Positive Edges in the Volatility Space.

Leonardo Valencia: leov@anixcorp.com. Aug 10 2016.

Introduction

In any transaction of any type, an edge is the implicit advantage that one party might have over another one. For instance, in a pure and “fair” transaction, there should be no edge at all for either the seller or the buyer. On the other hand, in a casino for instance, we know that the house has an implicit positive edge in each and every game offered (yet people keep flocking to them). So conceptually, the definition of a positive edge is easy to understand. Mathematically, we can define a positive edge as having a net positive expectation of return. In other words, the odds of winning multiplied by the winning amount should be higher than the odds of losing multiplied by the losing amount.

Edge = Win*prob of win - Loss*Prob of loss

Of course, traders are always looking to have positive edges in every transaction; that is the only way to make money and be successful in the long run. But the question is this: Are there any positive edges still left in the market?

This is an important question, as technology advances and the speed of communication links, processor power, and software complexity gets better and better, the equities market in the USA gets closer and closer to the academic definition of an efficient market. Be mindful that I’m talking about the equities market, which is the one that has seen the most impressive technological development compared with other ones (for instance, most of the futures market is still highly inefficient). And if you remember, in an efficient market there are no edges left at all; in other words, each transaction becomes a fair one. So at first glance, it looks that all hope is lost, and no edges are left for traders out there - at least in the pure price space.

Now don’t get me wrong: in order to make the market efficient, of course we need someone to smooth away the inefficiencies. As you can imagine, that's a job that pays handsomely, but it is mostly reserved for the big banks, big funds, and big trading desks that have the capital, access, and technology to exploit all the inefficiencies away (and in the process, they make lots of money). Unfortunately,there is no space for retail traders for the kinds of trades that remove transient inefficiencies from the market, so we need to find opportunities elsewhere.

The curious case of the Options Market.

The trading of options in the equities market deserves a more detailed look. This is particularly true if we are searching for edges that are still left around. Options are very peculiar. If you recall, options are derivatives of some underlying, and as a derivative, there are a few interesting properties about them:

• The supply of options is not constrained. In other words, options can be created out of thin air and thus the usual rules of supply and demand don’t apply to them.

• Because options are derivatives, their price must in some way follow the price of the underlying that they are tracking. This is, there are constraints on the set of possible values that an option might take, and again, these don't follow the usual rules of supply and demand.

• In modern option pricing theory, the final price of the options depends heavily on a single, unobservable parameter called implied volatility. Given that implied volatility is an estimate of future realized volatility, it follows that option prices carry information about the future. However, because no one can really predict the future, it means that option prices contain a big arbitrary component in their price that can turn out to be terribly wrong.

• Finally, the market for options tends to be asymmetrical. This is because option buyers tend to assume full risk, either because they are using options to speculate, or alternately, because they are using options to hedge, a buyer desires a full risk exposure. However, almost all option sellers are always risk neutral. They don’t really care or want to take the risk of the position; they just want to capture the bid/ask spread in the market or the volatility differential. This asymmetry in the market creates opportunities, as option prices strive to be fair in the risk-neutral world only.

As you can see, options present interesting opportunities. In particular, the third and fourth points open the door to all sorts of strategies that might have a positive edge.

Variance Risk Premium

Much has been written about the persistent Variance RIsk Premium embedded into option prices, more markedly in Index options. A simple way to define the VRP is that it is basically the difference between the realized volatility of an underlying and the one that was implied by the price of a particular set of options (at the money options, for instance).

If options were priced fairly, then the difference would be zero. However, that is not the case in the US options market. Countless studies have found that implied volatility always tends to be higher than the future realized volatility. This has been explained in many different ways, one plausible explanation being that option sellers want to be compensated for the risk that they are assuming with their positions. This rings especially true if you consider that the central tenet of option pricing theory is not attainable in practice - that tenet being the ability to continuously hedge the position without costs. In practice, we can’t have an infinite amount of hedging transactions, and also, no matter how cheap it is, trading activity always incurs transaction costs. Therefore, option sellers are stuck with imperfect hedging, and they therefore displace part of that risk into option prices.

