Return Dynamics With Lévy Jumps: Evidence from Stock and Option Prices

Abstract

We examine the performances of L evy jump models and a ne jump-di usion models in capturing the joint dynamics of stock and option prices. We discuss the change of measure for in nite-activity L evy jumps and develop e cient Markov chain Monte Carlo methods for estimating model parameters and latent volatility and jump variables using stock and option prices. Using daily returns and option prices of the SP E-mail: htli@umich.edu; phone: (734)-764-6409. 2Wells is from the Department of Biological Statistics and Computational Biology and the Department of Social Statistics, Cornell University, Ithaca, NY 14853; E-mail: mtw1@cornell.edu; phone: (607)-255-4388; he gratefully acknowledges the support of NSF Grant DMS 02-04252. 3Yu is from the Department of Statistics, Iowa State University, Ames, IA 50011; E-mail: cindyyu@iastate.edu; phone: (515)-294-3319. We would like to thank Torben Andersen and Luca Benzoni for their help in the early stage of this study. We thank Yacine A t-Sahalia (the editor), Antje Berndt, Peter Carr, Bjorn Eraker, Bob Jarrow, Nour Meddahi, Ray Renken, Sidney Resnick, Ernst Schaumburg, Neil Shephard, George Tauchen, Liuren Wu, an anonymous referee, and seminar participants at Cornell University, the 2004 CIREQ/CIRANO nancial econometrics conference, the 2004 International Chinese Statistical Association Applied Statistics Symposium, and the 2004 Institute of Mathematical Statistics Annual Meeting/6th Bernoulli World Congress for helpful comments. We are responsible for any remaining errors. Return Dynamics with L evy Jumps: Evidence from Stock and Option Prices ABSTRACT We examine the performances of L evy jump models and a ne jump-di usion models in capturing the joint dynamics of stock and option prices. We discuss the change of measure for in nite-activity L evy jumps and develop e cient Markov chain Monte Carlo methods for estimating model parameters and latent volatility and jump variables using stock and option prices. Using daily returns and option prices of the S&P 500 index, we show that models with in nite-activity L evy jumps in returns signi cantly outperform a ne jump-di usion models with compound Poisson jumps in returns and volatility in capturing both the physical and the risk-neutral dynamics of the S&P 500 index. Modeling the dynamics of stock returns is a key issue in modern asset pricing. A realistic model of return dynamics is essential for option pricing, portfolio analysis, and risk management. One of the most popular continuous-time models for return dynamics in the current literature is the a ne jump-di usion (hereafter AJD) models of Du e, Pan, and Singleton (2000) (hereafter DPS). In AJD models, stock returns are driven by a ne di usions and compound Poisson processes. AJD models capture important stylized behaviors of index returns and are highly tractable. They allow closed-form pricing formulae for a wide range of equity and xed-income derivatives. Despite the successes of AJD models, Brownian motion and compound Poisson process (the two main building blocks of AJD models) are only two special cases of L evy processes, which are continuous-time stochastic processes with stationary and independent increments. L evy processes are much more exible than Brownian motion and compound Poisson process for modeling purposes. For example, L evy processes allow non-normal increments (compared to normal increments of Brownian motion) and much richer jump structures than compound Poisson process. Moreover, Carr and Wu (2004) show that L evy processes are as tractable as AJD models for pricing purposes: Closed-form pricing formulae are available for a wide range of derivative securities under L evy processes. These appealing features of L evy processes have spurred a fast-growing literature that models return dynamics using L evy processes in recent years.1 This new development, however, has raised some challenging theoretical and empirical issues in the current literature. While existing studies of L evy processes have mainly focused on either the physical or the risk-neutral return dynamics, a key remaining question is whether L evy processes have signi cant empirical advantages over AJD models in modeling the joint return dynamics.2 This is an important question because the ultimate See Wu (2006) for an excellent review of the current literature on L evy processes. See A t-Sahalia (2004) and A t-Sahalia and Jacod (2004) on some fundamental issues on statistical inferences of L evy processes. Madan, Carr, and Chang (1998), Carr, Geman, Madan, and Yor (2002), Huang and Wu (2003), and Carr and Wu