## 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. 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
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.

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.

#### 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.