# HOW TO PROFIT FROM THE OUT OF THE MONEY CALL AND PUT OPTIONS PRICE SKEW

Jerry Felsen, Ph.D.

## SUMMARY/ABSTRACT

There is a big difference between prices of equally out-of-the-money call and put options. We call this price difference the skew. In this paper we briefly describe a trading system that profits from the skew.

### INTRODUCTION/PROLOGUE

As we can see in Figure 4.1, there is a large difference between the prices of equally Out-of-the Money (OM) call and put options. For example, on 18 April 2012, when the price of the underlying stock, QQQ, was near 67, the price of the QQQ May 73 call was near .02 and the price of the QQQ May 61 put was near .20. We call this price difference the skew.

Both options are equally OM and have the same time to expiry, but the price of the put is about 10 times greater than the price of the corresponding call. For many actively traded options this price skew exceeds ten.

We may quantify this skew as the ratio of equally OM put/call prices with the same time to expiry. In the above example the skew is .2 / .02 = 10. We consider such large skew to be a symptom of inefficiency of the options market.

We note that stock prices advance about 40% of the time, and they decline about 30% of the time. And for small X the probability a stock will jump X% is almost the same as the probability it will slump X%. Therefore the probabilities that both options will be profitable are about the same. So the prices of equally OM call and put options should also be approximately the same. Therefore we consider such large skew to be a symptom of inefficiency of the options market. So we’ll now describe the conceptual and general design of a trading system that attempts to profit from this skew.

### 1. HOW TO PROFIT FROM THE SKEW

In general, we profit from the skew by trading mainly favorably skewed options. By trading “favorably skewed options” we mean buy/hold OM calls and sell/write OM puts. And we consider all at-the-money and near-the-money options to be favorably skewed.

We note that prices of at-the-money calls and puts are usually approximately equal, and the more OM the options, the bigger their price skew, The OM call is relatively cheap compared to an equally OM put. Therefore far OM calls should preferably not be written (sold short), but they may be bought and held; and the opposite is true for far OM puts—we prefer to write them and we minimize buying and holding them.

In brief, we profit from the skew by emphasizing buying and holding far OM calls, and minimizing writing them. And we maximize writing and minimize buying/holding far OM puts.

We’ll now describe a trading system that will put these theoretical concepts into practice.

### 2. A PROFITABLE TRADING SYSTEM

We’ll now summarize the conceptual and general design of a trading system, which profits from the skew, and for which we can prove that it should make money.

The Provable Option Trading System (POTS) described in this article is based on three pillars. In other words, the three most important features of our POTS are the State-Subsystem Mapping (SSMap), the Growth Position (GroP), and we Continually Follow the Current Trend (CoFT).

#### 2.1 The State Subsystem Mapping

By its current volatility and trendiness we identify five states of the market, and we have a trading subsystem for each state. In other words, for each state of the market we try to trade the currently most profitable subsystem, i.e., the subsystem that has the highest expected return with the lowest risk if the current market state continues.

In fact, almost everything we do depends on the current state, i.e., everything is done as a function of the state (fate). In other words, before we trade, we try to identify the current state of the market. Then we find the subsystem that best fits the current state, and we trade that subsystem by making sure our current trades are commensurate with the current state.

We can automate the SSMap even though we do not know the current state of the market. That is, we assume the market is a random walk. Therefore we are not sure about the current state, and we know even less about the future. Nevertheless, as we’ll see later, we can almost guarantee that most of the time we trade the subsystem that best fits the current state.

#### 2.2 The Growth Position

There is a dream in options trading to find a strategy with a sure profit. We’ve news for you about this—both good and bad. The good news here is that we can approximate complex hedged positions that can only grow—Growth Positions (GroPs). But the bad news is that such positions exist only temporarily. Therefore, for some states we must continually trade to keep our POTS portfolio to approximate a GroP.

By GroP we mean a complex hedged position that has a positive expected return with a low risk for most states of the market. A key to success and a constraint here is that the GroP should be formed using mainly favorably skewed options.

We emphasize here that any single option or complex hedged positions—if held until expiry—has a negative expected return (equal to zero minus transaction costs). So the only way we can get a positive expected return is to continually trade to follow the current state and in some states trade to keep our Option Trading (OT) portfolio to approximate a GroP.

There are many ways we may approximate a GroP; we have described some of them in our recent book, e.g., Felsen (2010). One way is to hold a straddle and write enough OM iron condors to get a credit balance. This position will grow if there is a small up trend, or a small downtrend, or no trend (choppy, trend less market). But it has a relatively high risk. And one way to minimize risk is to CoFT. So CoFT is our next pillar.

#### 2.3 Continually Follow the Current Trend

To Continually Follow the current Trend (CoFT) is the best way to minimize risk. So we CoFT—but without taking any losses, i.e., the unprofitable trend-following positions are gradually converted into GroPs. (This means that before we trade, we must design the current GroP and make sure we trade mainly options from the GroP). And as the state of the market changes, we continually trade to adjust our OT portfolio accordingly.

