Mathematical Edge

Establishing Stock Targets

Before embarking on any type of comparative analysis regarding the best option spreads to trade, we need to establish some reasonable estimate as to where a stock price is likely to be on some future date. We can then assess all possible permutations of Buy and Sell Strikes for an option spread using this stock target to find the combination that offers the highest return.

So what constitutes a reasonable target for a stock price at some arbitrary date in the future; let's say one week from today. A logical starting point would be to ask, "how much does the stock typically move over the course of one week?". We could determine this my examining historical weekly returns and computing the Mean Absolute Deviation (MAD) of those returns. This would tell us the average amount a stock moves in a week without regard to the direction of that move (up versus down). With a long history of weekly returns you will find that the stock has an equal chance of moving either up or down in price over such a short time frame so the Mean return will come out to zero. MAD Formula As elegant as this approach may be, it has some serious short comings. For one, the range of movement may change as a company matures or might vary at different times. This begs the question, "what is the appropriate look back period for calculating historical returns?". Another issue concerns accounting for dividend payouts. It is a known phenomenon that a stock tends to drop by the amount of its dividend on xDividend Date. Stock volatility also tends to increase around the time earnings are being reported. Such events will cause distortions in the future volatility of the underlying stock which would not be reflected in a smoothed out set of historical returns.

Fortunately, there is an alternative approach to estimating the amount of movement one can expect in a given stock. It is based on the notion that option premiums reflect the anticipated movement in the underlying stock. If an earnings event is on the horizon, option premiums will rise to reflect the anticipated increase in volatility. If an xDividend date is pending before an option expires, then Put premiums will be higher than Call premiums in anticipation of the stock dropping by the amount of the dividend. Calculating the volatility of a stock based on the premiums of the underlying options is referred to as calculating the "Implied Volatility". The following algorithm will serve to illustrate how this can be accomplished using the Black-Scholes Option Pricing Model.

  1. Get the actual market price of an option
  2. Start with an arbitrarily high estimate of implied volatility
  3. Pass this volatility guess along with the option's attributes into the Black-Scholes Model
  4. Compare the computed price of the option to the actual price
    • If they are the same we are done and the implied volatility is the actual volatility
    • If the computed price is higher than the actual price, lower the volatility estimate
    • If the computed price is lower than the actual price, raise the volatility estimate
  5. Go back to step 3
After passing in all the call options (or put options) available on a particular stock we might end up with an Implied Volatility Surface that looks something like the following.

Volatility Surface

By applying special interpolation and weighting techniques to the implied volatilities of options at various strike prices, we can get a pretty good idea as to the distribution of anticipated future stock returns for a given expiry date. What we need to find is the average anticipated up or down move for the stock expressed in standard deviation units. This will give us a lower boundary below which 25% of possible future returns will lie and an upper boundary above which 25% of possible future returns will lie. We can simply look up these values from a z-score table for a Standard Normal Distribution. The cutoff values are -.6745 and +.6745 standard deviations.

Inter Quartile Range

Finding The Best Strike Price To Buy

Using actual option quotes for the S&P 500 ETF (SPY) expirying on July 21, 2017 (at the close of trading on June 5, 2017), we calculated the maximum returns for buying either a simple Call or Put option at each strike price that was within our projected lower and upper stock target. The maximum returns were calculated based on the assumption that the stock would close exactly at our projected target on expiry date. For both Calls and Puts, the highest expected returns occur with a strike price that is close to the current stock price. We found this pattern to be fairly consistent when testing over multiple stock symbols across different expiry dates. The fact that options close to the current stock price have higher expected returns than options farther in or out of the money should make sense at an intuitive level. For example, buying a deep in the money option would require a a large cash outlay there by lowering your return. An option that is far out of the money would be expected to expire worthless so an at the money option is the best compromise between these extremes. If you take a look at a financial website that displays the implied volatility in its option chains (e.g. Yahoo Finance), you will notice that in general, the implied volatility decreases as the strike price moves closer to the current stock price. this is another way of saying that these options are the cheapest.

Best Option Strike Prices

Finding The Best Spread To Trade

Using actual option quotes we paired all possible spread combinations for the S&P 500 ETF (SPY) Calls expirying on March 17, 2017 where the strike prices were between the lower and upper projected targets for the stock. These options had about 4 weeks of time remaining when the analysis was conducted. In the left image below we show only the set of strike prices paired with a buy strike of 230. However, for all combinations tested; the returns increased as the sell strike was pushed out further toward the stocks projected upper target. The table on the right summarizes the results for each buy strike. As you can see, the highest return is obtained by selecting a sell strike just inside the stock's upper target and a buy strike exactly one strike interval below it.

Best Call Spread Strike Prices

For put options the same pattern emerges. The highest return is obtained when the sell strike is equal to the the stock's lower target and the buy strike is one strike price interval above that.

Best Put Spread Strike Prices

From the above analysis it should come as no surprise that we focus on Option Spreads rather than Simple Call and Put options. The returns we can expect if we are correct on the direction of the underlying stock will be substantially higher.

The maximum returns depicted above represent a very aggressive strategy since the stock needs to move beyond the Buy Strike in order for the Spread to retain some value at expiry. At Expiry Week LLC we take a somewhat more conservative approach. Namely, we select a Buy Strike that is closest to the Current Stock Price for both Put and Call Spreads. We then select the Sell Strike that is closest to the Stock's Lower Target for a Put Spread, and the Sell Strike that is closest the the Stock's Upper Target for a Call Spread. With this approach we are in effect selecting the same Buy Strike that we would select were we to Buy a simple Call or Put Option. We then lower the cost of that position by writing an Option at the price that we expect the stock to end up at on expiry date. This gives us the lowest cost we can acheive for a Spread without sacrificing any of the expected movement in the underlying stock.