*Introduction*

2018 will be remembered by many investors as a difficult year. In sharp contrast to 2017, when S&P500 was performing spectacularly (+19.4% YoY), 2018 brought a bitter disappointment to many. Not only in terms of absolute performance (-7% YoY), driven heavily by the impressive sell-off in the last quarter of the year, but also in terms of volatility.

In this article we would like to discuss volatility and analyse arguments supporting two seemingly contradictory strategies that employ betting on future volatility. Both of them are bullish with respect to the S&P500 performance. Where they differ is the approach towards utilizing VIX futures. One of the strategies tries to hedge exposure towards a possible recession by adding a rolling hedge using long position in futures on VIX. Those securities are traded on the CBOE futures exchange since 2004. The other trading idea assumes short position on VIX in order not to hedge but to enhance the returns of the portfolio. The innovative element here is the attempt to minimize the tail risk associated with destructive volatility spikes that usually accompany market crashes. The reason for using VIX futures (for this analysis 1 month-maturity futures are used, rolled down to provide continuity) and not VIX spot prices is explained in the following paragraphs.

*CBOE VIX Index *

In order to start the analysis, a formal introduction to the VIX index is required. According to CBOE, VIX Index is the measure of “market expectations of near-term volatility conveyed by S&P 500 stock index option prices”. What does it mean in practice?

The VIX Index is a measure of the expected volatility of the US stock market. It is a continuous, 30-day expected volatility of the stock market, derived from real-time, mid-quote prices of S&P 500 Index call and put options. The model used for calculating daily values of VIX is described below but the simple intuition is as following: market participants express their beliefs regarding future market moves by generating demand for put and call options, This provides a market price for those instruments, which in turn can be used to calculate implied volatility using Black-Scholes model. Hence, VIX is directionally neutral, and moves in tandem with market sentiment.

*History*

The concept of creating an index that would measure implied volatility was first proposed by Prof. M. Brenner and Prof. Dan Galai in 1986 in their papers “New Financial Instruments for Hedging Changes in Volatility” and “Hedging Volatility in Foreign Currencies”. However, VIX was born only after the crunch of 1987 when the need to efficiently hedge volatility exposure led CBOE (Chicago Board Options Exchange) to create the CBOE Market Volatility Index, known today as VIX, in 1993. When the VIX was first introduced, it used S&P 100 at-the-money options. In 2003 the underlying asset was changed to S&P 500 options in order to better resemble broad market volatility.

However, VIX on its own is not a tradable security as it encapsulated a rolling basket of options as its underlying. To place bets on the behaviour of the index next degree derivative must have been created. Legend holds that Mark Cuban called Goldman Sachs in 2002 looking for a way to hedge his market risk. He wanted to go long on VIX as this index tended to soar when the market was plummeting. In an usual Goldman Sachs – fashion this instrument was quickly created benefiting from the wave of post-deregulation creativity. New VIX formula was proposed and implemented by CBOE. Eventually, the VIX futures were launched in 2004, followed by the VIX options in 2006.

However, it was only with the Great Recession in 2008 that the VIX saw huge growth in trading volumes. This was largely a psychological effect as the VIX hit an all-time-high of $80, when every other asset was crashing. At the same time, competing products entered the market, both ways, from the long-vol (VXX, UVXY, TVIX) to the short-vol (XIV, SVXY, ZIV) with varying levels of leverage.

The aftermath of the crisis saw a long period of economic healing, boosted by low interest rates, inflation and volatility. Short-vol became the overcrowded trade of the decade, luring investors with seemingly ludicrous returns (XIV 10-year return: 1,518%).

As trades picked nickels, the metaphorical steam-roller finally came crashing. On the February 5th 2018 Dow Jones plummeted by 1175 points, worst absolute daily downturn in history. The crash and following panic led to an astonishing spike in the VIX as it more than doubled from $17.31 to $37.32. The sweet dreams of the short-vol traders came to an end, with the XIV falling more than 90% in a single day. The issuer Credit Suisses came in and sterilize the product. Traders had to take a 90% haircut.

*Mechanics*

Here we would like to get deeper into the underlying model for the VIX spot price, being also the expected annualised variation of the broad market.

Cboe Options Exchange calculates the VIX Index using standard and weekly SPX options that are listed for trading on Cboe Options. Only SPX options with Friday expirations are used to calculate the VIX Index (Cboe lists SPX options that expire on days other than Fridays as well, but non-Friday SPX expirations are not used to calculate the VIX Index). Only SPX options with more than 23 days and less than 37 days to the expiration are used to calculate the VIX Index. These SPX options are then weighted to yield a constant, 30-day measure of the expected volatility of the S&P 500 Index.

