Correlation Between the VVIX and VIX indices

The VIX index is an important market indicator that everyone is watching. VVIX, on the other hand, receives less attention. In this post, we are going to take a look at the relationship between the VIX and VVIX indices.

While the VIX index measures the volatility risks, VVIX measures the volatility-of-volatility risks. Its calculation methodology is similar to the VIX’s except that instead of using SPX options it uses VIX options.

To study the relationship between these 2 indices, we first calculated the rolling 20-days correlation of the VIX and VVIX returns from January 2007 to March 2020. The median value of correlation is 0.807 and 25% quantile is 0.66

The figure below presents the rolling 20-days VIX/VVIX correlation for the last 2 years. We also superimposed SPY on the chart. We observe that the correlation is usually high but there are periods where it decreases significantly. The current period is one of those.

Correlation Between the VVIX and VIX indices
Correlation between the VVIX and VIX indices

The next figure shows the scatter plot of VVIX returns vs. VIX returns. It’s observed that there is a significant population where VIX and VVIX returns are of opposite signs.  We subsequently calculated the number of instances where VIX and VVIX move in the opposite direction. This indeed happens 22% of the time.

VVIX returns vs. VIX returns
VVIX returns vs. VIX returns

Some implications of this study are:

  • Although the correlation between VIX and VVIX appears to be high, there is a significant number of instances where VIX and VVIX move in the opposite direction. So it’s fair to say that VVIX follows separate price dynamics which is different from the VIX. In other words, VVIX prices in different risks.
  • Long VIX options or SPX back spreads are not always a good hedge for an equity portfolio. The hedge can break down.
  • At times it’s cheaper to hedge a long equity portfolio using SPX options; at times it’s cheaper using VIX options.
  • The speed of VIX mean reversion is greater when VIX is high as compared to when VIX is low.

Differences Between the VIX Index And At-the-Money Implied Volatility

When trading options, we often use the VIX index as a measure of volatility to help enter and manage positions. This works most of the time. However, there exist some differences between the VIX index and at-the-money implied volatility (ATM IV). In this post, we are going to show such a difference through an example. Specifically, we study the relationship between the implied volatility and forward realized volatility (RV) [1] of SP500. We utilize data from April 2009 to December 2018.

Recall that the VIX index

  • Is a model-independent measure of volatility,
  • It contains a basket of options, including out-of-the-money options. Therefore it incorporates the skew effect to some degree.

Plot below shows RV as a function of the VIX index.

Volatility trading strategies volatility arbitrage

We observe that a high VIX index will usually lead to a higher realized volatility. The correlation between RV and the VIX is 0.6397.

For traders who manage fixed-strike options, the use of option-specific implied volatilities, in conjunction with the VIX index, should be considered. In this example, we calculate the one-month at-the-money implied volatility using SPY options. Unlike the VIX index, the fixed-strike volatilities are model-dependent. To simplify, we use the Black-Scholes model to determine the fixed-strike, fixed-maturity implied volatilities.  The constant-maturity, floating-strike implied volatilities are then calculated by interpolation.

Plot below shows RV as a function of ATM IV.

Volatility trading strategies implied volatility

We observe similar behaviour as in the previous plot. However, the correlation (0.5925) is smaller. This is probably due to the fact that ATM IV does not include the skew.

In summary:

  • There are differences between the VIX index and at-the-money implied volatility.
  • Higher implied volatilities (as measured by the VIX or ATM IV) will usually lead to higher RV.

Footnotes

[1] In this example, forward realized volatility is historical volatility shifted by one month.

Is Asset Dynamics Priced In Correctly by Black-Scholes-Merton Model?

A lot of research has been devoted to answering the question: do options price in the volatility risks correctly? The most noteworthy phenomenon (or bias) is called the volatility risk premium, i.e. options implied volatilities tend to overestimate future realized volatilities.  Much less attention is paid, however, to the underlying asset dynamics, i.e. to answering the question: do options price in the asset dynamics correctly?

Note that within the usual BSM framework, the underlying asset is assumed to follow a GBM process. So to answer the above question, it’d be useful to use a different process to model the asset price.

