A jump diffusion model for VIX volatility options and futures

Abstract

Implied volatility indices are becoming increasingly popular as a measure of market uncertainty and as a vehicle for developing derivative instruments to hedge against unexpected changes in volatility. Although jumps are widely considered as a salient feature of volatility, their implications for pricing implied volatility options and futures are not yet fully understood. This paper provides evidence indicating that the time series behavior of the VIX equity implied volatility index is well approximated by a mean reverting logarithmic diffusion with jumps. This process is capable of capturing stylized facts of VIX dynamics such as fast mean-reversion at high levels, level effects of volatility and large upward movements during times of market stress.Based on this process, we develop closed form valuation models for volatility futures and options and show that incorrectly omitting jumps may cause considerable problems to pricing and hedging.a Lecturer (Corresponding author), Manchester Accounting

Mean-Reverting Logarithmic Modeling of VIX

“…2.2.4 Psychoyios, Dotsis, and Markellos [21] In Psychoyios [21] , the authors assume that V IX t follows a mean-reverting logarithmic jump process (MRLRJ) as below…”

“…Grunbichler and Longstaff [8] , Detemple and Osakwe [3] and Psychoyios [21] , where meanreverting square-root and mean-reverting logarithmic processes with or without jumps are adopted to characterize VIX.…”

“…Psychoyios and Skiadopoulos [22] and Psychoyios [21] recommend that the mean-reverting logarithmic model (denoted as MRLR) serves better than the meanreverting square root models (denoted as MRSR) in both aspects of fitting VIX historical data under the objective measure and calibrating VIX options under martingale measure.…”

“…The logarithmic models proposed in Psychoyios and Skiadopoulos [22] and Psychoyios [21] assumes that logarithm of VIX follows a OU process as Vasicek [26] . By adding upward jumps that follow exponential distribution into the MRLR model to construct mean-reverting logarithmic jump model (denoted as MRLRJ), Psychoyios [21] successfully get an explicit pricing formula for VIX option expressed by characteristic function of log-VIX in the MRLRJ model.…”

“…By adding upward jumps that follow exponential distribution into the MRLR model to construct mean-reverting logarithmic jump model (denoted as MRLRJ), Psychoyios [21] successfully get an explicit pricing formula for VIX option expressed by characteristic function of log-VIX in the MRLRJ model.…”