Tuesday, August 20, 2013

Guest Post: A qualitative review of VIX F&O pricing and hedging models

By Azouz Gmach

VIX Futures & Options are one of the most actively traded index derivatives series on the Chicago Board Options Exchange (CBOE). These derivatives are written on S&P 500 volatility index and their popularity has made volatility a widely accepted asset class for trading, diversifying and hedging instrument since their launch. VIX Futures started trading on March 26th, 2004 on CFE (CBOE Future Exchange) and VIX Options were introduced on Feb 24th, 2006.

VIX Futures & Options

VIX (Volatility Index) or the ‘Fear Index’ is based on the S&P 500 options volatility. Spot VIX can be defined as square root of 30 day variance swap of S&P 500 index (SPX) or in simple terms it is the 30-day average implied volatility of S&P 500 index options. The VIX F&O are based on this spot VIX and is similar to the equity indexes in general modus operandi. But structurally they have far more differences than similarities. While, in case of equity indices (for example SPX), the index is a weighted average of the components, in case of the VIX it is sum of squares of the components. This non-linear relationship makes the spot VIX non-tradable but at the same time the derivatives of spot VIX are tradable. This can be better understood with the analogy of Interest Rate Derivatives. The derivatives based on the interest rates are traded worldwide but the underlying asset: interest rate itself cannot be traded.

The different relation between the VIX derivatives and the underlying VIX makes it unique in the sense that the overall behavior of the instruments and their pricing is quite different from the equity index derivatives. This also makes the pricing of VIX F&O a complicated process. A proper statistical approach incorporating the various aspects like the strength of trend, mean reversion and volatility etc. is needed for modeling the pricing and behavior of VIX derivatives.

Research on Pricing Models

There has been a lot of research in deriving models for the VIX F&O pricing based on different approaches. These models have their own merits and demerits and it becomes a tough decision to decide on the most optimum model. In this regards, I find the work of Mr. Qunfang Bao titled ‘Mean-Reverting Logarithmic Modeling of VIX’ quite interesting. In his research, Bao not only revisits the existing models and work by other prominent researchers but also comes out with suggestive models after a careful observation of the limitations of the already proposed models. The basic thesis of Bao’s work involves mean-reverting logarithmic dynamics as an essential aspect of Spot VIX.

VIX F&O contracts don’t necessarily track the underlying in the same way in which equity futures track their indices. VIX Futures have a dynamic relationship with the VIX index and do not exactly follow its index. This correlation is weaker and evolves over time. Close to expiration, the correlation improves and the futures might move in sync with the index. On the other hand VIX Options are more related to the futures and can be priced off the VIX futures in a much better way than the VIX index itself.

Pricing Models

As a volatility index, VIX shares the properties of mean reversion, large upward jumps & stochastic volatility (aka stochastic vol-of-vol). A good model is expected to take into consideration, most of these factors.

There are roughly two categories of approaches for VIX modeling. One is the Consistent approach and the other being Standalone approach.

        I.            Consistent Approach: - This is the pure diffusion model wherein the inherent relationship between S&P 500 & VIX is used in deriving the expression for spot VIX which by definition is square root of forward realized variance of SPX.

      II.            Standalone Approach: - In this approach, the VIX dynamics are directly specified and thus the VIX derivatives can be priced in a much simpler way. This approach only focuses on pricing derivatives written on VIX index without considering SPX option.
Bao in his paper mentions that the standalone approach is comparatively better and simpler than the consistent approach.

MRLR model

The most widely proposed model under the standalone approach is MRLR (Mean Reverting Logarithmic Model) model which assumes that the spot VIX follows a Geometric Brownian motion process. The MRLR model fits well for VIX Future pricing but appears to be unsuited for the VIX Options pricing because of the fact that this model generates no skew for VIX option. In contrast, this model is a good model for VIX futures.

MRLRJ model

Since the MRLR model is unable to produce implied volatility skew for VIX options, Bao further tries to modify the MRLR model by adding jump into the mean reverting logarithmic dynamics obtaining the Mean Reverting Logarithmic Jump Model (MRLRJ). By adding upward jump into spot VIX, this model is able to capture the positive skew observed in VIX options market.

MRLRSV model

Another way in which the implied volatility skew can be produced for VIX Options is by including stochastic volatility into the spot VIX dynamics. This model of Mean Reverting Logarithmic model with stochastic volatility (MRLRSV) is based on the aforesaid process of skew appropriation.
Both, MRLRJ and MRLRSV models perform equally well in appropriating positive skew observed in case of VIX options.


Bao further combines the MRLRJ and MRLRSV models together to form MRLRSVJ model. He mentions that this combined model becomes somewhat complicated and in return adds little value to the MRLRJ or MRLRSV models. Also extra parameters are needed to be estimated in case of MRLRSVJ model.

MRLRJ & MRLRSV models serve better than the other models that have been proposed for pricing the VIX F&O. Bao in his paper, additionally derives and calibrates the mathematical expressions for the models he proposes and derives the hedging strategies based on these models as well. Quantifying the Volatility skew has been an active area of interest for researchers and this research paper addresses the same in a very scientific way, keeping in view the convexity adjustments, future correlation and numerical analysis of the models etc. While further validation and back testing of the models may be required, but Bao’s work definitely answers a lot of anomalous features of the VIX and its derivatives.

Azouz Gmach works for QuantShare, a technical/fundamental analysis software.

My online Mean Reversion Strategies workshop will be offered in September. Please visit epchan.com/my-workshops for registration details.

Also, I will be teaching a new course Millisecond Frequency Trading (MFT) in London this October.