Arbitrage-free market models for option prices : The multi-strike case

This paper studies modelling and existence issues for market models of option prices in a continuous-time framework with one stock, one bond and a family of European call options for one fixed maturity and all strikes. After arguing that (classical) implied volatilities are ill-suited for constructing such models, we introduce the new concepts of local implied volatilities and price level. We show that these new quantities provide a natural and simple parametrization of all option price models satisfying the natural static arbitrage bounds across strikes. We next characterize absence of dynamic arbitrage for such models in terms of drift restrictions on the model coeffcients. For the resulting infinite system of SDEs for the price level and all local implied volatilities, we then study the question of solvability and provide suffcient conditions for existence and uniqueness of a solution. We give explicit examples of volatility coeffcients satisfying the required assumptions, and hence of arbitrage-free multi-strike market models of option prices.