The post VIX Mean Reversion After a Volatility Spike appeared first on Relative Value Arbitrage.
]]>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 implication of this study is that
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]]>The post A Simple System For Hedging Long Portfolios appeared first on Relative Value Arbitrage.
]]>Short at the close when Close of today < lowest Close of the last 10 days
Cover at the close when Close of today > lowest Close of the last 10 days
The Table below presents results for SPY from 1993 to the present. We performed the tests for 2 different volatility regimes: low (VIX<=20) and high (VIX>20). Note that we have tested other lookback periods and VIX filters, but obtained qualitatively the same results.
Number of Trades | Winner | Average trade PnL | |
All | 455 | 24.8% | -0.30% |
VIX<=20 | 217 | 23.5% | -0.23% |
VIX>20 | 260 | 26.5% | -0.37% |
It can be seen that the average PnL for all trades is -0.3%, so overall shorting SPY is a losing trade. This is not surprising, since in the short term the SP500 exhibits a strong mean reverting behavior, and in a long term it has a positive drift.
We still expected that when volatility is high, the SP500 would exhibit some momentum characteristics and short selling would be profitable. The result indicates the opposite. When VIX>20, the average trade PnL is -0.37%, which is higher (in absolute value) than the average trade PnL for the lower volatility regime and all trades combined (-0.23% and -0.3% respectively). This result implies that the mean reversion of the SP500 is even stronger when the VIX is high.
The average trade PnL, however, does not tell the whole story. We next look at the maximum favorable excursion (MFE). Table below summarizes the results
Average | Median | Max | |
VIX<=20 | 0.83% | 0.44% | 10.59% |
VIX>20 | 1.62% | 0.73% | 24.25% |
Despite the fact that the short SPY trade has a negative expectancy, both the average and median MFEs are positive. This means that the short SPY trades often have large unrealized gains before they are exited at the close. Also, as volatility increases, the average, median and largest MFEs all increase. This is consistent with the fact that higher volatility means higher risks.
The above result implies that during a sell-off, a long equity portfolio can suffer a huge drawdown before the market stabilizes and reverts. Therefore, it’s prudent to hedge long equity exposure, especially when volatility is high.
An interesting, related question arises: should we use options or futures to hedge, which one is cheaper? Based on the average trade PnL of -0.37% and gamma rent derived from the lower bound of the VIX, a back of the envelope calculation indicated that hedging using futures appears to be cheaper.
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]]>The post Is a 4% Down Day a Black Swan? appeared first on Relative Value Arbitrage.
]]>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 |
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]]>The post Correlation Between SPX and VIX appeared first on Relative Value Arbitrage.
]]>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.
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]]>The post Mean Reverting and Trending Properties of SPX and VIX appeared first on Relative Value Arbitrage.
]]>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
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]]>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 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.
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.
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]]>The post Are Short Out-of-the-Money Put Options Risky? Part 2: Dynamic Case appeared first on Relative Value Arbitrage.
]]>As a reference, results for the static case are replicated here:
ATM (K=100) | OTM (K=90) | |||||
Margin | Return | Variance | VaR | Return | Variance | VaR |
100% | 0.0171 | 0.0075 | 0.1940 | 0.0118 | 0.0031 | 0.1303 |
50% | 0.0370 | 0.0292 | 0.3844 | 0.0206 | 0.0133 | 0.2783 |
15% | 0.1317 | 0.3155 | 1.2589 | 0.0679 | 0.1502 | 0.9339 |
Table below summarizes the results for the dynamically hedged case
ATM (K=100) | OTM (K=90) | |||||
Margin | Return | Variance | VaR | Return | Variance | VaR |
100% | -0.0100 | 1.9171E-05 | 0.0073 | -0.0059 | 1.4510E-05 | 0.0062 |
50% | -0.0199 | 7.6201E-05 | 0.0145 | -0.0118 | 5.8016E-05 | 0.0121 |
15% | -0.0660 | 8.7943E-04 | 0.0480 | -0.0400 | 6.5201E-04 | 0.0424 |
From the Table above, we observe that:
It is important to note that given the same notional amount, a delta-hedged position is less risky than a static position. For example, the VaR of a static, cash-secured (m=100%) short put position is 0.194, while the VaR of the corresponding dynamically-hedged position is only 0.0073. This explains why proprietary trading firms and hedge funds often engage in the practice of dynamic hedging.
Finally, we note that while Value at Risk takes into account the tail risks to some degree, it’s probably not the best measure of tail risks. Using other risk measures that better incorporate the tail risks can alter the results and lead to different conclusions.
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]]>The post Are Short Out-of-the-Money Put Options Risky? appeared first on Relative Value Arbitrage.
