Multifactor Stochastic Variance Models in Risk Management: Maximum Entropy Approach and Levy Processes

Abstract

There is extensive empirical evidence that historical distributions of daily changes for stock prices, interest rates, foreign exchange rates, commodity prices and other underlyings have high peaks, heavy tails and non-zero skewness contrary to the normal distribution. These risk factors exhibit jumps, their volatility varies stochastically with clustering. Above distributional properties have significant impact on Risk Management, specifically on Value-at-Risk (VaR) calculations. This paper presents a class of multivariate models with the stochastic variance driven by Levy processes. The models with correlation structure in the stochastic variance allow for different shape and tail behavior of the marginal risk factor distributions, exact fit into the risk factor correlation structure, and proper non-linear scaling of VaR for different holding periods. In one-dimensional case, a pure jump Gamma process and other Levy processes for the stochastic variance are derived from the Maximum Entropy principle. Corresponding stochastic processes for the risk factors possess marginal distributions with wide range of heavy tails, from exponential to polynomial. Ornstein-Uhlenbeck type processes for the stochastic variance and corresponding term structure of the risk factor kurtosis and quantiles are investigated. In multi-dimensional case, the effective calibration and Monte Carlo simulation procedures are considered. Presented empirical evidence for different markets confirms a good agreement between the model and actual historical risk factor distributions.