The generalized formula used in the VIX Index calculation is:

Where:

σ | VIX divided by 100 |

T | Time to expiration |

F | forward index level derived from index option prices |

K_{0} |
first strike below the forward index level, F |

K_{i} |
strike price of i^{th} out-of-the-money option; a call if K_{i} > K_{0} and a put if K_{i} < K_{0}; both put and call if K_{i} = K_{0} |

K_{i} |
interval between strike prices – half the difference between the strike on either side of K_{i}:
(Note: K for the lowest strike is simply the difference between the lowest strike and the next higher strike. Likewise, K for the highest strike is the difference between the highest strike and the next lower strike.) |

R | Risk-free interest rate to expiration |

Q(K_{i}) |
The midpoint of the bid-ask spread for each option with strike K_{i} |

More on mechanics of the calculation can be found in the VIX paper by CBOE.

Given that implied volatility, in general, exceeds realized volatility, the VIX has a positive skew. Statistically speaking, this implies a greater chance of positive outcomes. At the same time, VIX’s delta is convex to negative returns. This means that positive movements give little or no changes to the prices, while negative movements causes larger spikes. This agrees with behavioural economics in that investors reach more to negative changes.

The index measures market expectations towards volatility for the following 30 days with at-the-money options. This is backed mathematically by the Black-Scholes model pricing options, which famously states that implied volatility plays a huge role in determining the price.

Interestingly, empirical data has shown VIX’s tendencies for mean reversion, abandoning extremi for the medium-term average. That said, this average fluctuates incessantly.

The VIX futures term structure details the implied volatility over the next few months. Like other asset classes, VIX futures in normality is in contango, with a positive gradient as the volatility increases with maturity. However, backwardation can occur when markets are in crises, as short-term volatility exceeds long-term volatility. Also, VIX futures are different from futures with standard, tradable underlying. Since it is not possible to directly trade VIX at the spot price the price of VIX futures has much smaller predictive value. This is due to the fact that without tradable underlying, arbitrage-based futures pricing does not apply; hence the price of futures reflects future spot VIX in more relaxed fashion. Instead of assuming that VIX spot in 1 month will be trading at the level of today’s 1-month maturity future price, it is often a reverse relation. In 1 month futures will converge to today’s vix spot price. This is due to aforementioned mean reversion as well as the usual market tendency to overstate volatility.

Finally, VIX level has a very straightforward interpretation. It can be well illustrated with an example. VIX level of, lets say, $20, indicates that the term structure and pricing of options on S&P500 indicate a price movement of 20% of the underlying (S&P500) during the next year. To get monthly variance VIX has to be divided by the square root of 12, and for daily changes it is a square root of 262 (average number of trading days during one year). Therefore, the VIX at $20 is a bet on monthly variance of 5.77% and daily variance of 1.23%. Given this formula we can derive that the spike in 2008 up to $80 was representing market belief that S&P500 will drop by 4.94% the next day and 23.09% during the following month.

*First Strategy: Long broad market + Long VIX Futures *

First approach to VIX comes from the paper by W.J. Heslinga (2012) “Tactical Asset Allocation with VIX Futures”. Proposed portfolio focuses on taking the advantage of VIX futures negative correlation with the broad market. Long position on the broad market is taken by constructing theoretical portfolio as following:

- 60.5% Equity represented by The S&P 500 Index (S&PCOMP)
- 30.5% Bonds represented by Barclays Capital US Aggregate Index (LHAGGBD)
- 1.3% High Yield Bonds represented by Barclays Capital US High Yield Index (LHYIELD)
- 1.2% Hedge Funds represented by HFRX Global Hedge Fund Index (HFRXHF$)
- 0.4% Commodities represented by S&P GSCI Index (GSCITOT)
- 4.5% Real Estate represented by S&P US REIT Index (SBBRUSL)
- 1.6% Private Equity represented by S&P Listed Private Equity Index (SPLPEI$)

Portfolio weights were motivated by Markowitz Mean-Variance analysis. For the details we recommend checking the original paper. Analysis is conducted using data spanning from 2005 to 2011. First date is when VIX futures obtained liquidity sufficient for semi-frictionless trading. The other leg of this strategy is long position in VIX futures. For the sake of this analysis two long VIX futures positions are discussed. First of them assumes allocation of 2.5% of capital to this security, the other analyses 10% allocation. Historical prices of this securities can be traced using the ticker CVXCS00 on the Reuters DataStream.

*Motivation*

Biggest advantage of implementing VIX contracts for hedging purposes is their negative correlation with the broad market. Not only it is negative, it gets more negative during the times of financial stress.

From the perspective of an investor a very interesting statistical relation can be observed. Correlation between asset classes experience changes between normal, calm market and crisis, distressed market. First regime is described by numbers without brackets, second regine in brackets. For example: correlation between bonds and equity under first regime is equal to -0.29. This enables to decrease overall beta of the portfolio by diversification. However, under crisis regime this correlation decreases to -0.18. It means, that the **hedging effect of diversification is decreasing precisely when it is most needed. **

The opposite happens with VIX correlation. In this case, comparing VIX-equity correlation, we can see that first regime produces strongly negative correlation of -0.76. This result is favourable for hedging on its own. However, when crisis began in 2008 this negative correlation jumped to even more negative -0.87. This means, that VIX was the only security that benefited from the crisis in terms of providing hedge for the discussed portfolio.

