A Simple System For Hedging Long Portfolios

In this post, we are going to examine a trading system with the goal of using it as a hedge for long equity exposure. To this end, we test a simple, short-only momentum system. The rules are as follows,

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.

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

Are Short Out-of-the-Money Put Options Risky? Part 2: Dynamic Case

This post is the continuation of the previous one on the riskiness of OTM vs. ATM short put options and the effect of leverage on the risk measures. In this installment we’re going to perform similar studies with the only exception that from inception until maturity the short options are dynamically hedged. The simulation methodology and parameters are the same as in the previous study.

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:

  • Similar to the static case, delta-hedged OTM put options are less risky than the ATM counterparts. However, the reduction in risk is less significant. This is probably due to the fact that delta hedging itself already reduces the risks considerably (see below).
  • Leverage also increases risks.

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.

 

Are Short Out-of-the-Money Put Options Risky?

Traders often debate whether short out-of-the-money (OTM) or at-the-money (ATM) puts are riskier. The argument for OTM put options being riskier is that their Speeds (or dGamma/dspot) are higher than the ATMs’ ones, thus the Gamma, which is negative, can increase (in absolute value) substantially during a market downturn.

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,

short put option

where Pt0 and PT 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.

Using a Market Timing Rule to Size an Option Position, A Static Case

In the previous installment, we discussed the use of a popular asset allocation/market timing rule (10M SMA rule hereafter) to size a short option position. The strategy did not work well as it was the case in traditional asset allocation. We thought that the poor performance was due to the fact that the 10M SMA rule is more of a market direction indicator that is not directly related to the PnL driver of a delta hedged position.

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

options trading strategies using market timing

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:

  • While researching the literature on this subject, I came across a similar study presented by E. Sinclair [2]. He showed that, for a delta hedged short strangle position, market timing based on the VIX index improved the results significantly. Since the VIX is a measure of volatility, its good performance is consistent with our understanding that for a delta hedged position, we should use a market timing indicator based on volatility and not on direction.
  • Pavel Bambásek also published similar studies recently. He used 200-Days SMA to time the market: http://www.bluetrader.cz/delta-hedging-ano-ne/

 

References

[1]  N.N. Taleb, Dynamic Hedging: Managing Vanilla and Exotic Options, Wiley, 1997

[2] E. Sinclair, Volatility Trading, Wiley, 2nd edition, 2013

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).

Using a Market Timing Rule to Size an Option Position

Position sizing and portfolio allocation have not received much attention in the options trading community. In this post we are going to apply a simple position sizing rule and see how it performs within the context of volatility trading.

An option position can be sized by using, for example, a Markov Model  where the size of the position can be a function of the regime transition probability [1]. While this is a research venue that we would like to explore, we decided to start with a simpler approach. We chose an algorithm that is intuitive enough for both quant and non-quant portfolio managers and traders.

We utilize the market timing rule proposed by Faber [2] who applied it to different asset classes in the context of portfolio allocation. The rule is as follows

Buy when monthly price > 10M SMA (10 Month Simple Moving Average)

Sell and move to cash when monthly price < 10M SMA

This remarkably simple timing rule has been used successfully by Faber and others.  It has proved to significantly improve portfolios’ risk-adjusted returns [3].

Within the context of volatility trading, we compare 2 option strategies

1-NoTiming: Sell 1-Month  at-the-money  (ATM) put option on every option expiration Friday.  The option is held  until maturity, i.e. for a month.  The position is kept delta neutral, i.e. it is rehedged at the end of every day.

2-10M-SMA: Similar to the above except that Faber’s timing rule is applied, i.e. we only sell an ATM put option if the closing price of the underlying is greater than its 10M SMA. Note, however, that unlike Faber,  here we define the end of month as the option expiration Friday, and not the calendar end of month.

A short discussion on the rationale for choosing a market timing rule is in order here. Within the context of portfolio allocation, the 10M SMA rule is used for timing the direction of the market, i.e. the PnL driver is mostly market beta. Our trade’s PnL driver is, on the other hand,  the dynamics  of the implied/realized volatility spread. But as shown in a previous post, the IV/RV volatility dynamics correlates highly with the market returns.  Therefore, we thought that we could use a directional timing strategy to size an options portfolio despite the fact that their PnL drivers are different, at least theoretically.

We tested the 2 strategies on SPY options from February 2007 to November 2016. Table below provides a summary of the trade statistics (average PnLs, winning/losing trades and drawdowns are in dollar).

Strategy NoTiming 10M-SMA
Number of Trades 115 81
Percent Winners 0.68 0.69
Average P&L 18.84 14.87
Largest losing trade -269.79 -248.50
Largest winning trade 243.54 154.22
Profit Factor (W/L)            1.77            1.64
Worst drawdown -633.24 -339.91

Graph below shows the equity curves of the 2 strategies

As it is observed from the Table and the Graph, except for the worst drawdown, we don’t see much of an improvement when the 10M-SMA timing rule is applied.  Although the 10M-SMA strategy avoided the worst period of the Global Financial Crisis, overall it made less money than the NoTiming strategy.

The non-improvement of Faber’s rule in the context  of volatility selling probably relates to the fact that we are using a directional timing algorithm to size a trade whose PnL driver  is the volatility dynamics . A position sizing algorithm based directly on the volatility dynamics would have a better chance  of success.  We are currently extending our research in this direction;  any comment, feedback is welcome.

 

References:

[1]  C. Donninger, Timing the Tail-Risk-Protection of the SPY with VIX-Futures by a Hidden Markov Model. The Wool-Milk-Sow Strategy. April 2017, http://www.godotfinance.com/pdf/TailRiskProtectionHMM.pdf

[2]  M. Faber, A  Quantitative  Approach  to  Tactical  Asset  Allocation, Journal  of Investing , 16, 69-79, 2007

[3] See for example A. Clare, J. Seaton, P. Smith and S. Thomas, The Trend is Our Friend: Risk Parity, Momentum and Trend Following in Global Asset Allocation, Aug 2012, https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2126478