Model Independent Multi-Asset Volatility Smile with Empirical Confirmation

Abstract

This paper presents a new formalism to price European options in all asset classes that fits the market data remarkably well. We use a model-independent representation of European Option prices as path integrals over all of the underlying asset price from inception to maturity. The no arbitrage condition of the path integral representation generates a consistency condition upon the risk neutral probability density function to maturity. When solving for the risk neutral density function to maturity that satisfies the consistency condition, the European volatility smile is obtained. As an illustration of the formalism, we show that when the underlying asset price changes at constant volatility (standard deviation), the option price satisfies the Black-Scholes-Merton model without the pre-assumption that the underlying asset price satisfies a Brownian motion. More generally, without any pre-assumption on the underlying asset’s stochastic process, the risk neutral density function for a given expiration date is determined by the option prices of three strikes with the same expiration. We offer a parameterization of the volatility smile with a closed-form expression using pre-calculated tables. Comprehensive empirical evidence is provided, confirming that the new model accurately matches the market option prices in equities, commodities, currencies and interest rates. In addition, a formalism to obtain the conditional probability transfer density function from the vanilla options term structure is suggested.