General smile asymptotics and a multiscaling stochastic volatility model

In this thesis we discuss several aspects of the implied volatility surface. We first derive some model independent results, linking tail probabilities to option price and implied volatility. We then apply these results to a specific stochastic volatility model, obtaining a complete picture of the asymptotic volatility smile for bounded maturity. In Chapter 1 we present an extended summary of all the results obtained in this thesis. The details are contained in the following chapters, that are structured as follows. In Chapter 2 we show that, under general conditions satisfied by many models, the probability tails of the log-price under the risk-neutral measure determine the behavior of European option prices and of the implied volatility, in the regime of either extremes strike (with bounded maturity) or short maturity. Our results provide a powerful extension of previous work by Benaim and Friz (2009). We discuss the application to some popular models, including Carr-Wu finite moment logstable moment, Heston's model and Merton's jump diffusion model. In Chapter 3 we devote ourselves to the analysis of the implied volatility for a specific model, that has been recently proposed by Andreoli, Caravenna, Dai Pra and Posta (2012) to reproduce the multiscaling of moments and clustering of volatility observed in many financial indexes. Based on Chapter 2, this amounts to give sharp estimates on the tails of the log-price distribution. Although the moment generating function of the log-price is not known explicitly, we show that the tails can be well estimated via Large Deviation techniques, notably the Garter-Ellis theorem. In Chapter 4 we propose a possible enrichment of the model, adding jumps to the log-price in order to take account of the so called leverage effect. We prove some basic results and we describe a natural one-parameter family of martingale measures for this enriched model. We also show that the price of European options can be expressed through a generalization of the celebrated Hull & White formula, by averaging the usual Black & Scholes formula with respect to both a random volatility and a random spot price. Finally, in Chapter 5 we describe a numerical algorithm to price European option under the enriched model presented in Chapter 4, exploiting the generalized Hull & White formula. The algorithm uses a stratification method in order to improve the speed. Some preliminary results on the calibration of the model with real data, taken from the DAX index, are presented and discussed.