Price the Product

Question

I pay you, in dollars, the product of two fair dice rolled independently. What is the fair price of this game?

Accepted Answer

Independence factorizes the expectation: mathbbE[XY] = mathbbE[X],mathbbE[Y] = 3.5 times 3.5 = 12.25. Enumeration agrees: sum_a,b ab = left(sum_a aright)left(sum_b bright) = 21 times 21 = 441 over 36 outcomes = 12.25. The contrast that matters: the square of one die is also a "product", the die with itself, perfectly correlated: mathbbE[X^2] = frac916 approx 15.17 > 12.25, and the gap is exactly the variance: mathbbE[X^2] - (mathbbEX)^2 = frac916 - frac494 = frac3512 approx 2.92. Interviewers ask these back-to-back to check that mathbbE[XY] = mathbbE[X],mathbbE[Y] + operatornameCov(X,Y) is a reflex, not a formula you look up: independent legs price at the product of means, positively correlated legs are worth more, negatively correlated less. Trading relevance: product payoffs are everywhere, size × price move, fee × volume, quanto FX × equity. The first question a desk asks about any product payoff is "what's the correlation between the legs?" because that covariance term is the mispricing risk. Here the answer is exact, so your market should be tight: 12 bid at 12.5 is plenty wide for a known fair value of 12.25.

Price the Product

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