volatility arbitrage – Relative Value Arbitrage

When Correlations Break or Hold: Strategies for Effective Hedging and Trading

It’s well known that there is a negative relationship between an equity’s price and its volatility. This can be explained by leverage or, alternatively, by volatility feedback effects. In this post, I’ll discuss practical applications to exploit this negative correlation between equity prices and their volatility.

A Trading Strategy Based on the Correlation Between the VIX and S&P500 Indices

This paper [1] examines the strong correlation in the S&P 500 and identifies trading opportunities when this correlation weakens or breaks down.

Findings

-The study covers the period from January 1995 to October 2020, utilizing 6,488 daily observations of the VIX and S&P500 indexes.

– In scenarios where the options market indicates increased drawdown risk with higher implied volatility but negative returns have not yet occurred, consider shorting the market.

– The signal to short the market occurs when the negative correlation between the S&P 500 and VIX is broken, and they start exhibiting a positive correlation.

– The test setup involves identifying one or two consecutive days with positive co-movement between the VIX and S&P 500, then setting the transaction date for the day after or at the close of the chosen date.

– Empirical results show that the strategy outperforms the S&P500 index over the 25-year period, achieving higher returns, lower systematic risk, and reduced volatility.

-The findings provide evidence that excess returns can be generated by timing the market using historical data, even after accounting for trading costs.

Reference

[1] Tuomas Lehtinen, Statistical arbitrage strategy based on VIX-to-market based signal, Hanken School of Economics

Optimal Hedging for Options Using Minimum-Variance Delta

Contrary to the first paper, Reference [2] focuses on the strong correlation between the S&P 500 and its volatility, designing an efficient scheme for hedging an options book.

The authors developed a so-called minimum variance (MV) delta. Essentially, the MV delta is the Black-Scholes delta with an additional adjustment term.

Findings

-Due to the negative relationship between price and volatility for equities, the minimum variance delta is consistently less than the practitioner Black-Scholes delta.

-Traders should under-hedge equity call options and over-hedge equity put options compared to the practitioner Black-Scholes delta.

-The study demonstrates that the minimum variance delta can be accurately estimated using the practitioner Black-Scholes delta and the historical relationship between implied volatilities and asset prices.

-The expected movement in implied volatility for stock index options can be approximated as a quadratic function of the practitioner Black-Scholes delta divided by the square root of time.

-A formula for converting the practitioner Black-Scholes delta to the minimum variance delta is provided, yielding good out-of-sample results for both European and American call options on stock indices.

-For S&P 500 options, the model outperforms stochastic volatility models and models based on the slope of the volatility smile.

-The model works less well for certain ETFs

Reference:

[2] John Hull and Alan White, Optimal Delta Hedging for Options, Journal of Banking and Finance, Vol. 82, Sept 2017: 180-190

Closing Thoughts

These two papers take opposing approaches: one exploits correlation breakdown, while the other capitalizes on the correlation remaining strong. However, they are not mutually exclusive. Combining insights from both can lead to a more efficient trading or hedging strategy.

Educational Video

This seminar by Prof. J. Hull delves into the second paper discussed above.

Abstract

The “practitioner Black-Scholes delta” for hedging equity options is a delta calculated from the Black-Scholes-Merton model with the volatility parameter set equal to the implied volatility. As has been pointed out by a number of researchers, this delta does not minimize the variance of a trader’s position. This is because there is a negative correlation between equity price movements and implied volatility movements. The minimum variance delta takes account of both the impact of price changes and the impact of the expected change in implied volatility conditional on a price change. In this paper, we use ten years of data on options on stock indices and individual stocks to investigate the relationship between the Black-Scholes delta and the minimum variance delta. Our approach is different from earlier research in that it is empirically-based. It does not require a stochastic volatility model to be specified. Joint work with Allan White.

Examining Contango and Backwardation in VIX Futures

In this post, I will continue exploring various aspects of the volatility index and the associated volatility futures.

Data

To conduct this study, data is essential. Below are the data sources:

Spot VIX: Yahoo Finance provides data but no longer allows direct downloads. With some programming, a workaround can be found, but the most convenient option is to use Barchart.

https://www.barchart.com/stocks/quotes/$VIX/price-history/historical

VX Futures: CBOE offers historical data in CSV format.

https://www.cboe.com/us/futures/market_statistics/historical_data/

Short-Term Futures Index:

While not directly utilized in this issue, I use this data to validate other ideas. For completeness, here is the download link:

https://www.spglobal.com/spdji/en/indices/indicators/sp-500-vix-short-term-index-mcap/#overview

Statistics for spot VIX and VX Futures

The table below provides statistics for the S&P 500 (tracked by SPY), spot VIX, and VX futures. It shows the percentage of days the S&P 500 index is up and the percentage of days VX futures are in contango, where the front-month futures price is lower than the next-month futures price.

From January 2013 to November 2024, the S&P 500 index was up 54.9% of the time, while VX futures were in contango most of the time (85.2%).

The next table presents the number of days VX futures were in backwardation while the spot VIX was in contango. Spot VIX in contango is defined as the 1M spot VIX being less than the 3M spot VIX.

