: A three parameter stochastic process, termed the variance gamma process, that generalizes Brownian motion is developed as a model for the dynamics of log stock prices. The process is obtained by evaluating Brownian motion with drift at a random time given by a gamma process. The two additional parameters are the drift of the Brownian motion and the volatility of the time change. These additional parameters provide control over the skewness and kurtosis of the return distribution. Closed forms are obtained for the return density and the prices of European options. The statistical and risk neutral densities are estimated for data on the S&P500 Index and the prices of options on this Index. It is observed that the statistical density is symmetric with some kurtosis, while the risk neutral density is negatively skewed with a larger kurtosis. The additional parameters also correct for pricing biases of the Black Scholes model that is a parametric special case of the option pricing model d...

We investigate the relative importance of diffusion and jumps in a new jump diffusion model for asset returns. In contrast to the standard modelling of jumps for asset returns, the jump component of our process can display finite or infinite activity, and finite or infinite variation. Empirical investigations of time series indicate that index dynamics are essentially devoid of a diffusion component, while this component may be present in the dynamics of individual stocks. This result leads to the conjecture that the risk-neutral process should be free of a diffusion component for both indices and individual stocks. Empirical investigation of options data tends to confirm this conjecture. We conclude that the statistical and risk-neutral processes for indices and stocks tend to be pure jump processes of innite activity and finite variation.

This paper shows how the fast Fourier Transform may be used to value options when the characteristic function of the return is known analytically.

Three processes re°ecting persistence of volatility are initially formulated by evaluating three L¶evy processes at a time change given by the integral of a mean reverting square root process. The model for the mean reverting time change is then generalized to include Non-Gaussian models that are solutions to OU (Ornstein-Uhlenbeck) equations driven by one sided discontinuous L¶evy processes permitting correlation with the stock. Positive stock price processes are obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating these processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. In general, mean corrected exponentiation performs better than employing the stochastic exponential. It is observed that the mean corrected exponential model is not a martingale in the ¯ltration in which it is originally de¯ned. This leads us to formulate and investigate the important property of martingale marginals where we seek martingales in altered ¯ltrations consistent with the one dimensional marginal distributions of the level of the process at each future date. 1

pp.567-611. Contents

As is well known, the classic Black-Scholes option pricing model assumes that returns follow Brownian motion. It is widely recognized that return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to non-normal return innovations. Second, return volatilities vary stochastically over time. Third, returns and their volatilities are correlated, often negatively for equities. We propose that time-changed Lévy processes be used to simultaneously address these three facets of the underlying asset return process. We show that our framework encompasses almost all of the models proposed in the option pricing literature. Despite the generality of our approach, we show that it is straightforward to select and test a particular option pricing model through the use of characteristic function technology.

A partial integro-differential equation (PIDE) ∂tu + A[u] = 0 for European contracts on assets with general jump-diffusion price process of Lévy type is derived. The PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the θ-scheme in time and a wavelet Galerkin method with N degrees of freedom in space. The full Galerkin matrix for A can be replaced with a sparse matrix in the wavelet basis, and the linear systems for each time step are solved approximatively with GMRES in linear complexity. The total work of the algorithm for M time steps is bounded by O(MN(ln N) 2) operations and O(N ln(N)) memory. The deterministic algorithm gives optimal convergence rates (up to logarithmic terms) for the computed solution in the same complexity as finite difference approximations of the standard Black-Scholes equation. Computational examples for various Lévy price processes (VG, CGMY) are presented. 1

We investigate some portfolio problems that consist of maximizing expected terminal wealth under the constraint of an upper bound for the risk, where we measure risk by the variance, but also by the Capital-at-Risk (CaR). The solution of the mean-variance problem has the same structure for any price process which follows an exponential Levy process. The CaR involves a quantile of the corresponding wealth process of the portfolio. We derive a weak limit law for its approximation by a simpler Levy process, often the sum of a drift term, a Brownian motion and a compound Poisson process. Certain relations between a Levy process and its stochastic exponential are investigated.

Abstract The price of an American style contract on assets driven by L'evy processes with infinite jump activity is expressed as solution of a parabolic variational integro-differential inequality (PIDI). A Galerkin discretization in logarithmic price using a wavelet basis is presented with compression of the moment matrix of the jump part of the price process ' Dynkin operator. An iterative solver with wavelet preconditioning for the resulting large matrix inequality problems is presented and its efficiency is demonstrated by numerical experiments. Keywords: L'evy processes, integro-differential operators, variational inequalities, Galerkin discretization, bi-orthogonal wavelet basis, wavelet preconditioning

We consider American options in a market where the underlying asset follows a Variance Gamma process. We give a sufficient condition for the failure of the smooth fit principle for finite horizon call options. We also propose a second order accurate finite-difference method to find the American option price and the exercise boundary. The problem is formulated as a Linear Complementarity Problem and numerically solved by a convenient splitting. Computations have been accelerated with the help of the Fast Fourier Transform. A stability analysis shows that the scheme is conditionally stable, with a mild stability condition of the form k = O(|log(h) | −1). The theoretical results are verified numerically throughout a series of numerical experiments. Keywords: Integro-differential equations, Variance Gamma, finite differences, FFT.