Arbitrage-Free Mix Var Volatility Surfaces and Applications

Abstract

The implied volatility surface being a mapping from Black-Scholes prices, necessary and sufficient conditions for the surface to be free from static arbitrage must be defined in terms of the properties and limits of the Black-Scholes formula. Acknowledging this argument, we develop a parametric interpolation and extrapolation of the implied volatility surface to prevent arbitrage both in space and time. Expressing the price surface in terms of moneyness and variance time rather than standard calendar time, we decompose the market option prices into weighted sums of strike shifted Black-Scholes counterparts, combined with a term structure interpolation of implied total variance. As a result, static arbitrage is satisfied by construction while the dynamics of the implied volatility is taken into consideration, allowing for proper dynamic risk management. This simple model, intended to be used by practitioners, allows an analytical computation of the Greeks, the Skew and the Curvature of the fitted implied volatility surface. At last, we generate meaningful stress scenarios for risk management purpose by stress testing the model parameters while preserving the necessary and sufficient conditions for the call price surface to be free from arbitrage.

Keywords: implied volatility surface, static arbitrage, calibration, single parametric model, analytic Greeks, analytic stress scenarios