Modelling Subordinated Stochastic Processes with Student's t and Generalized Secant Hyperbolic Increments: Empirical Study of Speculative Energy Markets

Abstract

Regulatory decisions, such as the U.S. Commodity Futures Modernization Act, increased the number of market participants via the electronic trading platform in the new millenium. Volatile prices in the futures market has challenged current margin-setting methodologies for protection against default risk. Margin shortfall of futures contracts depends on the tail shape of their returns. GARCH model has not been recommended due to its too extreme margin levels and high frequency of margin changes (Lam et. al. 2004). This paper instead models returns in futures market as a subordinated stochastic process (SSP). Its use in modelling asset returns was often studied under the assumption of stable and Student’s t distributions. Most recently, stochastic volatility in energy futures market was modelled as a Student’s t Levy process with which trading volume was found to explain well its returns variability (Hu 2009). We extended this body of research in stochastic finance by changing independent increments assumption from Student’s t to Generalized Secant Hyperbolic distribution(GSHD). The family approaches uniform and Cauchy distributions in its limits, for which all moments exist and its CDF is invertible. We found by likelihood ratio test that the overall fit to empirical futures returns were comparable with Student’s t for large sample size (1992-2004). However, it has nicer properties than that of Stable and Student’s t and therefore lends to robust estimation with smaller sample size requirements for efficiency and reduced estimation error in calculation of risk metrics such as value-at-risk. As robust adaptive location tests have been recently developed for GSHD (Kravchuk and Hu 2008), shape a priori, knowable from trade volume, would be useful in detecting mean shift in returns away from zero.