Make me a market on the absolute difference of two dice

Question

Two fair six-sided dice are rolled. A contract pays X - Y in dollars. Compute the fair value exactly and make a two-sided market.

Accepted Answer

Count the 36 ordered outcomes by their gap d = X - Y: d outcomes count --- --- --- 0 (1,1)\dots(6,6) 6 1 pairs differing by 1 10 2 8 3 6 4 4 5 (1,6),(6,1) 2 The counts sum to 6 + 10 + 8 + 6 + 4 + 2 = 36, good. Then E[X-Y] = \frac{0\cdot 6 + 1\cdot 10 + 2\cdot 8 + 3\cdot 6 + 4\cdot 4 + 5\cdot 2}{36} = \frac{0 + 10 + 16 + 18 + 16 + 10}{36} = \frac{70}{36} \approx 1.94. The market. Fair \approx 1.94. Quote for example 1.80 bid, at 2.10. Notice the mode is d = 1 (10 outcomes) while d = 0 (a double) happens 6/36 = 1/6 of the time, so a big chunk of the mass sits at small gaps and the payoff is rig…

Make me a market on the absolute difference of two dice

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