The article below is provided by Michael McCarty. Mr. McCarty is the founding member and chief strategist of Differential Research.

The fair price of a derivative security is typically measured and determined against an economically equivalent replicating portfolio, e.g. a security or portfolio that has the same cash flows as the derivative security.

For example, the price of an S&P 500® Index (S&P 500) future is equal to the present value (including interest and dividends) of a properly weighted portfolio of 500 component stocks delivered at the time of the futures expiration. Any difference in the prices of the future and the replicating portfolio of stocks is rapidly erased by stock-index arbitrageurs taking positions in lower priced assets while simultaneously selling more expensive ones. This concept is well established and forms the basis for the calculation and broad media coverage of the pre-market S&P 500 Index Fair Value.

With the proliferation of derivative instruments based on the S&P 500 Index, a myriad of arbitrage strategies exist involving the S&P 500 future, portfolio stocks in the S&P 500 Index, the SPDR® S&P 500 ETF (SPY) and, for purposes of this discussion, the future and the reversal/conversion market for the S&P 500 Index (SPX) options market. Different margin considerations may yield slightly different values for the future or forward value for the S&P 500 Index.

Much like how the S&P 500 Index became recognized as the benchmark for equity returns, the CBOE Volatility Index® (VIX®) has become the benchmark for equity market implied volatility.

VIX Futures

While an S&P 500 Index futures contract represents a portfolio of stocks, the VIX futures contract represents a portfolio of SPX options. As a consequence, the construction of a portfolio whose returns mimic a VIX future is not as intuitive and the calculation of its value not as direct.

Sources: CFE, Differential Research, LLC

www.cboe.com/micro/vix/ vixwhite.pdf. First, a description of what value a VIX futures contract represents will be provided and a discussion about deconstructing the VIX calculation to reveal its components will be provided. We will then introduce how a replicating portfolio could theoretically be constructed and priced. Next, we will describe how a close approximation can be reproduced with SPX options and publically available data. We will close with a more detailed examination of the limitations of our model and suggest avenues for potential refinement.

While the purpose of this discussion is to provide an alternative price for a VIX futures contract, during the process we will have also laid the foundation for a simplified volatility arbitrage model. Todays stock-index arbitrageurs who trade only a portion of the 500 stocks that comprise the S&P 500 Index consider their basket to have a competitive advantage. Similarly, volatility arbitrageurs seek to gain a competitive advantage through a refinement of this model and its limitations.

When discussing volatility, we start with the standard deviation (or the square-root) of the variance of a distribution of daily lognormal returns expressed on an annualized basis. For historical or realized volatility that equation is straightforward:

With the assumption that the mean return on a daily basis is not significantly different than zero, the equation is often reduced for simplicity to the zero-mean version:

Future Volatility

One variable that figures into options pricing is the markets estimate for future or implied volatility. However, the shape of the future distribution of lognormal returns is unknown. Consequently, any attempt to derive an estimate for future volatility from options implied volatilities using a model which assumes a known distribution is inaccurate.

The VIX calculation provides that each successive out-of-the-money options price properly weighted reflects the incremental contribution to volatility for returns exceeding the strike price regardless of the distributions shape.

Further, a VIX futures contract reflects the markets expectation for the final 30-days implied volatility derived from a portfolio of SPX options. But how is that portfolio created and valued?

Deconstructing a Single VIX Expiration

We start exactly 30-days before the expiration of SPX options, which is when a VIX future expires.

The VIX settlement value is determined by applying the VIX methodology as outlined in the VIX White Paper to a broad series of SPX options expiring the following month. A series of steps is used to identify which options will be used in the VIX calculation. Those options are then priced and then weighted according to the VIX calculation.

Where…

First, we find the options implied forward price (F) for the S&P 500 Index at that option series expiration by identifying the at-the-money SPX options, at which the absolute difference between the call and put price is smallest using the formula from the VIX White Paper:

Next, we choose the strike prices (Ki) for inclusion in our calculation selecting all puts below K0, the strike price below (F) the forward price, and calls greater than K0. For the strike K0 both the price for the call and the put are included. Successively lower puts and higher calls are selected until two strikes in a row have zero bids. No further options are included. This can be done using a Microsoft Excel spreadsheet and inputting a series of embedded IF statements.