We found an interesting article on this subject [1].  Instead of using GBM, the authors used a process where the asset returns are auto-correlated and then developed a closed-form formula to price the options. Specifically, they assumed that the underlying asset follows an MA(1) process,

volatility trading strategies mean reverting asset

where β represents the impact of past shocks and h is a small constant. We note that and in case β=0 the price dynamics becomes GBM.

After applying some standard pricing techniques, a closed-form option pricing formula is derived which is similar to BSM except that the variance (and volatility) contains the autocorrelation coefficient,

volatility trading strategies trending asset

From the above equation, it can be seen that

  • When the underlying asset is mean reverting, i.e. β<0, which is often the case for equity indices, the MA(1) volatility becomes smaller. Therefore if we use BSM with σ as input for volatility, it will overestimate the option price.
  • Conversely, when the asset is trending, i.e. β>0, BSM underestimates the option price.
  • Time to maturity, τ, also affects the degree of over- underpricing. Longer-dated options will be affected more by the autocorrelation factor.

References

[1] Liao, S.L. and Chen, C.C. (2006), Journal of Futures Markets, 26, 85-102.

VIX Mean Reversion After a Volatility Spike

In a previous post, we showed that the spot volatility index, VIX, has a strong mean reverting tendency. In this follow-up installment we’re going to further investigate the mean reverting properties of the VIX. Our primary goal is to use this study in order to aid options traders in positioning and/or hedging their portfolios.

To do so, we first calculate the returns of the VIX index. We then determine the quantiles of the return distribution. The table below summarizes the results.

Quantile 50% 75% 85% 95%
Volatility spike -0.31% 3.23% 5.68% 10.83%

We next calculate the returns of the VIX after a significant volatility spike. We choose round-number spikes of 3% and 6%, which roughly correspond to the 75% and 85% quantiles, respectively. Finally, we count the numbers of occurrences of negative VIX returns, i.e. instances where it decreases to below its initial value before the spike.

Tables below present the numbers of occurrences 1, 5, 10 and 20 days out. As in a previous study, we divide the volatility environment into 2 regimes: low (VIX<=20) and high (VIX>20). We used data from January 1990 to December 2017.

VIX spike > 3%
Days out All cases VIX<=20 VIX>20
1 56.1% 54.9% 58.1%
5 59.7% 58.4% 61.8%
10 60.3% 57.0% 65.8%
20 61.6% 57.0% 69.5%

 

VIX spike > 6%
Days out All cases VIX<=20 VIX>20
1 58.2% 56.9% 60.3%
5 62.5% 62.0% 63.3%
10 64.0% 61.7% 67.6%
20 65.9% 61.4% 73.2%

We observe the followings,

  • The greater the spike, the stronger the mean reversion. For example, for all volatility regimes (“all cases”), 10 days after the initial spike of 3%, the VIX decreases 60% of the time, while after a 6% volatility spike it decreases 64% of the time,
  • The mean reversion is stronger in the high volatility regime. For example, after a volatility spike of 3%, if the VIX was initially low (<20), then after 10 days it reverts 57% of the time, while if it was high (>20) it reverts 66% of the time,
  • The longer the time frame (days out), the stronger the mean reversion.

The implication of this study is that

  • After a volatility spike, the risk of a long volatility position, especially if VIX options are involved, increases. We would better off reducing our vega exposure or consider taking profits, at least partially,
  • If we don’t have a position prior to a spike, we then can take advantage of its quick mean reversion by using bounded-risk options positions.

Is a 4% Down Day a Black Swan?

On February 5, the SP500 experienced a drop of 4% in a day. We ask ourselves the question:  is a one-day 4% drop a common occurrence? The table below shows the number of 4% (or more) down days since 1970.

   4% down 4% down and bullish
From 1970 40 5

 

On average, a 4% down day occurred each 1.2 years, which is probably not a rare occurrence.

We next counted the number of days when the SP500 dropped 4% or more during a bull market. We defined the bull market as price > 200-Day simple moving average.  Since 1970 there have been 5 occurrences, i.e. on average once every 10 years. We don’t know whether this qualifies as a black swan event, but a drop of more than 4% during a bull market is indeed very rare.

The table below shows the dates of such  occurrences. It’s interesting to note that before the February 5 event, the last two 4% drops when price> 200-day SMA occurred around the dot-com period.

