formula for the volatility surface is obtained in terms of the Mellin transform of the call option price with respect to the strike price, and a numerical algorithm is provided. Results of numerical simulations are presented and the stability of the method is numerically verified. The proposed Mellin transform approach provides a simpler and more direct fitting of generalised forms of the volatility surface given previously in the literature.