### 3. THE CONCEPT OF STATE

The word state includes both the state of the market (e.g. our underlying stock QQQ) and the state of our OT portfolio, see Figure 4.2. Therefore, if we mean only “state of the market” or only “state of our OT portfolio,” then we must say so explicitly.

#### 3.1 State of the Market

The two most important market state features include its current trend and volatility. The trend can be up, down, or flat. And volatility can be high/fast or low/slow. By combining these two state features, we come up with six market states: (1) fast uptrend, (2) fast downtrend, (3) slow uptrend, (4) slow downtrend, (5) fast choppy market, and (6) slow choppy market. The last two states are usually equivalent because we usually map them on the same trading subsystem. So we combine them into one state called ChoST—Choppy, Slow and/or Trend less or Trading range type of market.

The market is about 40% of the time in an up trend, about 30% of the time in a downtrend, and is ChoST also about 30% of the time. These are the Random Walk state probabilities.

#### 3.2 State of our Portfolio

The two most important features describing the state of our OT portfolio are Portfolio Delta (Pelt) and Theta. As we’ll see later, both features depend on the state of the market.

#### 3.3 State Continuation Principle

In the short-term time horizon we assume the market is an almost pure Random Walk. In other words, its future is unknown, uncertain, unpredictable, and unexpected (the four uns). The market can do almost anything at almost any time.

But we also assume the current market state will continue until there is some evidence that the state has changed. This is the State Continuation Principle (SCP). Please note that trend following is an instance of this principle.

In mathematical terms the SCP says that if the market is now in some state, then the probability that the future state will be the same as the current state is slightly greater than the probability that the future state is different. That is, if the current state of the market is Si, then P(future state is Si) > P(future state is Sk).

There is usually little or no correlation between market states, and usually current state probabilities are approximately equal to their Random Walk probabilities. But if the current state is Si, then the probability that the next state is Si is slightly greater than the corresponding Random Walk probability.

### 4. HIERARCHICAL STRUCTURE OF OUR POTS

Our POTS is a relatively complex system—complex to the extent of perhaps defying a full description. We deal with this complexity via the hierarchical approach: Our POTS is designed as a hierarchically organized structure of simpler subsystems that we can understand and describe.

We need a different trading subsystem for each state of the market. In Section 3.1 we have defined five states of the market, so we need at least five trading subsystems. We also need a main control subsystem that implements the State-Subsystem Mapping (SSMap), Risk Control (RiC), Conflict Resolution, etc. We organize these subsystems hierarchically with the Main Control Subsystem at the top. Figure 4.3 summarizes the hierarchical structure of our POTS.

#### 4.1 The Main Control Subsystem

Our current trading strategy is continually dynamically formed by the Main Control Subsystem (MCS). That is, our current trading scheme continually changes as a function of the market state. The main tasks of the MCS include implementing (1) the State-Subsystem Mapping, (2) Risk Control, and (3) Conflict Resolution.

The MCS is designed so that the POTS should perform well without the need to predict future market states, but the ability to predict the future should boost its performance.

We sometimes realize the SSMap by simultaneously trading several complementary and synergizing subsystems, but we emphasize the subsystem that best fits the current state. For example, because we usually don’t know the current and future state, we sometimes simultaneously trade at least a trend-following subsystem to guarantee we’ll profit from every big trend, and if there seems a ChoST, then we also trade a subsystem for the ChoST state.

If we trade options, we can easily quantify our risk, and we can set an upper limit on the risk we take. Of course, the best way to minimize risk is to continually follow the current trend.

A major function of the MCS is also to resolve conflicts in situations where one trading rule says do this, while at the same time another rule says do the opposite.

We need different trading subsystems for different market states. For example, the subsystem we use to follow fast down trends is quite different from the subsystem we use to follow fast up trends. And the subsystems needed to follow slow trends differ from the subsystems for fast trends. And, of course, the subsystem we trade if there is a ChoST is thoroughly different from our trend-following subsystems. So we need several trend-following subsystems and at least one subsystem for the ChoST state. We try to integrate these complimentary trading subsystems in a way that makes the total system better (more profitable) than the sum of its parts. In other words, our POTS should benefit from synergy in a complex system.

### THE GAME OF CHESS IS A GOOD MODEL OF OUR POTS

Our option trading is similar to the chess game where the trader corresponds to a chess player and the market for the underlying corresponds to his opponent. The trader estimates the state of the market and makes his move by making a trade (i.e., deploying a subsystem) corresponding to the current state of the market. The market then responds by moving into another state. The trader then tries to identify the new state of the market, makes another move corresponding to the new state by making his next trade, and this cycle repeats indefinitely.

