As noted in previous articles, there has been some talk over the potential skew existing over VIX’s implied volatility curves. The implied volatility over index options is often observed as having a skewed curve with a minima set around ATM strikes. But what would then occur over VIX options, which are themselves a proxy-measure for the implied volatility over S&P500 index options (SPX)?
How does the implied volatility of implied volatility actually measure up in reality?
Reviewing Some Fundamentals
Black and Scholes (1973) does not try and model different changes in volatility. Indeed, one of its core assumptions is that the distribution of an underlying price can be described as following a log-normal distribution with a set volatility estimate. This assumption would therefore imply that volatility is constant across all option strikes.
However, as has been strongly observed over time, equity market security price distributions are far from log-normal. Ongoing discussion regarding the actual distribution of underlying prices is never-ending. Most recently, the debate has pursued some weight towards confronting both kurtosis and skew parameters beyond the standard log-normal price distribution assumptions. Nassem Taleb comes to mind here.
But before we delve further down this route, it is useful to understand the current usage and application of Black & Scholes option pricing assumptions. The continuous and closed-form solution provided by the Black & Scholes model remains both highly used in contemporary finance and valuable as a founding base for derivative pricing and arbitrage theory.
The log-normal pricing assumption, if applied in current markets, therefore push volatility estimates away from a historical parameters and into the realm of core pricing metrics. The financial services industry has responded to this requirement for additional price-discovery with ongoing research and, in effect, a sub-market of supply and demand for volatility.
Illustrating the Skew
Goldman Sachs’ quantitative department presented both an illustration and a response to the issues confronting volatility skews on options market back in the early 90s. The following two diagrams illustrate the effect of such a skew by using binomial pricing trees as a reference point.
If one takes the tree branch length of a volatility tree as a constant probability then the following image illustrate how, from the market’s point of view, a skew can occur shifting the actual probability across each branch.
Had the distribution actually been modeled as above, but observed as below, then volatility measures across the higher branches would have recorded back as higher than those read from the lower branches, producing a volatility skew when plotted against underlying prices.
So How Does VIX Fit Into All This
As mentioned before, VIX is a constant maturity estimate over the underlying implied volatility of S&P 500 benchmark options (SPX). The constant maturity adjustment of VIX reflects another facet complicating the constant volatility assumption underpinning Black and Scholes. Term structures also impact variance measure in part due to changes of interest rate expectations, earnings profiles, et al., but also due to changing expectations of volatility over time.
The CBOE Volatility Index (VIX) now is an up-to-the-minute market estimate of expected volatility that is calculated by using real-time S&P 500 Index (SPX) option bid/ask quotes. VIX uses near-term and next-term out-of-the money SPX options with at least 8 days left to expiration, and then weights them to yield a constant, 30-day measure of the expected volatility of the S&P 500 Index.
VIX Methodology, CBOE
As for VIX options, their own implied volatility estimates are then further impacted. As the title of this post would suggest, the options over VIX are in essence measuring the implied volatility of implied volatility. Complications that arise are two folds: first, VIX is far afield from being log-normal – in fact it is mean-reverting, second, VIX options themselves have their own term structures that then compound back over VIX’ original complications.
Mean Reversion Over VIX Options
VIX, by definition, will never reach absolute or near zero unless one is willing to drop the reality of either a random-walk hypothesis completely or a Geometric Brownian Motion assumption that underpin stock market prices. Simply put, as long as the underlying SPX options are variable, which it is predisposed to be, then VIX will always come through above zero. This already pushes it apart from log-normality assumptions that imply the potential for low to near-zero pricing.
This then implies that there is a boundary state above zero which must exist and, furthermore, would therefore be more highly populated than suggested by standard log-normal distributions. In addition, VIX is observed as mean-reverting by exerting back a return motion due to the very nature of volatility.
VIX’s own volatility is, in effect, pulled back by the reality of the SPX’ own volatility: if volatility over the SPX were consistently higher then this would presume the possibility of sequential +/- 10% moves. However, though equity markets may move on an ongoing basis higher, their own tendency to be random implies the unlikely scenario of ongoing sequential large scale moves.
Yet, since 1990 the largest 1-day move in SPX has been -6.9%, and price changes of at least ±5% have occurred only 8 times.
CBOE VIX Options FAQ
Using Black & Scholes As Illustration of Skew
It might seem odd, therefore, that we should then revert back to the Black and Scholes option pricing model to better illustrate the actual skew over VIX options market. However, the Black and Scholes model provides a simplified construct under which to derive implied volatility given its direct impact on theoretical pricing estimates.
The following three graphs illustrate the effective skew in place over VIX options with near expiry and over subsequent expiry periods as of close of trade 11 March 2011. Implied volatility was calculated using the Black and Scholes model to match back a theoretical price with the current market cleared rate. Details of the calculation are available from the following results list.
The three dimensional volatility surface was constructed from the average of both Put and Call implied volatility, when both of these were already priced by the market, or, when market data was insufficient, were either interpolated across their term structure (highlighted in yellow in the table below) or across term structures (highlighted in red). No extrapolation was used to refine the curve beyond known priced strikes.
The Volatility ‘Smirk’ of Near Expiry
The Future Implied Volatility Skew
The Future Implied Volatility Skew – Changed Perspective
The Future Implied Volatility Skew – Tabular Form
Can a VIX Option Fair-Price be Theoretically Calculated?
Mean-reversion is not a phenomenon unique to VIX. Indeed, a number of other economic and financial variables are considered to have strong mean-reversion factors in their distribution patterns that can not be accounted for by standard log-normal distribution models. GDP growth rates and interest rates are but two examples that come to mind. Continue Reading…