Here is a question I found in a book I am not able to finish. Your help will be much appreciated! I also included where I have been so far.
Q: Team A plays team B in a series of 7 games, whoever wins 4 games first is the winner. You want to bet 100 that your team wins in which case you will receive 200 or 0 otherwise. However you can bet only on individual games (so game by game sequentially). You can bet X the day before the match and receive 2X if win otherwise 0 if lose. - Q1: How do you achieve the desired payout? - Q2: What amount do you bet on the 1st game?
For me this is like replicating a binary option paying 100 in case of success or losing 100 in case of default. The aim is to price this option (find this X). It seems to be a binomial tree of 7 states. We consider the first part of the tree (X, X_up, X_down). Probability of going up is a just the binomial probability of winning at least 3 games out of of 6 (we are already at the the state 1). So Proba_up = 1 - (1/2^6)*(1+15+6) = 21/32 Thus we can get the value of X_up = 100*21/32 - 100*(1-21/32) = 40.625 We then have Proba_down = 1 - Proba_up = 11/32 and then X_down = 100*11/32 - 100*(21/32) = -40.625
And now I am stuck: I cannot figure out the proba I need to use to discount these values. Many thanks for your help.