Relative Value Arbitrage

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.

 

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,

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.

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

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

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

 

 

The spot VIX index finished last Friday at 11.28, a relatively low number, while the SKEW index was making a new high. The SKEW index is a good proxy for the cost of insurance and right now it appears to be expensive. A high reading of SKEW means investors are buying out of the money puts for protection.

With the cost of insurance so high, is there a less expensive way for investors to hedge their portfolios?

One might think immediately of cross-asset hedging. However, if we use other underlying to hedge, we will then take on correlation (basis) risk and if not managed correctly, it can add risks to the portfolio instead of protecting it.

In this post we examine different hedging strategies using instruments on the same underlying.  Our goal is to investigate the cost, risk/reward characteristics of each hedging strategy. Knowing the risk/reward profiles will allow us to design a cost-effective portfolio-protection scheme.

We will use Monte Carlo (MC) simulation to accomplish our goal [1]. The parameters and assumptions of the MC simulation 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)	1
Number of paths	10000
Model	GBM [2]

 

The hedging strategies we’re investigating are:

1-NO HEDGE: no hedging is performed. The asset is allowed to evolve freely in a risky world. This would correspond to the portfolio of a Buy and Hold investor.

2-PPUT: protective put. We buy an at the money (ATM) put in order to hedge the downside. This strategy is the most common type of portfolio insurance.

3-GAMMA: convexity hedge.  We buy an ATM put, but we then dynamically hedge it. This means that we flatten out the delta at the end of every day.

The GAMMA hedging strategy is not used frequently in the industry. The rationale for introducing it here is that given a high price of a put option, we will try to partially recoup its cost by actively scalping gamma, while we still benefit from the positive convexity of the option. This means that in case of a market correction, the gamma will manufacture negative delta so that the hedging position can offset some of the loss in the equity portfolio.

We use 10000 paths in our MC simulation. At the end of 1 year, we calculate the returns (using a Reg-T account) and determine its mean and variance. We also calculate the Value at Risk at 95% confidence interval. The graph below shows the histogram of the returns for the NO HEDGE strategy,

Table below presents the expected returns, standard deviations and Value at Risks for the hedging strategies.

Strategies	Expected return	Standard Deviation	Value at Risk
NO HEDGE	0.075	0.048	0.318
PPUT	0.052	0.024	0.118
GAMMA	0.066	0.029	0.248

 

As it is observed from the table, hedging with a protective put (PPUT) reduces the risks. Standard deviation and VaR are reduced from 0.048 and 0.318 to 0.024 to 0.118 respectively. However, the expected return is also reduced, from 0.075 to 0.052. This reduction is the cost of the insurance.

Interestingly, hedging using gamma convexity (GAMMA strategy) provides some reduction in risks (Standard deviation of 0.029, and VaR of 0.248), while not diminishing the returns greatly (expected return of 0.066).

In summary, GAMMA hedging is a strategy that is worth considering when designing a portfolio insurance scheme. It’s a good alternative to the often used protective (and expensive) put strategy.

 

Footnotes

[1] We note that the simulations were performed under idealistic assumptions, some are advantageous, and some are disadvantageous compared to a real life situation. However, results and the conclusion are consistent with our real world experience.

[2] GBM stands for Geometric Brownian Motion.