Date %change
September 11, 1986 -4.8
October 13, 1989 -6.1
October 27, 1997 -6.9
April 14, 2000 -5.8
February 5, 2018 -4.1

 

Correlation Between SPX and VIX

Last week, many traders noticed that there was a divergence between SPX and VIX. It’s true if we look at the price series. Graph below shows the 20-day rolling correlation between SPX and VIX prices for the last year. We can see that the correlation has been positive lately.

volatility arbitrage trading strategies
20-day rolling correlation SPX-VIX prices, ending Jan 26 2018

However, if we look at the correlation between SPX daily returns and VIX changes, it’s more or less in line with the long term average of -0.79. So the divergence was not significant.

volatility trading strategies based on correlation
20-day rolling correlation SPX return -VIX changes ending Jan 26 2018

The implied volatility (VIX) actually tracked the realized volatility (not shown) quite well. The latter happened to increase when the market has moved to the upside since the beginning of the year.

Mean Reverting and Trending Properties of SPX and VIX

In the previous post, we looked at some statistical properties of the empirical distributions of spot SPX and VIX. In this post, we are going to investigate the mean reverting and trending properties of these indices. To do so, we are going to calculate their Hurst exponents.

There exist a variety of techniques for calculating the Hurst exponent, see e.g. the Wikipedia page. We prefer the method presented in reference [1] as it could be related to the variance of a Weiner process which plays an important role in the options pricing theory. When H=0.5, the underlying is said to be following a random walk (GBM) process. When H<0.5, the underlying is considered mean reverting, and when H>0.5 it is considered trending.

Table below presents the Hurst exponents for SPX, VIX and VXX. The data used for SPX and VIX is the same as in the previous post. The data for VXX is from Feb 2009 to the present. We display Hurst exponents for 2 different ranges of lags: short term (5-20 days) and long term (200-250 days).

Lag (days) SPX VIX VXX
5-20 0.45 0.37 0.46
200-250 0.51 0.28 0.46

We observe that SPX is mean reverting in a short term (average H=0.45) while trending in a long term (average H=0.51). This is consistent with our experience.

The result for spot VIX (non tradable) is interesting. It’s mean reverting in a short term (H=0.37) and strongly mean reverting in a long term (H=0.28).

As for VXX, the result is a little bit surprising. We had thought that VXX should exhibit some trendiness in a certain timeframe.  However, VXX is mean reverting in both short- and long-term timeframes (H=0.46).

Knowing whether the underlying is mean reverting or trending can improve the efficiency of the hedging process.

References

[1] T. Di Matteo et al. Physica A 324 (2003) 183-188

Statistical Distributions of the Volatility Index

VIX related products (ETNs, futures and options) are becoming popular financial instruments, for both hedging and speculation, these days.  The volatility index VIX was developed in the early 90’s. In its early days, it led the derivative markets. Today the dynamics has changed.  Now there is strong evidence that the VIX futures market leads the cash index.

In this post we are going to look at some statistical properties of the spot VIX index. We used data from January 1990 to May 2017. Graph below shows the kernel distribution of spot VIX.

volatility trading strategies: VIX distribution is not normal
Kernel distribution of the spot VIX index

It can be seen that the distribution of spot VIX is not normal, and it possesses a right tail.

We next look at the Q-Q plot of spot VIX. Graph below shows the Q-Q plot. It’s apparent that the distribution of spot VIX is not normal. The right-tail behavior can also be seen clearly. Intuitively, it makes sense since the VIX index often experiences very sharp, upward spikes.

volatility arbitrage: Q-Q plot of spot volatility index
Q-Q plot of spot VIX vs. standard normal

It is interesting to observe that there exists a natural floor around 9% on the left side, i.e. historically speaking, 9% has been a minimum for spot VIX.

We now look at the distribution of VIX returns. Graph below shows the Q-Q plot of VIX returns. We observe that the return distribution is closer to normal than the spot VIX distribution. However, it still exhibits the right tail behavior.

Relative value arbitrage: distribution of VIX returns
Q-Q plot of VIX returns vs. standard normal

It’s interesting to see that in the return space, the VIX distribution has a left tail similar to the equity indices. This is probably due to large decreases in the spot VIX after sharp volatility spikes.

The natural floor of the spot VIX index and its left tail in the return space can lead to construction of good risk/reward trading strategies.

