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 Spread To Trade

Best Buy Strike
A quick glance at any financial website that displays option prices along with greek values, will reveal that strike prices that are close to the underlying stock price are cheaper (lower implied volatility) than those that are farther out of the money. A practical value oriented approach would therefore suggest that the buy side of an option spread should be placed near the price of the underlying stock. A simple thought experiment yields similar results. If you were to buy a call option that is far above the stock price, the odds of the stock actually reaching that price would be low. Conversely if you were to buy an option that was far in the money, it would cost a lot and your expected return would be low. The best compromise between these two extremes would be to by an At-The-Money option.

Best Sell Strike
When you buy a call spread instead of just a simple call option, you are effectively lowering the stock price at which you will break even as well as the stock price at which your investment will double (assuming the strike prices are far enough apart). This however, comes at the cost of capping your potential gains at the strike price of the option sold. It is therefore important to pick a sell strike that is not to close and not to far away from the underlying stock price. This is where our projected target for the underlying stock comes into play as it represents the best guess as to where the stock price will be on expiry day if it moves in the anticipated direction.