Mean-Reverting Logarithmic Modeling of VIX

Abstract

Since March 26, 2004, when the CBOE Futures Exchange (CFE) began trading futures written on S&P500 volatility index (VIX), volatility has become a widely accepted asset class as trading, diversifying and hedging vehicle by traders, investors and portfolio managers over the past few years. On February 24, 2006, CBOE introduced options written on VIX index and since then VIX option series has now become the most actively traded index option series on CBOE. This thesis focuses on mathematical modeling of spot VIX with standalone approach. Unlike the consistent modeling approach in literature, which starts with specifying joint dynamics for SPX index and its instantaneous stochastic volatility then derives expression for spot VIX and price VIX derivatives based on this expression, standalone approach starts with directly specifying dynamics for spot VIX and prices VIX derivatives in this simpler framework. Although there is work in literature that studies the mean-reverting logarithmic model (MRLR), no work has been done in considering stochastic volatility in MRLR to capture the positive implied volatility skew of VIX option, nor have they compared the pure diffusion version of MRLR with its jump and/or stochastic volatility extensions. Furthermore, most of the literature only focus on static pricing formulas for VIX future and VIX option, no work has been done in investigating the dynamic feature of VIX future, calibration and hedging strategies of mean-reverting logarithmic models, as well as the convexity adjustment of VIX future from forward variance swap, which has a liquid variance swap market to back out the vol-of-vol information in mean-reverting logarithmic models. In this thesis, I present four versions of MRLR models. The first model is a pure diffusion model where spot VIX follows a mean-reverting logarithmic dynamics. Then I extend this basic MRLR model by adding jump or stochastic volatility into spot VIX dynamics to get MRLRJ and MRLRSV models. Finally, I combine jump and stochastic volatility together and add them into dynamics of spot VIX to get the fully specified MRLRSVJ model. For all the four models, I derive either transition function or conditional characteristic function of spot VIX. Based on those results, the pricing formulas for VIX future and VIX option are derived. In order to calibrate to VIX future term structure, I make the long-term mean of spot VIX be a time-dependent function and use the diffusion, jump and/or stochastic volatility parameters to calibrate VIX implied volatility surface. Two types of calibration strategies are suggested in this thesis. On the first stage of calibration, we need to calibrate all vol-of-vol parameters to convexity of spot VIX or VIX future. One strategy is to calibrate those parameters to VIX option implied volatility surface. Another strategy is to calibrate them to convexity adjustment of VIX future from forward variance swap, which can be replicated by liquid variance swaps. On the second stage of calibration, the long-term mean function of spot VIX is used to fit VIX futuer term structure given the vol-of-vol parameters calibrated on the first stage. In addition to the static pricing formula, dynamics of VIX future is also derived under all mean-reverting logarithmic models. The analysis in this thesis shows that VIX future follows geometric Brownian motion under MRLR model, jump-diffusion dynamics under MRLRJ model, stochastic volatility dynamics under MRLRSV model and stochastic volatility with jump dynamics under MRLRSVJ model. I develop the hedging strategies of VIX future and VIX option under mean-reverting logarithmic models. As spot VIX is not tradable asset, investors are unable to take positions on this index. Instead, research in literature has shown that a shorter-term VIX future has good power in forecasting movements of the subsequent VIX future. Therefore, hedging VIX future with a shorter-term VIX future is expected to perform well. Moreover, as VIX option can also be regarded as an option on a VIX future contract that has same maturity as VIX option, using the shorter-term VIX future contract as hedging instrument is a natural choice. In this thesis, I derive hedging ratios of VIX future and VIX option under the above hedging strategy. At last, numerical analysis in this thesis compares the four models in fitting VIX implied volatility surface. The results show that MRLR is unable to create positive implied volatility skew for VIX option. In contrast, MRLRJ and MRLRSV models perform equally well in fitting positive skew. However, the fully specified MRLRSVJ model adds little value in fitting VIX skew but incurs additional cost of calibrating more parameters and is subject to less stable parameters over maturities and over time.