We do not anticipate the next move of the market, i.e., our POTS is designed so that we do not need to predict the future of the market. We cannot predict the future any better than we can predict the next move of a chess player. We assume the future of the market is uncertain, unknown, unpredictable, and unexpected (the four uns). Our POTS performs well even if the market is a pure random walk. But the better we can predict the future, the higher its performance.

Like in chess, we assume that the market will probably try to hurt us. Most of the time, the market will do whatever it has to do to hurt us the most. If, by making a mistake, you give the market a chance to harm you, the market will usually do it, i.e., if you make a trading error, the market will immediately use it to hurt you—the trading version of the Murphy’s Law. In other words, if we give the market an opportunity to hurt us, it will. Therefore, when we trade, we hope for the best, but we always prepare for the worst.

The most important feature of our POTS is the State-Subsystem Mapping (SSMap): Everything is done as a function of the current state. In Section 3.1 we have identified five states of the market, and we have a Trading Subsystem (TraS) for each state. We trade a subsystem that best fits the current state—but we never take a loss—the unprofitable trades are gradually converted into a Growth Position (GroP) described in Section 2.2. For example, if there seems a trend, we’ll trade a trend-following subsystem. If a trend-following trade is profitable, we might take the profit; else, the unprofitable trend-following position is gradually converted into a GroP.

We START by trying to identify the current state of the market. Then we find that TraS that seems to best fit the current state. We then adapt the TraS to the current state by assigning specific values to its system parameters like strike prices and time to expiry. We trade mainly options from the current GroP. Usually we build GroPs by trend following. Specifically, we build the GroP in steps so that each step corresponds to a state of the market. For example, we might gradually form GroPs starting with a position for a fast trend and convert it into a GroP only if there is a ChoST. In other words, we might start by forming a position (that is a subset of the GroP), which can follow fast trends, and if there is a fast trend, then we build more such positions. But if there is not a fast trend, then we convert our current position into another position (which is also a proper subset of the GroP) that can follow slow trends, and if there is a slow trend, then we build more such positions. But if it turns out there is a ChoST, then we convert our current position into a GroP. If the ChoST continues, then we wait until the state changes; else, we return to START.

There are many ways the above conceptual and general design can be realized. So if you decide to use our POTS, then we strongly suggest that you either adapt our implementation so it fits your nature, or find another implementation that better fits your temperament. We also point out that our POTS trades very often and therefore it’s essential that you implement it using an electronic broker that minimizes your transaction costs.

### PROOF THAT POTS MAKES MONEY

There are at least two ways we can prove that our POTS should make money with relatively low risk—practically and theoretically.

We can practically prove the effectiveness of our POTS by actually trading it and comparing our returns to market returns. The average expected market return is about 10% to 11% per year, but to earn this return, we have to take a rather high market risk including occasional equity drawdowns of more than 50% during bear markets.

We have implemented our POTS using an electronic trading platform provided by Interactive Brokers. At the time of this writing, our transaction costs range from about 15 cents to \$1 per contract. We are now using this POTS to manage our First Alpha Fund. At the time of this writing we estimate our annualized return on margin is about (15% to 20%)/year, and the risk of our POTS is much less than market risk. For instance, during our limited testing time our maximum drawdown has not exceeded about 5%. And we can always set an upper limit on our risk.

Of course, this practical proof tells almost nothing about future performance. To prove that our POTS will continue to make money in the future can only be done theoretically.

The theoretical proof is based on the three pillars of our POTS described earlier in Section 2—SSMap, GroP, and CoFT. We can guarantee that almost all the time we trade a subsystem that fits the current state of the market. By definition, if the trading subsystem fits the current state of the market, then it makes money. So our POTS makes money almost all the time. Also, we Continually Follow the current Trend (CoFT)—but we never take a loss—the unprofitable trend-following trades are converted into a GroP. By definition, the GroP can only grow, at least temporarily. About one third of our trend-following trades are immediately profitable, and the unprofitable trades are converted into GroPs. So we almost never lose and we should be almost continually making money.

As we have shown earlier, CoFT is also the best way to minimize risk. So our OT portfolio value rarely suffers any large drawdowns, and our risk should be relatively low because theoretically we should not lose.

Moreover, because we only trade options, we can always keep risk tolerable, i.e., we can always set an upper limit on our risk. In fact, we can keep risk at almost any level including near zero. But, of course, by lowering risk, we also lower our expected return.

Please note that this is a “heuristic” proof. (The Webster’s Dictionary defines the word “heuristic” to mean: “providing aid or direction in the solution of a problem but otherwise unjustified or incapable of justification.”)

Finally, in conclusion we note that our POTS is rather complex. But this complexity is needed for the long-term survival of this POTS because we can easily prove that any simple trading system must self-destruct if too many traders use it. Only if our POTS is so complex that relatively few people will use it, then it may not self-destruct. In other words, this complexity is needed for its long-term survival.