From January 2013 to July 2024, this situation was very rare, occurring only 7% of the time. However, from August 1 to November 4, 2024, this divergence occurred with much higher frequency, 53% of the time.

This situation presented a high reward/risk trade opportunity. For instance, one could structure a trade to capitalize on the high likelihood of VX futures returning to contango and a decline in the overall volatility level. One potential trade is buying a put option in VXX. We’ll discuss this strategy in an upcoming webinar.

Seasonality of Volatility

With the holiday season approaching, the equity world often discusses the “Santa Rally.” This raises the question: Is there any seasonality in the volatility market?

The graph below shows the average and median monthly implied and historical volatilities. A clear seasonal pattern is observed, with low volatility between April and July and high volatility in October. However, for December, there is no discernable pattern—volatility can be either high or low during this period.

Educational Video

This webinar by Prof Andrew Papanicolaou covers fundamental concepts of VX futures, such as contango, backwardation, and roll yield. It also presents an approach to modeling the VX futures term structure.

Abstract:

We study VSTOXX, VSTOXX futures and VSTOXX exchange-traded notes (ETNs) econometrically. We find that different rates of mean reversion capture fluctuations in the short and long maturities, respectively. We fit an exponential Ornstein-Uhlenbeck (OU) model to the data and find it to capable of simulating ETN time series that have similar properties to the historical observed ETN time series. We compare these results to a similar study performed on ETNs and futures for VIX. We also look at the joint behavior of VIX and VSTOXX futures, and explore portfolio allocation strategies among ETNs for both markets.

Closing Thoughts

The volatility market offers unique insights and opportunities for investors. By understanding concepts like contango, backwardation, and seasonality, we can structure strategies with favorable risk-reward profiles.

Let me know your thoughts in the comments below.

Making Use of Information Embedded in VIX Futures Term Structures

With the U.S. election now over, the VIX futures term structure has normalized. It typically follows the spot VIX term structure. However, before the election, the futures term structure was in backwardation while the spot VIX was in contango most of the time. This is a rare occurrence. Below is a snapshot of the spot and futures term structures on September 26.

VIX futures term structure

In a future issue, I’ll present statistics and trade opportunities for such situations. In today’s issue, however, I will discuss two papers that develop trading systems for VIX futures.

Trading VIX Futures Using Neural Networks

Reference [1] explores the use of neural networks, a type of artificial intelligence, to trade VIX futures. The authors assume that the term structure of VIX futures follows a Markov model. An interesting aspect of this paper is that it made use of a utility function to generate trading signals. The authors also performed thorough out-of-sample testing using the k-fold cross-validation technique.

The Model

The trading strategy aims to maximize expected utility for a day-ahead horizon considering the current shape and level of the term structure.
Computationally, a deep neural network with five hidden layers models the functional dependence between the VIX futures curve, positions, and expected utility.
Out-of-sample backtests indicate that this method achieves good portfolio performance.

Validation

The standard procedure for training involves dividing the data into two blocks: one for in-sample training and the other for out-of-sample testing.
VIX futures curves from April 14th, 2008, to August 7th, 2019, are used for in-sample training, while the remaining curves from August 8th, 2019, to November 5th, 2020, are used for out-of-sample testing.
Since the out-of-sample test is based on a single portfolio run, good performance could be attributed to luck. Therefore the method of k-fold cross-validation is applied.

Reference

[1] M. Avellaneda, T. N. Li, A. Papanicolaou, G. Wang, Trading Signals In VIX Futures, Applied Mathematical Finance. 2021;28(3):275–298

Trading VIX Futures Using Machine Learning Techniques

Building on the first paper, Reference [2] investigates machine learning techniques for trading VIX futures. It proposed using Constant Maturity Futures (CMF) to generate trading signals for VIX futures. It applied machine learning models to create these signals.

Findings

The experiment results show that term structure features, such as μt and ∆roll, are highly effective in predicting the next-day returns of VIX CMFs and offer potential economic benefits.
The C-MVO strategy outperformed the benchmark rank-based long-short strategy in backtesting across most machine learning models, offering valuable insights for trading VIX CMFs.
Neural network models, particularly ALSTM, demonstrated the best performance in both prediction and backtesting.
Tree-based models showed no clear superiority, while the linear regression model, which only considers linear relationships, outperformed all other models.
The findings highlight the predictive power of term structure features for next-day returns in VIX CMFs.

Reference

[2] Wang S, Li K, Liu Y, Chen Y, Tang X (2024), VIX constant maturity futures trading strategy: A walk-forward machine learning study, PLoS ONE 19(4): e0302289

Closing Thoughts

These papers present trading systems developed using advanced techniques in machine learning and AI. As such, validation is critical. Techniques such as k-fold validation and walk-forward analysis should be carried out rigorously.

The research also suggests that there is valuable information embedded in the VIX futures term structure. In my opinion, “simple”, intuitive systems can be developed using VIX term structure that can provide decent risk-adjusted returns. Additionally, as I’ve discussed in one of my LinkedIn posts, the S&P 500 market generally leads the VIX market. Therefore, signals from the S&P 500 can also be used to trade VIX futures.

Let me know your thoughts in the comments below.

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.

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,

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