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Thinking in Probability Space

A linear instrument rewards you for thinking in price. A binary rewards you for thinking in probability, and traders who carry stock intuitions into event contracts make a predictable set of mistakes. This lesson is the coordinate change.

The geometry of a bounded payoff

A contract bought at price cc can make at most 1−c1 - c and lose at most cc. The ratio of those two numbers is the market's odds:

Odds against=1−cc\text{Odds against} = \frac{1 - c}{c}

At 50 cents you risk one to win one. At 20 cents you risk one to win four. At 95 cents you risk nineteen to win one. The same one-cent move in price means a completely different change in odds depending on where it happens: from 50 to 51 is a shrug, from 98 to 99 it halves the value of the No contract and nearly halves the Yes backer's profit per unit staked. Prices near the middle are coarse; prices near the boundary are microscopically sensitive, and intuition trained on the middle fails at the edge.

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