UPDATE: we plotted probability mass function of spot VIX on the log scale. Graph below shows that spot VIX spent most of its time in the 12%-22% (log(VIX)=2.5 to 3.1) region during the sample period.

options trading strategies: distribution of the volatility index
Kernel distribution on the log scale

Is Volatility of Volatility Increasing?

Last Wednesday, the SP500 index went down by just -1.8%, but in the volatility space it felt like the world was going to end; the volatility term structure, as measured by the VIX/VXV ratio, reached 1, i.e. the threshold where it passes from the contango to backwardation state. The near inversion of the volatility term structure can also be seen on the VIX futures curve (although to a lesser degree), as shown below.

VIX futures as at May 16 (blue line) and May 17 2017 (black line). Source: Vixcentral.com

With only -1.8% change in the underlying SPX, the associated spot VIX went from 10.65 to 15.59, a disproportional increase of 46%. The large changes in the spot and VIX futures were also reflected in the prices of VIX ETNs. For example, SVXY went down by about 18%, i.e. 9 times bigger than the SPX return. We note that in normal times SVXY has a beta of about 3-4 (as referenced to SPY).

So was the spike in volatility normal and what happened exactly?

To answer these questions, we first looked at the daily percentage changes of the VIX as a function of SPX returns. The Figure below plots the VIX changes v.s.  SPX daily returns. Note that we plotted only days in which the underlying SPX decreased more than 1.5% from the previous day’s close. The arrow points to the data point of last Wednesday.

Daily VIX changes v.s. SPX returns

A cursory look at the graph can tell us that it’s rare that a small change in the underlying SPX caused a big percentage change in the VIX.

To quantify the probability, we counted the number of occurrences when the daily SPX returns are between -2.5% and -1.5%, but the VIX index experienced an increase of 30% or greater. The data set is from January 1990 to May 19 2017, and the total sample size is 6900.

There are only 11 occurrences, which means that volatility spikes like the one of last Wednesday occurred only about 0.16% of the time. So indeed, such an event is a rare occurrence.

Table below presents the dates and VIX changes on those 11 occurrences.

Date VIX change
23/07/1990 0.515
03/08/1990 0.4068
19/08/1991 0.3235
04/02/1994 0.4186
30/05/2006 0.3086
27/04/2010 0.3057
25/02/2013 0.3402
15/04/2013 0.432
29/06/2015 0.3445
09/09/2016 0.3989
17/05/2017 0.4638

 

But what happened and what caused the VIX to go up that much?

While accurate answers must await thorough research, based on other results (not shown) and anecdotal evidences we believe that the rise in the popularity of VIX ETNs, and the resulting exponential increase in short interest, has contributed greatly to the increase in the volatility of volatility.

We also note that from the Table above, out of the 11 occurrences, more than half (6 to be precise) happened after 2010, i.e. after the introduction of VIX ETNs.

With an increase in volatility of volatility, risk management became more critical, especially if you are net short volatility and/or you have a lot of exposure to the skew (dGamma/dSpot).

Forward Volatility and VIX Futures

Last week the VIX index was more or less flat, the contango was favorable, and yet VIX ETF such as XIV, SVXY underperformed the market. In this post we will attempt to find an explanation.

As briefly mentioned in the footnotes of the blog post entitled “A Volatility Term Structure Based Trading Strategy”, VIX futures represent the (risk neutral) expectation values of the forward implied volatilities and not the spot VIX. The forward volatility is calculated as follows,

Using the above equation, and using the VIX index for σ0,t , VXV for σ0,T, we obtain the 1M-3M forward volatility as shown below.

1M-3M forward volatility from October 2016 to February 2017

Graph below shows the prices of VXX (green and red bars) and VIX April future (yellow line) for approximately the same period. Notice that their prices have increased since mid February, along with the forward volatility, while the spot VIX (not shown) has been more or less flat.

VXX (green and red bars) and VIX April future (yellow line) prices

If you define the basis as VIX futures price-spot VIX, then you will observe that last week this basis widened despite the fact that time to maturity shortened.

In summary, VIX futures and ETF traders should pay attention to forward volatilities, in addition to the spot VIX. Forward and spot volatilities often move together, but they diverge from time to time. The divergence is a source of risk as well as opportunity.