VIX and volatility derivatives: a survey of relevant financial instruments and pricing models

Abstract

Although it is relatively difficult to predict the exact return associated with a stock, one can compute, with a certain degree of precision, the volatility associated with the stock’s return by utilizing appropriate mathematical models. In order to account for volatility, a new metric called the Volatility Index (VIX), based on S&P 100 shares, was posited by the Chicago Board Options Exchange (CBOE). This definition was tweaked eventually to make S&P 500 shares (SPX) the basis for calculating VIX by averaging the prices of calls and puts on SPX shares (over a wide range of strike prices) considering a time frame of 30 days. Futures and options eventually emerged based on the VIX and have gone on to acquire a very critical role in financial markets worldwide, especially after the Global Recession. Thus, quite intuitively, one can infer that it is possible to decode the dynamics of SPX shares from the dynamics of the options based on them. One of the interesting features of VIX (which computes a close approximation of the 30 day variance swap rate) is that its computation is starkly different from the implied volatility derived from the Black-Scholes model used by traders. The rationale behind using such a model is to account for the “skew” effect occurring in stock prices which leads the Black-Scholes model to fail to account for random volatility and thereby predict option prices which fall with an increase in the strike price of the option. There have been two kinds of methods widely discussed in literature used for pricing VIX options: model-free approaches and model-dependent approaches. In particular, the modeldependent approaches consider the underlying process to be stochastic in nature so as to arrive at a feasible model for estimating VIX options prices wherein the model parameters are derived by analyzing past data pertaining to these options. As quite evident, there does exist a disconnect in approaches utilizing tweaked versions of the Black-Scholes model to price volatility options given that the computation of the VIX (i.e. the most accurate volatility estimate) is primarily done on the basis of the average bid-ask spread pertaining to these options. Secondly, the options and stocks are priced based upon forward-looking data while the underlying assets are not thereby further widening the afore-mentioned disconnect. Finally, very few models take bootstrapping of past market data on a relatively dynamic scale into consideration while arriving at the model parameters, whose validity may be established based on empirical studies.