The Right to Double the Stake

You and a rival flip a fair coin repeatedly; the first to win 2 flips wins the match and the loser pays the stake, which starts at \$1. You alone hold a one-time right to double the stake to \$2, which you may exercise before any flip. When you exercise it, your rival either accepts (the match continues for \$2) or folds immediately and pays the current \$1. Your rival plays optimally. What is that right worth to you?