Section 21.3

Betting, Bidding and Market Making

Kelly sizing, making markets on combinations, pricing options on dice, winner's-curse bidding.

Toolkit

These problems price bets, quotes and options by comparing expected value (or expected log-wealth) under the condition that a transaction actually occurs, and by exploiting replication/no-arbitrage across legs.

Synthetic quotes must be arbitrage-free against their legs: for a combination $C=A-B$ ,

\text{bid}_C=\text{bid}_A-\text{ask}_B,\qquad \text{ask}_C=\text{ask}_A-\text{bid}_B.

Kelly criterion: for a bet paying $b$ to $1$ with win probability $p$ , the growth-optimal stake fraction is

f^*=\frac{p(b+1)-1}{b},

reducing to $f^*=2p-1$ at even odds ( $b=1$ ); accept a wager iff its expected value (or expected log-wealth increment) is positive.

Fair value of a contingent payoff is its expectation under the true (or risk-neutral) distribution, summing payoff times probability over all outcomes in the money.
Winner’s curse: conditioning on winning an auction shifts the distribution of value — e.g. value uniform on $(0,V)$ given a bid $x$ wins becomes uniform on $(0,x)$ , so rational bids must discount expected value by the conditional expectation given victory, not the unconditional one.

Problems