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.

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.

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.

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.

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

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.

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.

It is interesting to observe that there exists a naturalfloor 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.

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.

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.

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.

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).

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 σ_{0,t} , VXV for σ_{0,T}, we obtain the 1M-3M forward volatility as shown below.

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.

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.

In previous blog posts, we explored the possibility of using various volatility indices in designing market timing systems for trading VIX-related ETFs. The system logic relies mostly on the persistent risk premia in the options market. Recall that there are 3 major types of risk premium:

1-Implied/realized volatilities (IV/RV)

2-Term structure

3-Skew

A summary of the systems developed based on the first 2 risk premia was published in this post.

In this article, we will attempt to build a trading system based on the third type of risk premium: volatility skew. As a measure of the volatility skew, we use the CBOE SKEW index.

According to the CBOE website, the SKEW index is calculated as follows,

The CBOE SKEW Index (“SKEW”) is an index derived from the price of S&P 500 tail risk. Similar to VIX®, the price of S&P 500 tail risk is calculated from the prices of S&P 500 out-of-the-money options. SKEW typically ranges from 100 to 150. A SKEW value of 100 means that the perceived distribution of S&P 500 log-returns is normal, and the probability of outlier returns is therefore negligible. As SKEW rises above 100, the left tail of the S&P 500 distribution acquires more weight, and the probabilities of outlier returns become more significant. One can estimate these probabilities from the value of SKEW. Since an increase in perceived tail risk increases the relative demand for low strike puts, increases in SKEW also correspond to an overall steepening of the curve of implied volatilities, familiar to option traders as the “skew”.

Our system’s rules are as follows:

Buy (or Cover) VXX if SKEW >= 10D average of SKEW

Sell (or Short) VXX if SKEW < 10D average of SKEW

The table below summarizes important statistics of the trading system

Initial capital

10000

Ending capital

40877.91

Net Profit

30877.91

Net Profit %

308.78%

Exposure %

99.50%

Net Risk Adjusted Return %

310.34%

Annual Return %

19.56%

Risk Adjusted Return %

19.66%

Max. system % drawdown

-76.00%

Number of trades

538

Winners

285 (52.97 %)

The graph below shows the equity line from February 2009 to December 2016

We observe that this system does not perform well as the other 2 systems [1]. A possible explanation for the weak performance is that VXX and other similar ETFs’ prices are affected more directly by the IV/RV relationship and the term structure than by the volatility skew. Hence using the volatility skew as a timing mechanism is not as accurate as other volatility indices.

In summary, the system based on the CBOE SKEW is not as robust as the VRP and RY systems. Therefore we will not add it to our existing portfolio of trading strategies.

Footnotes

[1] We also tested various combinations of this system and results lead to the same conclusion.

A recent post on Bloomberg website entitled Rising VIX Paints Doubt on S&P 500 Rally pointed out an interesting observation:

While the S&P 500 Index rose to an all-time high for a second day, the advance was accompanied by a gain in an options-derived gauge of trader stress that usually moves in the opposite direction

The article refers to a well-known phenomenon that under normal market conditions, the VIX and SP500 indices are negatively correlated, i.e. they tend to move in the opposite direction. However, when the market is nervous or in a panic mode, the VIX/SP500 relationship can break down, and the indices start to move out of whack.

In this post we revisit the relationship between the SP500 and VIX indices and attempt to quantify their dislocation. Knowing the SP500/VIX relationship and the frequency of dislocation will help options traders to better hedge their portfolios and ES/VX futures arbitrageurs to spot opportunities.

We first investigate the correlation between the SP500 daily returns and change in the VIX index [1]. The graph below depicts the daily changes in VIX as a function of SP500 daily returns from 1990 to 2016.

We observe that there is a high degree of correlation between the daily SP500 returns and daily changes in the VIX. We calculated the correlation and it is -0.79 [2].

We next attempt to quantify the SP500/VIX dislocation. To do so, we calculated the residuals. The graph below shows the residuals from January to December 2016.

Under normal market conditions, the residuals are small, reflecting the fact that SPX and VIX are highly (and negatively) correlated, and they often move in lockstep. However, under a market stress or nervous condition, SPX and VIX can get out of line and the residuals become large.

We counted the percentage of occurrences where the absolute values of the residuals exceed 1% and 2% respectively. Table below summarizes the results

Threshold

Percentage of Occurrences

1%

17.6%

2%

3.9%

We observe that the absolute values of SP500/VIX residuals exceed 1% about 17.6% of the time. This means that a delta-neutral options portfolio will experience a daily PnL fluctuation in the order of magnitude of 1 vega about 17% of the time, i.e. about 14 times per year. The dislocation occurs not infrequently.