*Results *

Over the analysed period our strategy produced following results:

It is very clear that adding VIX future position to analysed portfolio is very beneficial. Average return increased, variance decreased and portfolio has more positive skew. In addition, VIX futures experience smaller volatility than VIX spot itself and are more suited to provide hedge and diversification.

*Criticism*

This strategy has a very strong downside. It performs better than pure broad market long portfolio if and only if there is a financial stress on the market at some point. If not, longing VIX futures is an expensive process. Not only because of the negative correlation but also because of high prices attached to call options on relatively peaceful market. In other words, this strategy is expensive to implement and works well under specific circumstances.

*Second Strategy: Long S&P500 + Short VIX Futures*

The second strategy is originally proposed in the paper by A. Dondoni, D.M. Montagna and M. Maggi (2018) titled “Shorting volatility as a portfolio enhancing strategy”. Here the approach is radically different. Goal of the transaction is to benefit from well-performing S&P500 enhanced by continuously shorting VIX futures with 1 month to maturity.

*Why 1-month maturity future?*

VIX futures terms structure has specific properties. First of all, it is most of the time upward sloping so the futures are traded in contango. Because most of the time implied volatility captured by the VIX Index includes a risk premium which disappears at the time of maturity, selling those futures is, provided normal market environment, very profitable. What posed a great weakness for the first strategy is exactly the advantage of the second one.

To understand this behaviour behavioural economics might come in handy. The same as in the famous volatility smile, implied volatility of an underlying asset does not want to stay constant. Some might argue that high futures premium is due to the value of the insurance they provide against market shock. The truth lays probably in the middle with inherent risk-aversion of the investors that drives put options prices, and the VIX as a whole, up, and the tail risk that those contracts can hedge.

*The strategy*

Shorting VIX can be performed in 3 ways in order to enhance the gains on the overall portfolio. All 3 will be presented with the conclusion pointing at the most efficient one. All the calculations cover the period between 2005 and 2014.

*All-In*

First attempt is a classical “all-in” with VIX futures contract being a simple short rolling position in 1 month-maturity futures. Below are the results of this approach divided with respect of the percentage of capital allocated in the futures:

The results are positive. Adding short position on VIX futures with maturity of 1 month not only increases the return but also decreases the standard deviation of the portfolio.

*Short UX1, Long UX3*

Second approach tries to protect portfolio to adverse stock movements by adding long position on VIX futures with the maturity of 3 months. This way the benefit of limiting the losses is introduced and as long as term structure of the futures remains in contango, the time decay of value on the 3-month future (UX3) will be compensated by the gain on shorted UX1.

Here the results are weaker in absolute terms but variance of the portfolio decreases more providing very similar Sharpe ratios.

*Short UX1 with contrarian indicator*

Last strategy aspires to be the best of both worlds. In normal times it is exactly the same as the “all-in”, however it has one more parameter. The strategy monitors the difference between UX1 and VIX. This value is obtained for each day of trading. Another step is to calculate the long moving average of this parameter. Here 1-year windom was used (260 sessions). Finally, we calculate shorter moving average. Best results were obtained using 15 days MA. The signal is defined as the moment when the short moving average gets more distant from the long average by 2 standard deviation. If the signal is triggered our position on futures gets flipped for the duration of the next month (ie. if we were shorting volatility we are now longing it).

Thanks to the “flip” indicator the strategy would avoid heavy losses of 2008. Signals were send on October and November of that year producing the return of 45% instead of the loss of the same amount. However, signal was also send on November 2014 resulting in the loss of 14%.

This strategy is by all means the most profitable and obtains very favourable results compared to the two explained before.

*Conclusion*

Today, more than one year after the deadly spike, shorting VIX once again is incredibly popular. The number of open contracts is now minus 150 thousand, indicating net 150 thousand short position. It seems that market participants are more optimistic about the market in general. Of course, on the 3rd of February 2018 everyone was probably equally as optimistic.

The choice looks quite simple. If you believe in the strong bull market to come in the foreseeable future, short VIX. This is what most of market participants have been doing for some time now, with the record high short-volatility position. In particular, it has been for some time now subject of study whether the hedge fund industry can be considered as a “short put on volatility” (see Ang (2014) and Jurek and Stafford (2015)). The fact that all kinds of hedge funds (from macro to event driven, from relative value to merger arbitrage) but long-short ones have correlations which are statistically significant suggest this might be the case. It is also worth remembering how occasional falls in the stock market lead to huge losses in such cases, also causing such institutional players to default. Today’s situation, in any case, seem to suggest that record-high stock valuations are here to stay according to hedge funds players. So far, they proved right: at the beginning of the year volatility expectation was higher than the one that was actually witnessed by the market and thanks to a steep VIX futures curve the second strategy proposed in the article would have outperformed the S&P500. On the contrary, if you are not sure if you will be able to stomach the afternoon that stock market crashes, hedge and go long. One thing you have to remember: today investors are looking for returns. Safety seems to be just a background concern. Only the future will show how those approaches will perform.

Previously on volatility by Bocconi Students Investment Club: