My Process - Outlined
I trade options as if they were insurance products for two outcomes; protection against loss of opportunity (calls) and protection against loss of capital (puts). In this way, options should be more expensive the greater the risk to the issuer/writer of them expiring in-the-money (ITM). This is because the writer shoulders the risk of having to go into the market and buy shares or lose his already owned shares should he be called away, which can happen at any time before or upon expiration. The same goes for puts, instead the writer would have to buy shares at a huge premium to the current price should his contract be exercised. In exchange, the writer collects a premium, which should be higher the greater the risk of assignment.
If option prices are conceptually priced based on their probability of expiring ITM, how does one determine that probability and therefore a specific price? Well, Black-Scholes for one, but you don't need to know anything about that formula to understand the probabilities.
There are five sensitivities that option prices react to, known as the Greeks. Only three of them are of any real importance, with one of them only coming in to play when there is a short time before expiration. They are delta, theta, vega, gamma, and rho. Delta, theta, and vega play the largest role in determining the probability of options expiring ITM.
In order to demonstrate this precisely, I will use graphics of Normal Distributions:
Standard Distribution
The standard normal distribution is nothing more than a simplified histogram, in which the curve seen above is just "draped" on top of random observances. The gray line shows where the mean of the data lie, which is always the center so that 50% of observances are above the mean and 50% are below. The shape of the curve is determined by the standard deviation (sigma) of the data. The larger the value of sigma, the shorter and fatter the distribution will be, as there is a greater chance of observing a value that is more distant from the mean value.
Delta
Delta has the effect of shifting where the distribution lies on the number line by changing the mean value. When evaluating probabilities in the stock market, I always use the current going price of the stock as the mean value, with trading software calculating the value of sigma. When the price of the stock goes down, the mean value does too, and so the entire distribution of observances over a given periodicity moves accordingly. However, all else equal, the shape of the distribution is unchanged because the value of sigma remains constant. You will notice this effect with no math behind it by simply observing the graphic above. Notice how the old mean value (through the light blue curve) lies at the tail end of the new distribution. Whereas under the old distribution this value was the mean, now it is a distant outlier under the new distribution, and therefore under said new distribution, the likelihood of observing that value in the data is diminished. Thus, that value as represented as a call strike price should now be much lower under the new distribution than under the old distribution, as the likelihood of said strike becoming ITM upon expiration has been reduced greatly.
Theta
Out-of-the-money strikes (OTM) have no intrinsic value, only extrinsic value, sometimes referred to as time value. The longer the periodicity of expiration, the greater the probability that the underlying stock will move far enough to become ITM and/or expire ITM. Thus, options with longer dated expirations should, and do, have higher premiums than do those with shorter expirations. These premiums for OTM strikes decay geometrically and at an increasing rate as expiration approaches, with the most rapid decay occuring once the expiration date becomes the "front month" contract, i.e. with 45-30 days to expiration (DTE). Because observances that are farther from the mean are more likely to be seen with a longer periodicity than with a shorter periodicity, the value of sigma changes over time, all else equal, and this changes the shape of the distribution. As seen above, the distribution appears as short and fat with say, 90 DTE, whereas the distribution appears tall and skinny with 30 DTE. Observances far from the mean become ever more unlikely to be seen as the distribution is squeezed, and thus the premium on those OTM strikes decreases over time.
Vega
Vega refers to implied volatility (IV or vol) of the underlying, which is essentially the same thing as variance, which is used to calculate sigma. Higher IV = higher variance = larger sigma = short and fat distribution. Thus, because observances farther from the mean become increasingly likely to be seen as IV increases, the premium of those strikes (really ALL strikes) increases. This is separate from the effect of theta that continuosly squeezes the distribution, and will most of the time supercede said effect. This is because time decay happens at a predictable rate whereas IV changes constantly from day to day. Therefore, even though there may only be 10 or 15 DTE, the distribution can be relatively short and fat, so long as IV is high. And again, as IV contracts, all else equal, the premium of strikes declines, and vise versa.
Calculating Probability of Profit (PoP)
Simultaneously to calculating the probability of an option expiring ITM, one can calculate their PoP with the simple Z-score method. It only depends on which side of the transaction one is on, buyer or writer. Because this is so rudimentary, a single, short example will suffice:
Z = (X - mean)/sigma
A Z-score is a value that represents how far your observance is from the mean, as measured by the number of standard deviations (sigmas) it is from the mean. In this case, Z = 1.25, meaning 110 is 1.25 sigmas from the mean. We can then use this to calculate the PoP for both the writer and buyer of the 110 strike call using a Z-score calculator online, which is 89.43% for the writer and 10.56% for the buyer. For the seller, we use the area to the left of the strike for such a calculation, because the writer does not want the price to exceed 110 upon expiration. For the buyer, the right side is used, because he wants to see the price of the underlying exceed 110 upon expiration. However, it should be noted that this simple example is not perfect as it does not account for the premium of the hypothetical 110 strike call. To really calculate the PoP, you would need to find your break even point upon expiration and use that value as your observation in the Z-score formula.
From Understanding to Trading
The process then for trading options follows directly from these statistical concepts, but first you have to make one assumption; stock prices are random and unpredictable, thus we are able to use normal distribution in evaluating probabilities of prices occurring in the future. In short, what I do is sell OTM strikes when IV is high relative to itself with about 45 DTE in liquid underlyings, and in doing so I attempt to remain "delta neutral" on said underlyings.
Now for the explanation. First, you have to start with what "IV high relative to itself" means. IV is a percentage value that reflects how much the underlying is expected to range from the current price over the next 52 week period. This percentage value of course has its own normal distribution, as IV percentages are random if we assume that stock prices are random. Because of this, I seek to sell IV when it is high, i.e. when it is on the tail end of the distribution (above the 50th percentile). Doing so provides ample premium, due to the vega effect. As time passes, IV is more likely to contract towards the 50th percentile, or the mean, than it is to continue to increase. This process is known as "mean reversion," and is explained by simply re-explaining what a mean is. The mean is the most frequent occurrence/observation in a distribution, hence why it is located in the center and at the peak of the distribution. Because it is the most frequent observation, as time progresses, observances are more likely to come down towards the mean than they are to go up away from it, and vise versa. Thus, the premium for the OTM strikes declines and as the writer, I gain from IV contraction.
In that explanation, I mentioned the passage of time twice. As the writer of the contract, I gain from the passage of time. So, simultaneously, I gain from time passing and IV contracting. I look to sell as close to 45 DTE as possible because, as outlined earlier, that is when the time decay accelerates the most.
Liquid underlyings simply refers to underlyings where the spread between the bid and ask quotes on the contracts is not so large that I cannot quickly and "painlessly" enter and exit into positions. You don't want to end up with large unrealized gains, measured by your cost basis against the mid-price of the contract or spread, only to have them eaten away when attempt to realize them by exiting the position at a price inevitably higher or lower than the mid-price.
Remaining "delta neutral" is paramount to my strategies because as noted earlier, I assume stock prices are random and unpredictable. Because I don't know where the price of a stock is going to go for any periodicity, it is best to remain delta neutral and profit from the passage of time and the contraction of volatility, as those two processes are much more reliable. In doing so, the discretionary element of which trade to take is completely removed. One can easily screen with proper software for stocks with high IV and 45 DTE, whereas searching for directional opportunities will inevitably require your own interpretation. Often times, you will see patterns that don't actually exist, and attempting to trade from that perspective will only result in maybe a 50% success rate, if you're decent and a 55-60% success rate if you are great. With my process, success rates are likely to range from 70-80% over the long term. Why?
Returning to statistical analysis, we know that roughly 68% of all observances fall between the -1 and +1 sigma values for any normal distribution, regardless of shape. So, selling a +1 sigma strike call and a -1 sigma strike put to form a "strangle" spread will have a 68% PoP at current IV levels and DTE. As time passes and IV contracts, all else equal, the PoP of such spread can dramatically increase, as illustrated below:
The red lines mark the +1/-1 sigma range under the light blue distribution, where IV and DTE are both higher than under the dark blue distribution. As both IV and DTE decrease, the distribution becomes taller and skinnier and, all else equal, the probability of observing the light blue +1/-1 sigma values under the dark blue distribution decreases. Ergo, what was once a 68% probability range becomes much higher.
Doing this process over and over is a practical application of the "central limit theorem." A single sample of 10 trades that, regardless of IV, are 1 sigma strangles should have a success rate of 68%. However, due to the randomness of markets, it may only be that 4 were successful as opposed to the anticipated 6.8 or 7. The next sample of 10 such trades may produce 6 successful trades, and the next after that produces 9, then 5, then 7, and so on. In essence, it is more likely that 100 such trades will be closer to the anticipated 68% success rate than a sample of 10. If such trades are done in underlyings with high and then contracting IV, the success rate will inevitably increase from 68%. That is the edge of my process, plain and simple.
In order to apply the CLT, lots of trades have to be made over the course of time. Each position should risk a small and consistent portion of the capital at hand, which could be measured with margin requirements for the positions or by their defined risks if such spreads are made. It should then be expected that over time you will have more "winners" than "losers." And finally, due to the possible utilities of options, one can reduce the amount lost from the losing trades so that the overall process is quite profitable over time.
Wrapping Up
There are many more elements to my process in regards to trade selection and risk management, but this is only an outline. None of the information expressed above is stolen from any investor in some book or trading course found online. It is just the application of statistical analysis in trading assets that, in my view, were designed with such analysis in mind. It is purely quantitative and eliminates the discretionary element from trading, which means that I do not rely on chart patterns or indicators. I put this understanding of options trading together myself with the assistance of supplemental educational material found online. None of it is plagiarized. Please do not plagiarize this work yourself.
Written by Brandon Powers, Bachelor of Arts in Economics, Ohio State University
June 7, 2016
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