The Table also shows that divergence greater than 2% occurs less frequently, about 3.8% of the time. This year, 2% dislocation happened during the January selloff, Brexit and the US presidential election.

Most of the time this kind of divergence is unpredictable. It can lead to a marked-to-market loss which can force the trader out of his position and realize the loss. So the key in managing an options portfolio is to construct positions such that if a divergence occurs, then the loss is limited and within the allowable limit.

Footnotes:

[1] We note that under different contexts, the percentage change in VIX can be used in a correlation study. In this post, however, we choose to use the change in the VIX as measured by daily point difference. We do so because the change in VIX can be related directly to Vega PnL of an options portfolio.

[2] The scope of this post is not to study the predictability of the linear regression model, but to estimate the frequency of SP500/VIX divergence. Therefore, we applied linear regression to the whole data set from 1990 to 2016. For more accurate hedges, traders should use shorter time periods with frequent recalibration.

In previous posts, we presented 2 volatility trading strategies: one strategy is based on the volatility risk premium (VRP) and the other on the volatility term structure, or roll yield (RY). In this post we present a detailed comparison of these 2 strategies and analyze their recent performance.

The first strategy (VRP) is based on the volatility risk premium. The trading rules are as follows [1]:

Buy (or Cover) VXX if VIX index <= 5D average of 10D HV of SP500

Sell (or Short) VXX if VIX index > 5D average of 10D HV of SP500

The second strategy (RY) is based on the contango/backwardation state of the volatility term structure. The trading rules are as follows:

Buy (or Cover) VXX if 5-Day Moving Average of VIX/VXV >=1 (i.e. backwardation)

Sell (or Short) VXX if 5-Day Moving Average of VIX/VXV < 1 (i.e. contango)

Table below presents the backtested results from January 2009 to December 2016. The starting capital is $10000 and is fully invested in each trade (different position sizing scheme will yield different ending values for the portfolios. But the percentage return of each trade remains the same)

RY

VRP

Initial capital

10000

10000

Ending capital

179297.46

323309.02

Net Profit

169297.46

313309.02

Net Profit %

1692.97%

3133.09%

Exposure %

99.47%

99.19%

Net Risk Adjusted Return %

1702.07%

3158.54%

Annual Return %

44.22%

55.43%

Risk Adjusted Return %

44.46%

55.88%

Max. system % drawdown

-50.07%

-79.47%

Number of trades trades

32

55

Winners

15 (46.88 %)

38 (69.09 %)

We observe that RY produced less trades, has a lower annualized return, but less drawdown than VRP. The graph below depicts the portfolio equities for the 2 strategies.

It is seen from the graph that VRP suffered a big loss during the selloff of Aug 2015, while RY performed much better. In the next section we will investigate the reasons behind the drawdown.

Performance during August 2015

The graph below depicts the 10-day HV of SP500 (blue solid line), its 5-day moving average (blue dashed line), the VIX index (red solid line) and its 5-day moving average (red dashed line) during July and August 2015. As we can see, an entry signal to go short was generated on July 21 (red arrow). The trade stayed short until an exit signal was triggered on Aug 31 (blue arrow). The system exited the trade with a large loss.

The reason why the system stayed in the trade while SP500 was going down is that during that period, the VIX was always higher than 5D MA of 10D HV. This means that 10D HV was not a good approximate for the actual volatility during this highly volatile period. Recall that the expectation value of the future realized volatility is not observable. This drawdown provides a clear example that estimating actual volatility is not a trivial task.

By contrast, the RY strategy was more responsive to the change in market condition. It went long during the Aug selloff (blue arrow in the graph below) and exited the trade with a gain. The responsiveness is due to the fact that both VIX and VXV used to generate trading signals are observable. The graph below shows VIX/VXV ratio (black line) and its 5D moving average (red line).

In summary, we prefer the RY strategy because of its responsiveness and lower drawdown. Both variables used in this strategy are observable. The VRP, despite being based on a good ground, suffers from a drawback that one of its variables is not observable. To improve it, one should come up with a better estimate for the expectation value of the future realized volatility. This task is, however, not trivial.

References

[1] T Cooper, Easy Volatility Investing, SSRN, 2013

In previous 2 articles, we explored a volatility trading strategy based on the volatility risk premium (VRP). The strategy performed well up until August 2015, and then it suffered a big loss during the August selloff.

In this article, we explore another volatility trading strategy, also discussed in Ref [1]. This strategy is based on the volatility term structure [2].

