Hedging a series bet with single-game wagers

Two teams play a best-of-seven series (first to four wins). Your client wants a $100 bet on the series: he wins $100 if his team takes the series and loses $100 otherwise. You may only place even-money bets on individual games, and each bet can depend on the score so far. How much should you stake on the first game to replicate the series bet exactly?

$31.25. Let $V(a,b)$ be the series payoff (in units of $100) when the score is $a$–$b$; with even-money bets, $V$ is the average of its two successors, and the stake at each node is half the difference between them. Backward induction gives $V(1,0)=\tfrac{5}{16}$, $V(0,1)=-\tfrac5{16}$, so the first-game stake is $\tfrac5{16}\times100=31.25$.