Make a market on the product of two dice

Question

Two fair six-sided dice will be rolled. The contract settles at the product of the two faces. The interviewer will buy or sell against your quote before the roll.

Make a market: give a bid and an ask, and justify the width.

Accepted Answer

Fair value. For independent dice the expectation of the product is the product of the expectations: E[XY] = E[X]\,E[Y] = 3.5 	imes 3.5 = 12.25. That is well below the naive "middle" of the 1-to-36 range, because the product distribution is heavily skewed: most rolls land low, a few (like 6 	imes 6 = 36) are far out in the tail. Dispersion. Using independence again, E[X^2] = 	frac{91}{6} \approx 15.17, so \mathrm{Var}(XY) = E[X^2]E[Y^2] - (E[XY])^2 = \left(	frac{91}{6}ight)^2 - 12.25^2 \approx 230.0 - 150.1 = 79.9, giving \sigma \approx 8.9. That is a huge standard deviation relative to …

Make a market on the product of two dice

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