There are also structural market sources for overpriced options, especially when talking about index puts. They might be of a regulatory character - for instance, when certain type of funds are mandated to hedge their positions, and they tend to hedge with index puts that create  price pressure on the out of the money side of put strikes. There are also many other sources of hedging that can put pressure on index puts. For instance, they could be just normal funds that, although not mandated by regulation, hedge their positions as part of a well-defined risk mitigation strategy. And finally, with the advent of more advanced volatility derivatives, hedging and speculation in that space translates into index option trading activity that can put pressure on prices, too.

Independent of the sources, it is indisputable that there is a Variance Risk Premium embedded in option prices. The question then becomes, given that everyone knows about this, can it still be considered an edge?

The question is very tricky, because there is a belief out there in market lore that a true edge must be some kind of secret inefficiency that only a handful of people know about, and that this is why it remains profitable. However, if the VRP were merely an inefficiency of the option market, it would have disappeared long ago (right about the time of its discovery). The fact that it is still present means that this is something beyond efficient markets -more of a structural phenomenon. In other words, option markets are efficient yet still contain a structural VRP.

But coming back to the original question - can the VRP still be considered an edge - we should only care about the mathematical definition of an edge. Fortunately, this is something we can check ourselves, but we need to come up with a trading strategy that monetizes the VRP first. Once we have the strategy, we can check past performance and see if we have a positive expectation of return (a positive edge).

Monetizing the VRP

To address the elephant in the room: Yes, I’m giving away a trading strategy for free.

However, very soon you will see why you should never trade this strategy yourself. There is the reason why it is free. I’m presenting it here because it serves a very good educational purpose, but no sane trader should ever attempt this strategy.(I know that after all these warnings some of you will start raising money for a fund and try doing this anyway). Besides, the strategy has a lousy Sharpe ratio of 0.11, so I doubt outside investors would put up with that level of volatility.

To design this strategy, let’s combine the 3rd and 4th points I made about the options market. The VRP covers the 3rd point already, now for the 4th. Remember that options are priced (overpriced it seems) for a risk-neutral world. Let’s exploit that fact and run the strategy in the full-risk world (so it gains or loses even more). Finally, because we know that there is an excess Variance Risk Premium, it means that we will want to sell it. In other words, we are going to become net sellers of options in this strategy, and because of that, we need to make a very important digression here.

Some thoughts about selling Gamma

Buying options and selling options are two very different endeavours, and the participants in each activity are very different as well. Option buyers tend to assume full risk, be it for speculation or for hedging purposes, while option sellers tend to be risk-neutral and more interested in capturing the bid ask spread - or they are volatility players betting on the differential between implied and realized volatility.

We are going to be even more different, since we want to sell options and also assume almost full risk (no dynamic delta hedging of the positions). One of the things we have to realize is that this strategy involves selling short dated options (weeklies), which in general are very rich in gamma and, by extension, very poor in vega. What this means is that short dated options react very quickly to movements in the underlying but are affected very little by changes in implied volatility.

I know that some of you must be familiar with strategies that involve selling short dated options, like selling weekly iron condors or selling strangles that use out of the money strikes. I can’t say it strongly enough that those strategies are tremendously inefficient because their execution is contrary to their thesis. Any time we sell options, and in particular when we sell short dated options, we are selling gamma. If we are pursuing a gamma selling strategy, the idea is to maximize the amount of gamma we sell while also minimizing the gamma risk incurred.

Please take a look at the following plot of gamma per strike for a particular expiration. This graph is from my book so it is a little old, but the distribution of gamma remains the same in general.

The red line marks the at the money value of SPX at the time it was produced. Please notice how gamma changes with the strikes. Gamma is very high near the at the money strikes but becomes very small for strikes outside that range.

