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Why VIX Options are superior to S&P 500 Options - A comparison of the volatility surface

In a recently published article, the benefits of using VIX call options over SPX put options as hedging tools were discussed (remember: VIX and SPX have an inverse relationship, so an SPX put would correspond to a VIX call). While I did not go into detail, I mentioned that the outerperformance mainly stems from the unique volatility surface of the VIX options. Implied volatility is a major price driver, as it is an important input variable of the Black-Scholes model, with higher implied volatility pushing the price of an option up and vice versa. This article aims to shed more light on the behavior of the volatility surface of SPX put and VIX call options and shows why the very unique volatility surface of VIX options gives them a massive advantage over SPX put options.


The volatility surface – what it is and why we care

For those not familiar with volatility surfaces, here is a quick recap: The volatility surface shows the level of implied volatility (y-axis) for options on the same underlying, with different deltas (x-axis) and expiration dates (z-axis). The graph below illustrates the volatility surface of an SPX put option.



The delta of an option shows the sensitivity of its price with respect to a movement in the underlying. Consequently, an at the money option would have a delta of 0.5, and the further they are in the money, the higher the delta (and vice verse). Usually only at- and out-of the money options are displayed, but since the purpose of this article is to compare VIX and SPX options as hedging tools, only the surface of Put (for SPX) and Call (for VIX) options is displayed.

The z-axis shows how much time is left till the option expires. This article only examines options with expiration dates between 1 month and 2 months. The reason for choosing this time frame is that expirations between 0 days and 1 months would be very difficult to model with my methodology, and VIX options with an expiration of more than 2 months show a much smaller sensitivity to the underlying than options with a closer expiration, making them very ineffective hedging tools. The methodology is described in more detail in the appendix.

Having this sorted out, it is about time to examine the different factors that affect the volatility surface more closely: time, delta and market movements.


Impact of time till expiration on the surface

As the appeal of options is their asymmetric pay off profile (theoretically unlimited profits compared to a limited loss), the implied volatility of an option approaching expiration decreases, since the odds of making this (at least theoretically) unlimited profit decrease as well. As a result, the decreasing level of implied volatility accelerates the time decay of the option. This is a behavior that can be observed for options on most underlyings, such as the S&P 500 (=SPX) and the volatility surface below displays quite clearly, how implied volatility drops across different delta levels for SPX put options.



This observation does not hold true however for the surface of VIX options, which is shown below. The implied volatility of VIX options increases, the closer they are to their expiration date. This behavior is triggered by the combination of the mean reverting nature of the VIX index and the European-style exercise. The mean reversion of the index implies that even if the index moves massively in favor of the option, if the expiration is far away in the future, the index might revert back to its long-term mean by the time the option expires. Moreover, VIX options are always priced based on the respective VIX future, and as the sensitivity of VIX futures to the VIX spot price decreases, the later the futures expire, this also means that the option is less sensitive to a movement in the spot price. As a result, for VIX options it can be said that in case of a massive market move in favor of the option, an option that is closer to expiration is more valuable to the buyer.



Impact of delta on implied volatility levels

As the graphs already show, implied volatility is not constant across delta levels, making it necessary to examine its behavior across different levels. The first thing that becomes obvious is that the volatility surface usually has the steepest rise (or fall) for deep out of the money options. This is triggered by the fact that even if the option is deep out of the money, there is still a small probability of it moving back into the money.

With SPX options, again displayed below, the volatility surface is working against the investor, since the decrease in implied volatility accelerates the time decay of the option. As the volatility surface gets flatter when options are in the money, the impact of the decreasing implied volatility gets smaller, which leads towards a less severe time decay of the option. On average, between 2008 and 2018, the implied volatility for options with an expiration of 2 months was 1.17% higher than the implied volatility of options with a 1 month expiry, when the option has a delta of 0.25, 0.62% higher, with a delta of 0.5 and 0.22% higher with a delta of 0.75.



For VIX options, shown below, the exact opposite is true: The volatility surface is working in favor of the investor, since the increase in implied volatility curbs the time decay of the option. As the steepest increase can be experienced for out of the money options, a portfolio that is hedged with out of the money VIX options yields the best result. Just like the SPX surface, the surface of the VIX options is the steepest for out of the money options. In the simulated timeframe, options with a delta of 0.25 have on average an implied volatility level that is 10.89% higher for options with one month till expiration than for options with two months till expiration. The difference for options with a delta of 0.5 is 7.59% and for options with a delta of 0.75 is 6.41%. These numbers not only demonstrate that the volatility surface of VIX options is inverted, but they also show that the increase in implied volatility when approaching expiration is much steeper than the drop in implied volatility for SPX options.



