Assigning future smile surfaces: conditions for uniqueness and absence of arbitrage
Dr Riccardo Rebonato, Royal Bank of Scotland

Abstract:
I explore in this paper to what extent future implied volatility surfaces (smile surfaces) can be assigned by a trader without incurring the risk of model-independent arbitrage. I show that, if the future smile is assumed to be floating (in a precise sense to be defined) then there exists a single future smile surface that does not allow arbitrage and is consistent with today's prices. I then introduce some conditions on the nature of the stochastic future smiles and show that, if they are met, it is always possible to find a deterministic (equivalent) future smile surface such that the prices of calls and put are exactly the same in the deterministic and stochastic settings. Finally, I analyse popular models (jump-diffusions, stochastic-volatility) etc and comment on the extent to which the conditions mentioned above are met in these cases.