So if we want to maximize selling gamma, why are we selling OTM options? It makes no sense really. We are actually selling very little gamma, and also from a risk management point of view, it is a terrible proposition. Imagine for a second selling gamma for the 2020 option strike. Gamma is very small in the first place, but what happens if SPX moves in the direction of that strike? You guessed it right: the graph clearly shows that gamma increases dramatically for options that start moving into the at the money region, so if our 2020 strike becomes at the money at some point, its gamma will shoot through the roof. Consequently, we will also be losing a lot of money because we are short. That is what happens when people use those kinds of selling strategies. Selling OTM options blindly is a sure way to go broke (if done naked) and/or is a very inefficient way of selling gamma. When you sell OTM gamma because it has a “high” probability of success, you are collecting very little money and assuming very high risk.

Now let’s think about this in a different way. Let’s sell the 1950 Strike call. Right off the bat, we are selling the maximum gamma possible of all the strikes for that expiration. Not only that, but what happens if SPX moves abruptly one way or another? From the graph, it is clear that if the option goes deep - either in the money or out of the money - its gamma will decrease dramatically. In other words, from a risk management point of view, we will be in much better shape than in the OTM selling case, because something we are short is actually lower than what we sold it for.

So there you have it. Because of those considerations, our strategy will involve selling weekly at the money options. But what exactly are we going to sell ?

Weekly straddles as an expression of the VRP

As the headline says, we are going to sell weekly straddles. The question is: why? And the answer is simple: Even though we said that we were going to assume full risk, it is better to compromise a bit and use a strategy that is neutral (at least at inception) and also one that remains statically hedged during its lifetime, so it is a compromise between full risk and dynamic hedging. We are not selling the straddle because we think that the limits imposed by positive PL mean anything. Instead, we are doing it because we think that the price of those ATM options contains a variance risk premium that can be extracted by letting the straddle run until expiration and pocketing the difference between the actual move (from Friday to Friday) and the implied move (which is a risk neutral move, by the way). Also, we are selling the ATM straddle because we want to sell close to maximum gamma and be in better shape if our thesis is wrong (because of gamma dynamics).

Now you can see why I’m giving the strategy away for free. Of course, one has to be crazy to sell a naked straddle every week, at least in principle, because you are subject to step losses if SPX drops or moves up a lot. Now part of the thesis of this strategy is that the at the money options are priced with a buffer for those kind of moves, and that we should do fine in the long run. However, it is very scary to have positions that could accumulate great potential losses and only deliver a fixed profit.

The full VRP strategy

Last warning here: please don’t even think about trying to execute this strategy as part of any serious money operation. Be aware that you are exposed to great risk by doing this. With that out of the way, here is the detailed strategy execution:

1. Every Friday at 4:00PM, sell the weekly ATM straddle in SPX. Of course, rarely is SPX at a level that aligns with available strikes, so for this strategy, we are going to pick the nearest straddle where the put leg is out of the money. In other words, if SPX is at 2188, we are going to pick the 2185 straddle (as the put is OTM at that strike).

2. Be mindful that the weekly expiration needs to be a PM one, so make sure that it expires at 4:00PM. If the next week expiration is a 3rd week one (classic SPX), then we will use SPXPM options instead.

3. Let the straddle run all the way to expiration without doing absolutely anything. No hedging, no adjustments, just let it expire.

4. Repeat this every Friday during the whole year.

Simple and scary at the same time. Remember that this trading strategy is only used to evaluate if there is a true edge in index options in terms of selling the excess Variance Risk Premium that is always present in them. The analysis of the results is very simple, as the profit and loss of a straddle is easy to compute:

PL = premium received - abs(strike-spx closing price)

So without further ado, here are the results of an arbitrary three-year period of running this strategy, from Sept 20, 2013 until Aug 05, 2016. There is not really anything mysterious about that period of time, other than I had a dataset of detailed option prices handy going back to Sept 2013. I have other databases with longer coverage, but they are more cumbersome to convert and use for this exercise. In total, there are 144 observation points in the period for this strategy, which are numerous enough to see if an edge presents itself. Also, the period covers at least 3 major events that produced downside moves of significant magnitude. Finally, the period covers several regimes of implied volatility, including extensive low implied volatility during 2013 and early 2014, and also high implied volatility during half of 2015.

The following graph shows the profile of Profit and Losses of the strategy. For simplicity, the PL is represented in SPX points per straddle (+50, -50 etc).