Behavior of the volatility surface when markets are in distress

Another point that has to be made is that the volatility surface is not static. This means that not only are the levels of implied volatility affected by market movements, but also their relative difference. The previous analysis of the volatility surface has been made under normal market circumstances and these observations are true for most of the time. It is still important however, to demonstrate the impact of an extreme market movement on the volatility surface. The biggest one-day movement of the VIX (excluding the 1987 crash) happened on the 05.02.2018, which is why the volatility surface at this day is now examined more closely.

The graph below now shows the volatility surface of SPX put options at market close on that day. It immediately can be noticed that the volatility surface of SPX options became fully inverted and the implied volatility level of options expiring in one month now exceeds the implied volatility level of options expiring in two months. The reason for that is similar to the behavior of the volatility surface of the VIX under normal market conditions: The market expects the underlying to come back from its extreme level with the passing of time and reflects this in the increased level of implied volatility for the front-end options. Depending on the delta, it took between one week (out of the money deltas) and three weeks (in the money deltas) to revert back to the volatility surface that is observed under normal market circumstances. What this means for the portfolio is that during a period of elevated volatility, the time decay of the options is less severe than in a calm market environment.



When it comes to VIX options, the increase from the implied volatility level of two months options to the implied volatility level of one month options becomes even steeper for options that are either deep out of the money or deep in the money (displayed below). What is interesting however, is that implied volatility levels of options that are centered on the at-the-money level are now higher for the options which have their expiration date further in the future. While this might seem like a very unfavorable impact on the options in the portfolio, it is important to keep in mind that delta is dynamic: A VIX call option that had a delta of ~ 0.4 on the day prior to that event had a delta of ~ 0.8 at the close of the 5th February, due to the extreme VIX movement in its favor. The result is that in a time of elevated volatility, the increase from the 2 month implied volatility level to the 1 month implied volatility level becomes even steeper, acting in favor of the option buyer.



While it might on first sight seem like in periods of high volatility the SPX and the VIX option are both equally working in favor of the investor, the steepness of the increase should be kept in mind. For SPX options, the 1 month option implied volatility was higher by 7.28% for 0.25 delta options, 4.8% for 0.5 delta options and 3.4% for 0.75 delta options. This increase is already lower than the increase for VIX options under normal market circumstances, but the difference gets even more severe, when it is compared to the increase for VIX options in a market crash. The increase for VIX options on the 05.02.2018 was 9.27% for 0.25 delta options, -59.94% for 0.5 delta options and 116.75% for 0.75 delta options.


An important remark at this point is that when analyzing the volatility surface, it always has to be kept in mind that the delta does not stay constant over time. However, when the impact of time and the chosen delta was analyzed, the assumption was that the delta stays roughly the same, while the option is held in the portfolio. The reason for that is quite frankly that it takes a massive market move for a deep out of the money option (like the 0.25 delta VIX call), to change its delta. Therefore, it seems reasonable to assume that under normal market circumstances, the delta might change, but not enough to suddenly get into the money with a previously deep out of the money option. As a result, the impact of a massive market movement had to be analyzed separately.

Another important point I would like to mention before reaching the conclusion is that implied volatility is obviously not the only parameter that impacts the price of the option. This analysis was only made with respect to the volatility surface and left out other factors, such as the contango or backwardation of futures, as well as the general behavior of the option greeks over time.


Conclusion

The major factor that makes VIX options attractive, when compared with SPX options, is their volatility surface. Particularly out of the money options have an inverted volatility surface, where implied volatility gets higher, the closer the options come towards expiration. With implied volatility positively affecting the price of the option, the time decay of VIX options is curbed by their unique volatility surface, making them an attractive tool for hedging an equity portfolio.


Appendix: Methodology

The data used for the volatility surface are implied volatility levels for SPX puts and VIX calls with two months and one month till expiration for delta levels of (0.1, 0.25, 0.4, 0.5, 0.6, 0.75, 0.9), as displayed by bloomberg. Academia usually suggests to use a spline interpolation to generate the volatility surface, but since I only showed the volatility surface between 1 and 2 months till expiration, I used bilinear interpolation between the deltas and expiration dates. The effect discussed in this article is strongest for options with less than 2 months till expiration, which is why I left out options that have a later expiration date. Modelling the surface for less than 1 month till expiration would require not only more data points, but also a more sophisticated interpolation approach, since the decay (or rise) of implied volatility in the last few weeks before expiration is by no means a linear process.

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