Enrique ter Horst
Professor of Finance, School of BusinessColegio de Estudios Superiores de Administracion (CESA), Bogota, Columbia
A Lévy generalization of compound Poisson processes in Finance: Theory and applications
Since Black and Scholes (1973), Mathematical Finance has grown as a branch of mathematics in its own right. The rejection of the normality of asset return distributions by Mandelbrot (1963) led to consideration of Lévy-stable stochastic processes as an interesting alternative. Modelling asset returns through a stochastic volatility model, composed of a diffusion and general Lévy pure jump process, is described in Chapter 2. The pricing of European options is also considered in Chapter 2, leading to a derivation of conditions that allow us to the approach of Duffie et al. (2000), to transform a measure to a risk-neutral one. The equivalence between risk-neutrality and no-arbitrage is then guaranteed by Delbaen and Schachermayer (1994). In Chapter 3 we perform a Bayesian analysis of a stochastic volatility model of Barndorff-Nielsen and Shephard (2001), where we treat Lévy jump times and sizes as uncertain and are interested in their posterior distributions. We also find the posterior distribution of the parameters governing the law of the Lévy subordinator (driving the stochastic volatility model of O-U type). The computations are done through the Reversible Jump Markov Chain Monte Carlo approach of Green (1995), since we deal with a Lévy pure jump process that has an uncertain number of large jumps.