It is well known that volatilities exhibit a term structure which is similar to the yield curve in the interest rate market. The picture below depicts the volatility term structure for SP500 as at August 31 2016 [3].

Most of the time the term structure is in contango. This means that the back months have higher implied volatilities than the front months. However, during a market stress, the volatility term structure curve usually inverts. In this case we say that the volatility term structure curve is in backwardation (a similar phenomenon exists in the interest rate market which is called inversion of the yield curve).

The basic idea of the trading strategy is to use the state (contango/backwardation) of the volatility term structure as a timing mechanism. Specifically, we go long if the volatility term structure is in backwardation and go short otherwise. To measure the slope of the term structure, we use the VIX and VXV volatility indices which represent the 1M and 3M implied volatilities of SP500 respectively.

The trading rules are as follows,

Buy (or Cover) VXX if 5-Day Moving Average of VIX/VXV >=1 (i.e. backwardation)

Sell (or Short) VXX if 5-Day Moving Average of VIX/VXV < 1 (i.e. contango)

The Table below presents the results

All trades

Long trades

Short trades

Initial capital

10000

10000

10000

Ending capital

177387.15

19232.01

168155.15

Net Profit

167387.15

9232.01

158155.15

Net Profit %

1673.87%

92.32%

1581.55%

Exposure %

99.44%

6.64%

92.80%

Net Risk Adjusted Return %

1683.22%

1390.38%

1704.17%

Annual Return %

46.07%

9.00%

45.05%

Risk Adjusted Return %

46.33%

135.54%

48.54%

All trades

30

15 (50.00 %)

15 (50.00 %)

Avg. Profit/Loss

5579.57

615.47

10543.68

Avg. Profit/Loss %

13.29%

2.29%

24.28%

Avg. Bars Held

64.53

9.8

119.27

Winners

14 (46.67 %)

4 (13.33 %)

10 (33.33 %)

Total Profit

208153.11

36602.85

171550.26

Avg. Profit

14868.08

9150.71

17155.03

Avg. Profit %

36.58%

25.17%

41.14%

Avg. Bars Held

129.64

14.25

175.8

Max. Consecutive

3

1

4

Largest win

71040.59

18703.17

71040.59

# bars in largest win

157

35

157

Losers

16 (53.33 %)

11 (36.67 %)

5 (16.67 %)

Total Loss

-40765.96

-27370.84

-13395.12

Avg. Loss

-2547.87

-2488.26

-2679.02

Avg. Loss %

-7.09%

-6.02%

-9.45%

Avg. Bars Held

7.56

8.18

6.2

Max. Consecutive

5

6

2

Largest loss

-9062.89

-8222.29

-9062.89

# bars in largest loss

6

8

6

Max. trade drawdown

-28211.89

-15668.21

-28211.89

Max. trade % drawdown

-23.97%

-18.20%

-23.97%

Max. system drawdown

-32794.28

-26555.13

-37915.18

Max. system % drawdown

-50.07%

-90.85%

-34.31%

Recovery Factor

5.1

0.35

4.17

CAR/MaxDD

0.92

0.1

1.31

RAR/MaxDD

0.93

1.49

1.41

Profit Factor

5.11

1.34

12.81

Payoff Ratio

5.84

3.68

6.4

Standard Error

12109.91

6401.13

12526.9

Risk-Reward Ratio

1.49

0.15

1.36

Ulcer Index

11.25

42.48

8.24

Ulcer Performance Index

3.62

0.08

4.81

Sharpe Ratio of trades

0.8

0.53

0.97

K-Ratio

0.0745

0.0073

0.0683

The graph below shows the portfolio equity from 2009 up to August 2016.

The annual rerun is 46% and the drawdown is 50%. There are 2 interesting observations

This strategy did not suffer a large loss like the VRP strategy during the August selloff of last year

Long volatility trades are profitable

In the next installment we will compare the 2 strategies, volatility risk premium and roll yield, in details.

References

[1] T Cooper, Easy Volatility Investing, SSRN, 2013

[2] Note that there is a so-called term structure risk premium in the options market that is not often discussed in the literature. The strategy discussed in this post, however, is not meant to exploit the term structure risk premium. It uses the term structure as a timing mechanism.

[3] The volatility term structure presented here is calculated based on VIX futures, which are the expectation values of 30-day forward implied volatility. Therefore, it is theoretically different from the term structure of spot volatilities which are calculated from SP500 options. Practically speaking, the 2 volatility term structures are highly correlated, and we use the futures curve in this article for illustration purposes.