Variance of an Options Payoff Distribution

You model two options positions. Both return an average of $50 per contract. One has payoffs of $0 and $100 with equal probability. The other pays $49 or $51, also equally likely. The variance of an options payoff distribution tells you why the first can lose your budget while the second barely moves. Variance measures the average squared deviation from the mean. For the $0/$100 option, that is 2500 dollars squared. For the $49/$51 option, it is 1 dollar squared. Same expected value, but the spread is 2500 times larger. That is the answer to the question: two positions with similar average modeled returns can have very different outcome spread because variance depends on the squared distance of each outcome from the mean, not on the mean itself.

Variance is the average of the squared differences from the expected value. For a discrete random variable, you take each possible payoff, subtract the mean, square that number, multiply by the probability of that outcome, and sum them all up. The unit is the square of the payoff unit, dollars squared, not dollars. That is why standard deviation exists: to bring the measure back to original units. The formula is Var(X) = Σ p_i (x_i − μ)², where μ is the expected value. This comes directly from Blitzstein and Hwang's Introduction to Probability and Fisher's 1918 paper that first used the term 'variance'.

Why the Same Mean Does Not Mean the Same Spread

Consider two options. Option A: payoff $0 with probability 0.5 and $100 with probability 0.5. The mean is $50. The variance is 0.5 × (0−50)² + 0.5 × (100−50)² = 0.5 × 2500 + 0.5 × 2500 = 2500 dollars². Option B: payoff $49 with probability 0.5 and $51 with probability 0.5. The mean is also $50. The variance is 0.5 × (49−50)² + 0.5 × (51−50)² = 0.5 × 1 + 0.5 × 1 = 1 dollar². The means are identical. The variances differ by a factor of 2500. That is the entire explanation for why two positions with similar average modeled returns can have dramatically different outcome spread.

Work through a concrete calculation. You have an option that pays $10, $50, or $90 with probabilities 0.2, 0.6, and 0.2 respectively. First compute the mean: μ = (0.2×10) + (0.6×50) + (0.2×90) = 2 + 30 + 18 = $50. Same mean again. Now compute the variance: 0.2×(10−50)² + 0.6×(50−50)² + 0.2×(90−50)² = 0.2×1600 + 0.6×0 + 0.2×1600 = 320 + 0 + 320 = 640 dollars².

Now change the probabilities to 0.4 for $10, 0.2 for $50, and 0.4 for $90. The mean becomes 0.4×10 + 0.2×50 + 0.4×90 = 4 + 10 + 36 = $50. The variance becomes 0.4×1600 + 0.2×0 + 0.4×1600 = 640 + 0 + 640 = 1280 dollars². Same three payoffs, same mean, but the variance doubled because more probability mass moved to the extremes. This is the core mechanics at work: variance is driven by how probability is distributed across the outcomes, not by where the center sits.

To compute variance by hand, follow these steps. Step one: find the mean by multiplying each payoff by its probability and summing. Step two: for each payoff, subtract the mean and square the result. Step three: multiply each squared deviation by its probability. Step four: sum those products. The result is the variance in squared units. If you are working with a sample of actual trades rather than a probability model, you use n−1 in the denominator, Bessel's correction, to get an unbiased estimate of the population variance. This is the source of the most common mistake: using n instead of n−1 causes sample variance to be biased low by a factor of (n−1)/n.

When you need to verify your calculation against a model, the options profit calculator supplies scenario outcomes while variance quantifies their dispersion. Input the same payoffs and probabilities, and the calculator returns the mean and variance. Compare your hand calculation to the calculator output. If they match, your method is correct. If they do not, check whether you squared the deviations before multiplying by probability, a common error, and whether you used the correct mean. A quick cross-check: for a two-outcome symmetric distribution, the variance is always p × (1−p) × (range)², where p is the probability of the higher outcome. For the $0/$100 example, that is 0.5 × 0.5 × 100² = 0.25 × 10000 = 2500 dollars². Works every time.

Variance has a serious weakness: it is sensitive to extreme outcomes because deviations are squared. A single large deviation can dominate the entire measure. In the $0/$100 example, removing the $100 outcome and replacing it with $60 at 0.5 probability and keeping $0 at 0.5 gives a mean of $30 and a variance of 0.5×(0−30)² + 0.5×(60−30)² = 0.5×900 + 0.5×900 = 900 dollars². The $60 outcome is not extreme, but the variance still reflects the spread from the mean. The real failure case is when the payoff distribution has infinite variance. The Cauchy distribution, for example, has no finite variance at all. If your modeled returns follow a Cauchy-like distribution, heavy tails that do not decay fast enough, variance is undefined and cannot be used to compare spread. In practice, this means you cannot rely on variance for strategies that involve rare but catastrophic losses, such as deep out-of-the-money options with very low probability of a massive payoff. Variance will be dominated by that extreme squared deviation, and the number you compute may not be stable across repeated sampling.