Arbitrage-free market models for option prices

Abstract

In this paper we construct arbitrage-free market models of stochastic volatility type for one stock, one bank account and a finite family of European call options with various strikes and maturities. We first introduce local implied volatilities and price level as market observables which parametrize the static arbitrage bounds of options across strikes and maturities in a natural way. Option prices are explicit functions of the local implied volatilities and price level, and absence of dynamic arbitrage in the market can be characterized in terms of drift restrictions on the model coefficients. Most importantly, we give sufficient conditions for existence and uniqueness of a solution to the corresponding system of SDEs, and we provide a class of explicit examples satisfying the no-arbitrage conditions. This allows us to construct arbitrage-free multi-option market models with a prespecified volatility structure.