Same Expected Return, Different Options Risk

Newcomers often assume that if two options trades have the same expected profit, they are roughly equivalent. This is the same expected return, different options risk problem: the average projected payoff can be identical while the actual outcome dispersion is radically different. A long call and a long put can each show an expected profit of $4, but one can lose money on a bad day while the other simply breaks even. The key is variance: it measures how spread out the possible payoffs are around that average. A trade with high variance can deliver a large gain or a large loss; a trade with low variance stays close to the expected value. The profit formula alone, max(0, S-K) minus premium for a call, or max(0, K-S) minus premium for a put, does not reveal this. You must examine the range of outcomes under different underlying prices at expiration.

Take two positions on the same underlying, each with an expected profit of $4. Strategy A is a long call with a $10 strike and a $2 premium. At expiration, there are two equally likely scenarios: the underlying price is $22 (profit = max(0,22-10)-2 = $10) or the underlying price is $0 (profit = max(0,0-10)-2 = -$2). The expected profit is 0.5×10 + 0.5×(-2) = $4. Variance = 0.5×(10-4)² + 0.5×(-2-4)² = 0.5×36 + 0.5×36 = 36. The standard deviation is $6, so a one-standard-deviation move spans from a $2 loss to a $10 gain. Strategy B is a long put with a $10 strike and a $2 premium. Scenarios: underlying at $8 (profit = max(0,10-8)-2 = $0) or underlying at $0 (profit = max(0,10-0)-2 = $8). Expected profit is again $4. Variance = 0.5×(0-4)² + 0.5×(8-4)² = 0.5×16 + 0.5×16 = 16. Standard deviation is $4, so a one-standard-deviation move spans from $0 to $8. Both trades have the same average modeled payoff; the difference is that the long call can lose $2, while the long put's worst case is breaking even. The variance calculator confirms this: Strategy A's variance of 36 is more than double Strategy B's variance of 16. The dispersion of outcomes is what distinguishes them.

Consider a bull spread and a collar, two multi-leg strategies from the Cboe Options Institute standard diagrams. A bull spread uses one long call and one short call at a higher strike. Assume a $10 strike for the long call, a $15 strike for the short call, and a net premium cost of $1. The payoff diagram has a plateau between $10 and $15: profit is capped at $4 (the distance between strikes minus premium) when the underlying is at or above $15. Below $10, the loss is capped at the premium of $1. A collar uses a long call at $10, a short call at $8, and a long put at $12, with net premium cost of $2. It has two plateaus: profit is capped at $2 on the upside and loss is capped at $2 on the downside. Both can be constructed to have the same expected profit of, say, $0.50, but the bull spread has a wider range of outcomes (from -$1 to +$4) while the collar's range is narrower (from -$2 to +$2). The variance is higher for the bull spread because the upside is uncapped relative to the short call's strike minus the long call's premium, whereas the collar's payoff is fully bounded. This is the practical meaning of same expected return, different options risk: two strategies that appear equally profitable on average can have very different risk profiles because of the shape of their payoff diagrams.

To compute variance for any option position, list the possible profit outcomes and their probabilities from the payoff diagram. The diagram is a piecewise linear function of the underlying price at expiration. For a single long call with strike K and premium P, profit at expiration is max(0, S-K) - P. The possible outcomes depend on the distribution of S. In the worked example above, two discrete scenarios were used, but in practice you would use a continuous range. The formula is Variance = Σ [P(S_i) × (Profit_i - E[Profit])²] for each scenario i. The unit is squared currency. The standard deviation (the square root) is in currency units and is more interpretable: for Strategy A it is $6, for Strategy B it is $4. A trader who ignores this calculation sees only the $4 average and picks the trade with the higher variance, exposing themselves to a $2 loss when the lower-variance trade would have lost nothing. The calculation also reveals the probability of profit. In Strategy A, there is a 50% chance of a $10 profit and a 50% chance of a $2 loss, so a 50% chance of profit. In Strategy B, there is a 50% chance of $8 profit and a 50% chance of $0 profit, also a 50% chance of profit, but the profit outcomes differ. The variance calculation is what separates these identical-looking probabilities.

The reason variance distinguishes positions with identical expected profit lies in the option's capped and uncapped payoff shapes. A long call has unlimited upside (the underlying price can rise arbitrarily) and a capped downside (loss limited to the premium). This asymmetry increases variance because the upside tail pulls the distribution wide. A long put also has a capped downside (loss limited to the premium) but its upside is also capped: the underlying price cannot fall below zero, so the maximum gain is strike minus premium. Variance for a long put is therefore lower than for a long call with the same expected profit. Multi-leg strategies like collars and bull spreads show the same pattern: any strategy with an uncapped leg (like a short call, which has unlimited loss) will have higher variance than a strategy where all legs are bounded. The OCC Characteristics and Risks document confirms that the payoff shape is the fundamental driver of outcome dispersion. Variance does not capture everything, it treats upside and downside deviations equally, but it is the single best measure for comparing the risk of two positions with the same average payoff.

The most common mistake is to treat low variance as low risk. A long put with low variance protects you from a loss, but it also caps your gain. A short call with the same expected profit can have similar variance but unlimited loss potential, the variance calculation cannot tell you about the asymmetry of the tail. For example, a short call with a $10 strike and a $2 premium has profit = premium - max(0, S-K). If the underlying jumps to $50, the loss is $2 - $38 = -$36. A long put's worst case is a loss of $2. Their variances might be similar if the probability of extreme moves is low, but the tail risk is completely different. Always look at the full payoff diagram, not just the variance number. Variance is a useful summary, but it is not a complete risk assessment for options.