Modelling the dynamics of implied volatility smiles and surfaces

Abstract

"Smile-consistent" no-arbitrage stochastic volatility models take today's option prices as given, and they let them to evolve stochastically in such a way as to preclude arbitrage. This allows standard options to be priced correctly, and enables exotic options to be valued and hedged relative to them. We study how to model the dynamics of implied volatilities, since this is a necessary prerequisite for the implementation of these models. First, we investigate the number and shape of shocks that move implied volatility smiles, by applying Principal Components Analysis. The technique is applied to two different metrics: the strike, and the moneyness. Three distinct criteria are used to determine the number of components to retain. Subsequently, we construct a "Procrustes" type rotation in order to interpret them. Second, we use the same methodology to identify the number and shape of shocks that move implied volatility surfaces. In both cases, we find that the number of shocks is the same (two), in both metrics. Their interpretation is a shift for the first one, and a Z-shaped for the second. The results have implications for both option pricing and hedging, and for the economics of option pricing. Finally, we propose a new and general method for constructing a "smile-consistent" no-arbitrage stochastic volatility model: the simulation of the implied risk-neutral distribution. An algorithm for the simulation is developed when the first two moments change over time. It can be implemented easily, and it is based on the idea of mixture of distributions. It can also be generalized to cases where more complicated forms for the mixture are assumed