On Lévy Processes for Option Pricing: Numerical Methods and Calibration to Index Options

Since Black and Scholes published their article on option pricing in 1973, there has been an explosion of theoretical and empirical work on the subject. However, over the last thirty years, a vast number of pricing models have been proposed as an alternative to the classic Black-Scholes approach, whose assumption of lognormal stock diffusion with constant volatility is considered always more flawed. One major reason is that since the stock market crash of October 19, 1987, deviations of stock index option prices from the benchmark Black-Scholes model have been extraordinarily pronounced. In fact, since then, to equate the BlackScholes formula with quoted prices of European calls and puts, it is generally necessary to use different volatilities, so-called implied volatilities, for different option strikes and maturities (the Black-Scholes model required a constant volatility based on the subjacent historical volatility). That feature suggests that the distribution perceived by market participant and incorporated into option prices is substantially negatively skewed (that is to say leptokurtic with a fat tail on the negative side), in contrast to the essentially symmetric and slightly positively lognormal distribution underlying the Black-Scholes model. The pattern formed by the implied volatilities across the strikes is then called volatility smile or skew, due to the fact that the implied volatility of in-themoney call options is pretty much higher than the one of out-of-the-money options. Typically, the steepness of the skew decreases with increasing option maturities. The existence of the skew is often attributed to fear of large downward market movements. The research of a new form of models able to incorporate the smile has been one of the most active fields of studies in modern quantitative finance. There are two assumptions that have to be made in order to price derivatives with the Black-Scholes model: returns are subject to a single source of uncertainty and asset prices follow continuous sample paths (a Brownian motion). Then, under these two assumptions, a continuously rebalanced portfolio can be used to perfectly hedge an options position, thus determining a unique price for the option. Therefore, extensions of the Black-Scholes model that capture the existence of volatility smile can, broadly speaking, be grouped in two approaches, each one relaxing one of these two assumptions. Relaxing the assumption of a unique source of uncertainty leads to the stochastic volatility family of models, where the volatility parameter follows a separate diffusion, as proposed by Heston . Relaxing the assumption of continuous sample paths, leads to jump models, where stock prices follow an exponential Lévy process of jump-diffusion type (where evolution of prices is given by a diffusion process, punctuated by jumps at random intervals) or pure jumps type. Jump models attribute the biases in Black-Scholes model to fears of a further stock market crash. They would interpret the crash as a revelation that jumps can in fact occur. Looking to a plot of a stock index time series, there is clear evidence that prices don’t follow a diffusion process and actually jump. This thesis deals with the study of L´evy processes for option pricing. L´evy processes are an active field of research in finance, and many models have been presented during the last decade. This thesis is not an attempt to describe all the Lévy models discussed in the literature or explain their mathematical properties. We focus on four famous models, two of jump-diffusion type (Merton normal jump-diffusion and Kou double-exponential jump-diffusion) and two pure jump models (Variance Gamma and Normal Inverse Gaussian) for which we describe their foremost mathematical characteristics and we concentrate on providing modeling tools.