As you can see from the Profit and Loss profile, the graph exhibits the typical PL shape of a short volatility strategy, which is usually normal-sized profits punctuated by heavy sporadic losses. In particular, it is not surprising that the Aug 24 event produced a huge loss.

Now, let’s check the density of the distribution of returns to see if it shows anything interesting. As you can see in the next graph, the PL distribution is very interesting. It is fat tailed towards the negative side as I would expect. However, it actually looks like we have a positive mean, and the peak of the distribution is also a positive number (not zero and not a negative number). What this is telling us is that the strategy has a big chance of having a positive edge.

Despite the fat tails, the strategy seems to tip towards a positive cumulative return. Finally, let’s take a look at the cumulative PL of this strategy using one more graph:

The cumulative graph is interesting because it shows that, despite the drawdowns, over the long run there is positive expectancy of return for the strategy. Look at the terrible performance around Sept 2014 and early part of 2015, and also notice the incredible volatility around Aug 2015 and towards the end of that year. Despite all this, the cumulative return managed to stay above zero.

Of note, too, is the amazing performance of the strategy during 2016, which is a good sign that for the most part of this year, options have been consistently overpriced (contrast that with late 2014). Of course, from the graph it is obvious that if we started trading right around Sept 2014, we would have suffered a massive drawdown for almost a whole year. If we had started in Aug 2015, we could have suffered a shorter drawdown before being profitable again. So it is clear that this is not an optimal strategy, as it exposes us to unexpected drawdowns (for those who are curious, the mean return for this strategy is 2.08% and the Sharpe ratio is 0.104, which is dismal). However, we can expect that the strategy will perform well after some time. More importantly, it is clear that the short straddle does indeed capture a portion of the VRP, at least for certain volatility regimes.  

The question then becomes, can we do a better job of capturing the VRP for all volatility regimes? The short answer is yes, but I won’t go into the details, as they are proprietary at this time. However, I’ll explain the gist of it for those of you who like to find hidden gems in the market, so that you at least have a way of knowing where to start looking.

The VRP is not constant

It is clear from the cumulative PL graph of our straddle strategy that the Variance Risk Premium is not always enough to compensate for the realized risk that is observed later on. This can be seen more clearly in the long underperformance period in late 2014 and early 2015. Clearly, the straddle was terrible priced for the kind of moves that actually happened. What this tells us is that option dealers and market makers are not infallible, and in fact, they can can misprice options under certain conditions. And that is where we need to start looking to understand the issues with the VRP. We need to look at the underperformance periods to see what is common to all of them.

The biggest problem with the VRP is that we can only quantify it after the fact. This is a very inconvenient consequence of humans not being able to predict the future. Because of this, it is hard for us to know in advance if the VRP is worth selling or not, so we need to find a proxy - some type of estimator of the VRP that can help us improve the strategy. That way, we will sell the straddle if the estimator shows us a fat VRP. Vice versa, if the estimator shows that options are underpriced, we can buy the straddle instead. Just for comparison, here is a PL graph of the optimized strategy in red, with the original strategy in blue:

The difference is notable. By using an estimator of future VRP, you can see that we have improved the returns during the original underperformance periods in such a way that the total performance of the new strategy is almost 3 times better than the original. The Sharpe ratio of the enhanced strategy is also better at 0.32, although it is still dismal by any standard.

Even with the estimator (which for obvious reasons is proprietary), using straddles is still a dangerous proposition because it subjects the strategy to uncapped losses. If you want to deploy a variation of this strategy in a professional setting, you need to do a few things:

1. Develop an estimator of VRP so the system can choose the correct moments to buy or sell the VRP.

2. Modify the system in such away that losses are capped up to a tolerable level.

3. Make sure that the new system has a Sharpe ratio > 1.0 ( the higher the number, the better).

Conclusion

In the long rung, it is possible to monetize the ever-present variance risk premium in Index options. It is better to be net sellers of ATM options than OTM options from a risk management and total profit point of view. And finally, it is indeed possible to design an estimator of future Variance Risk Premium that can be used in a trading system to improve returns and help mitigate downside risk.

But most important of all: Never try to use a toy system in an environment where real money is at risk.