]]>In this post, we will quantify and compare the risks of short OTM and ATM put options. We do so by performing Monte Carlo simulations and calculating the Value at Risk (VaR at 95% confidence interval) and variance of the return distribution. This strategy involves shorting unhedged puts. The return is determined as follows,
where P_{t0} and P_{T} denote the put prices at time zero and expiration respectively
K is the strike price; K=90, 100 for OTM and ATM options, respectively
m is a factor for margin. m=100% means that we sell a cash-secured put.
Note that the above equation takes into account the margin requirement in an approximate way. The exact formula for margin calculation depends on brokers, exchanges and countries. But we believe that using a more realistic margin calculation formula will not change the conclusion of this article.
We use the same simulation methodology and parameters as in the previous post. The parameters are as follows,
Parameter | Value |
Initial stock price | 100 |
Volatility | 20% |
Risk-free rate | 0.02 |
Drift | 0.07 |
Days in simulation | 252 |
Time step (day) | 1d |
Number of paths | 10000 |
Model | GBM |
It’s important to note that we focus here on the risks only. Hence we utilize the same values for the option’s implied volatility and the underlying’s realized volatility. In real life the puts implied volatilities are usually higher than the realized due to volatility and skew risk premia. This means that the strategy’s real-life expected return is normally higher. Our simulated return is more conservative.
The table below summarizes the risk characteristics of short put options.
ATM (K=100) | OTM (K=90) | |||||
Leverage | Return | Variance | VaR | Return | Variance | VaR |
100% | 0.0171 | 0.0075 | 0.1940 | 0.0118 | 0.0031 | 0.1303 |
50% | 0.0370 | 0.0292 | 0.3844 | 0.0206 | 0.0133 | 0.2783 |
15% | 0.1317 | 0.3155 | 1.2589 | 0.0679 | 0.1502 | 0.9339 |
We observe that for the same level of leverage, short OTM put positions are actually less risky than the ATM ones. For example, for m=100%, i.e. a cash-secured short put position, the variance and VaR of the OTM position are 0.0031 and 0.1303 respectively; they are smaller than the ATM option’s counterparts which are 0.0075 and 0.1940, respectively.
The risk comes from leverage. Let’s say, for example, a trader wants to sell OTM puts. Since he receives less premium for each put sold, he will likely increase the position size. For example, if he sells 2 OTM puts using leverage (m=50%), then the variance and VaR of his position are 0.0133 and 0.2783 respectively. Compared to selling 1 ATM cash-secured put, the risks increased substantially (VaR went from 0.194 to 0.2783)
In summary, ceteris paribus, a short OTM put option position is less risky than the ATM one. The danger arises when traders use excessive leverage.
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]]>The post Using a Market Timing Rule to Size an Option Position, A Static Case appeared first on Relative Value Arbitrage.
]]>Recall that an option position can be loosely divided into 2 categories: dynamic and static [1]
1-Dynamic: the option position is delta hedged dynamically; its PnL driver is the implied/realized volatility dynamics. The profit and loss at the option expiration depends on the volatility dynamics, but not on the terminal value of the spot price.
2-Static: the option position is left unhedged; the payoff of the strategy depends on the spot price at option expiration but not on the volatility dynamics, i.e. it’s path independent.
In this post, we will apply the 10M SMA rule to a static, unhedged position. All other parameters and rules are the same as in our previous post. Briefly, the trading rules are as follows
1-NoTiming: Sell 1-Month at-the-money (ATM) put option, no rehedge.
2-10M-SMA: we only sell an ATM put option if the closing price of the underlying is greater than its 10M SMA.
Our rationale for investigating this case is that because the payoff of a static, unhedged position depends largely on the direction of the market, the 10M SMA timing rule will have a higher chance of success.
Table below summarizes and compares results of the short put strategy with and without the application of the 10M SMA rule
Strategy | NoTiming | 10M-SMA |
Number of Trades: | 115 | 81 |
Percent Winners: | 0.77 | 0.77 |
Average P&L: | 65.69 | 62.77 |
Largest losing trade | -2702.50 | -1601.00 |
Largest winning trade | 652.00 | 451.50 |
Profit Factor (W/L): | 1.47 | 1.54 |
Worst drawdown | -5002.50 | -1897.00 |
Graph below shows the equity curves of the 2 strategies
As we can see from the Table and Graph, the 10M SMA rule performed better in this case. Although the win percentage and average PnL per trade remained approximately the same, the risks have been reduced significantly. The largest loss was reduced from $2.7K to $1.6K; drawdown decreased from $5K to $1.9K. As a result, the profit factor increased from 1.47 to 1.54.
In conclusion, the 10M SMA rule performs well in the case of a static, unhedged short put position. Using this rule, the risk-adjusted return of the trade was enhanced significantly.
Other related studies:
References
[1] N.N. Taleb, Dynamic Hedging: Managing Vanilla and Exotic Options, Wiley, 1997
[2] E. Sinclair, Volatility Trading, Wiley, 2nd edition, 2013
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]]>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).
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]]>