Each month on VIX settlement days, a modified opening procedure for SPX options is used to facilitate price discovery and opening prices. The opening prices for SPX options or in their absence the opening bid-ask midpoint, for the selected strikes are inserted into the VIX calculation.

Note that the VIX calculation is the square root of the general equation multiplied by 100.

While the general form of the VIX calculation references Q (Ki), at expiration the bid-ask midpoint is only used in the absence of an opening trade price. Today, with the growing volume in VIX futures, the modified opening procedure used for SPX options on volatility index settlement days is extremely active and covers a broad range of strikes.

cfe.cboe.com/data/ EOSpage.aspx )

Beyond Expiration

We are able to calculate the value of a VIX future that settles for cash, but what about at any other time? Consider the application of the VIX calculation to the same series of SPX options the following morning. The range between double no-bids is tighter, so fewer strikes would be used in the VIX calculation. There are fewer trades so midpoints are used and finally it is now less than 30 days. Arguably the result of the calculation is the theoretical price of a (non-existent) 29-day volatility future.

Similarly, the calculation can be applied to all series of SPX options from which a series of annualized volatility estimates can be constructed–using various terms each approximately 30-days apart.

www.cboe.com/data/ volatilityindexes/ volatilityindexes.aspx .

Sources: CFE, Differential Research, LLC

The volatility values that we arrive at however represent annualized volatility for the life of the SPX options contracts and not our desired target volatility for the final 30-days.

Sources: CFE, Differential Research, LLC

A Simple Model:

With data available for SPX option expirations approximately 30-days apart, the incremental contribution to volatility for the final 30-days is typically a function of the implied volatility of that series and the previous series. Unfortunately, we cant just subtract one from the other. First, we must de-annualize to arrive at simple period implied volatility and then square implied volatility to arrive at implied variance. Because we are looking at independent random variables, variance is additive. If we subtract-29 days of variance from 59 days of implied variance we have 30 final days implied variance for the longer dated SPX option series. Squared and re-annualized to reflect 30 days, we now have a reasonable approximation for prices of a replicating portfolio for a VIX future expiring 30-days prior to the longer dated SPX option series. The replicating portfolio for a VIX future is comprised of long position in a series of SPX options in the targeted month and short positions in previous monthly series.

In similar fashion a longer dated SPX option strip, or longer dated volatility, could be replicated using a series of VIX futures and the strip of SPX options for the series with maturity less than 30 days.

Model Limitations

Our model is limited by the results of our reliance on publicly available data. We generally are forced to rely on the bid/ask mid-point due to the infrequent trading of many out-of-the money SPX options. The breadth of strikes included in the calculation may be significantly reduced by the increased occurrence of zero-bid strikes. Historically, the near simultaneous calculation of the VIX Index which only uses midpoints at the time of VIX expiration has yielded a result below the VIX settlement value. This suggests that either the midpoint is below the true value or many zero-bid options have non-zero values. The reliance on bid/ask midpoints likely understates the true price of volatility.

The VIX calculation remains a discrete calculation and assumes implied volatility of the range between strikes is constant. The difference in modeled implied volatility between strikes would suggest otherwise. The effect of this difference is less clear but most-likely overstates true implied volatility.

Finally, if we stack VIX futures to hedge longer dated volatility exposures, we are assuming that at expiration the next future has 30-days until expiration. Typically, when a VIX future ascends to the front month position it expires in 4 weeks or 28 days, although a couple of times a year it will expire in 5 weeks or 35 days. Most of the time VIX futures overlap by two days and occasionally there is a five-day gap

Sources: CFE, Differential Research, LLC

The risk of loss in trading commodity futures and options is substantial. Before trading, you should carefully consider your financial position to determine if futures trading is appropriate. When trading futures and/or options, it is possible to lose more than the full value of your account. All funds committed should be risk capital. Past performance is not necessarily indicative of future results.