to forecasts of future price trends, the purpose of this text is to demonstrate how trading options according to current market conditions and directional volatility assumptions (rather than price assumptions) has allowed options sellers to consistently outperform the market.
This “edge” is not the result of some inherent market inefficiency but rather a trade‐off of risk. Recall the example long call trade from the previous section. Notice that there are more scenarios in which the short trader profits compared to the long trader. Generally, short premium positions are more likely to yield a profit compared to long premium positions. This is because options are assumed to be priced efficiently and scaled according to the perceived risk in the market, meaning that long positions only profit when the underlying has large directional moves outside of expectations. As these types of events are uncommon, options contracts go unused the majority of the time and short premium positions profit more often than long positions. However, when those large, unexpected moves do occur, the short premium positions are subject to potentially massive losses. The risk profiles for options are complex, but they can be intuitively represented with probability distributions.
Probability Distributions
To better understand the risk profiles of short options, this book utilizes basic concepts from probability theory, specifically random variables and probability distributions. Random variables are formal stand‐ins for uncertain quantities. The probability distribution of a random variable describes possible values of that quantity and the likelihood of each value occurring. Generally, probability distributions are represented by the symbol P , which can be read as “the probability that.” For example,
Let's begin with an example of a simple probabilistic system: rolling a pair of fair, six‐sided dice. In this case, if D represents the sum of the dice, then D is a random variable with 11 possible values ranging from 2 to 12. Some of these outcomes are more likely than others. Since, for instance, there are more ways to roll a sum of 7 ([1,6], [2,5], [3,4], [4,3], [5,2], [6,1]) than a sum of 10 ([4,6], [5,5], [6,4]), there is a higher probability of rolling a 7 than a 10. Observing that there are 36 possible rolls ([1,1], [1,2], [2,1], etc.) and that each is equally likely, one can use symbols to be more precise about this:
The distribution of D can be represented elegantly using a histogram. These types of graphs display the frequency of different outcomes, grouped according to defined ranges. When working with measured data, histograms are used to estimate the true underlying probability distribution of a probabilistic system. For this fair dice example, there will be 11 bins, corresponding to the 11 possible outcomes. This histogram is shown below in Figure 1.1, populated with data from 100,000 simulated dice rolls.
Figure 1.1 A histogram for 100,000 simulated rolls for a pair of fair dice. This diagram shows the likelihood of each outcome occurring according to this simulation (e.g., the height of the bin ranging from 6.5 to 7.5 is near 17%, indicating that 7 occurred nearly 17% of the time in the 100,000 trials).
Distributions like the ones shown here can be summarized using quantitative measures called moments.3 The first two moments are mean and variance.
Mean (first moment): Also known as the average and represented by the Greek letter μ (mu), this value describes the central tendency of a distribution. This is calculated by summing all the observed outcomes
For distributions based on statistical observations with a sufficiently large number of occurrences, the mean corresponds to the expected value of that distribution. The expected value of a random variable is the weighted average of outcomes and the anticipated average outcome over future trials. The expected value of a random variable X , denoted
(1.6)
In the dice sum example, represented with random variable D , the possible outcomes (2, 3, 4, …, 12) and the probability of each occurring (2.78%, 5.56%, 8.33%, …, 2.78%) are known, so the expected value can be determined as follows:
The theoretical long‐term average sum is seven. Therefore, if this experiment is repeated many times, the mean of the observations calculated using Equation (1.5) should yield an output close to seven.
Variance (second moment): This is the measure of the spread, or variation, of the data points from the mean of the distribution. Standard deviation, represented with by the Greek letter σ (sigma), is the square root of variance and is commonly used as a measure of uncertainty (equivalently, risk or volatility). Distributions with more variance are wider and have more uncertainty around future outcomes. Variance is calculated according to the following:4
(1.7)
When a large portion of data points are dispersed far from the mean, the variance of the entire set is large, and uncertainty on measurements from
