Expected Value of Binary Contracts

Expected value (EV) combines payoff size with probability. It answers what one trade contributes on average under the probability model, not what the next trade will realize.

General and binary formulas #

For outcomes $X_i$ with probabilities $P_i$,

$$EV=\sum_i P_iX_i.$$

For one YES share bought at $p$, with your probability $q=P(YES)$,

$$EV=q(1-p)+(1-q)(-p)=q-p.$$

$q$ is a model estimate; $p$ is the actual purchase price. The algebra works because $q+1-q=1$.

Complete 65% comparison #

Assume $q=0.65$ and no costs:

This table is the central lesson: one belief can imply an excellent trade, a marginal trade, a fair trade, or a bad trade depending on price.

Life example #

A warranty costs 20 and pays 100 with probability 0.10. Its expected payout is 10, so its monetary EV to the buyer is -10. It can still be rational if avoiding a rare loss has personal value; trading EV and utility are related but not identical.

Application to no-hedge #

No-hedge seeks $q-p>0$ by buying below an estimated complementary reference value. The edge is not locked: if the reference misstates $q$, becomes stale, or differs in settlement semantics, the EV calculation fails while the binary loss remains real.