McMc Estimation of Lévy Jump Models Using Stock and Option Prices

Abstract

We examine the performances of several popular Lévy jump models and some of the most sophisticated affine jump-diffusion models in capturing the joint dynamics of stock and option prices. We develop efficient Markov chain Monte Carlo methods for estimating parameters and latent volatility/jump variables of the Lévy jump models using stock and option prices. We show that models with infinite-activity Lévy jumps in returns significantly outperform affine jump-diffusion models with compound Poisson jumps in returns and volatility in capturing both the physical and risk-neutral dynamics of the S&P 500 index. We also find that the variance gamma model of Madan, Carr, and Chang with stochastic volatility has the best performance among all the models we consider.

Option pricing for stochastic volatility model with infinite activity Lévy jumps

“…Because model calibrations to market options data formulates a nonlinear optimization problem which often suffers from local minima difficulties, which leads to poor performance in analyzing the behavior of derivative markets, as addressed by Yu [25]. Therefore, it is significant to obtain an accurate parameter set that calibrates the cross-sectional data well for the proposed model so that the model can be effectively applied to pricing options.…”

Calibrating the Smile With Multivariate Time-Changed Brownian Motion and the Esscher Transform

“…Both historical asset prices and option prices are considered to calibrate the model. Parametric models introduced in Christoffersen et al (2010) and Yu et al (2011) and the nonparametric one studied in Barone-Adesi et al (2008) have found a tractable way to connect the historical with the risk-neutral measure.…”

The Analysis of Corporate Bond Valuation under an Infinite Dimensional Compound Poisson Framework

“…For long-maturity bonds, Collin-Dufresne et al [3] have argued that firms with good credit quality are likely to issue more debt, which leads to higher yield spreads. The authors in [4][5][6][7] have shown that models with incorporating jump risk explain a significant part of observed credit spreads. Liquidity risk is an important element of spreads, if the debt market is of lower liquidity.…”