A Statistical Deterministic Implied VolatilityModel

We considerthe implied volatility surface to characterise agentsbelief of the future evolution of stock price returns. However, today’s market prices do not provide us with the right future anticipations of the stock price process. This is because the impliedvolatility surface is neitherstationarynorMarkovian.It is thereforenatural to model the evolution of the implied volatility surface directly. Our goal is to model the implied volatility surface with general dynamics by relating its future evolution to an observable stochastic process and by adding noises. We choose to link the stock price process to the implied volatility which implies that the volatility surface is dynamically modified according to stock price realisations. We model the stock price process discretely and using conditional expectations we define its joint distributions. We calibrate the transition matrices to historical data augmenting our filtration set by adding past vanilla option prices. To satisfy the absence of arbitrage opportunities, we will impose that future smile surfaces are compatible with today’s prices of calls and puts. Also, defining a deterministic smile surface means defining a future deterministic density for stock process. Therefore, a natural condition would be to impose that the future density is actually a conditional density. That is what we will formalise as the Kolmogorovcompatibility condition. We then apply our approach to the pricing of forward start and cliquet options andshow thatthe forwardvolatility hasto behigherthanthe spotvolatility because of the risk attached to such products which is not taken into account in today’s information.