interview question : replication strategy of a betting game

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.

• This is almost identical to question 4.4 in Timothy Falcon Crack's book "Heard on the Street: Quantitative Questions from Wall Street Job Interviews". I suggest you get the book which I found overall very useful. – LocalVolatility Sep 10 '16 at 21:46
• Sorry - while this question sounded very familiar, question 4.4 in the above mentioned book is actually different in spirit. – LocalVolatility Sep 10 '16 at 21:55
• thanks: i will hav a look to the book . Meanwhile can you kindly let me know what do you think about my solution / a clue for the final answer? – phacoo Sep 10 '16 at 22:15
• Here is the correct reference: Your question is 100% identical to question 2.34 in the book by Mark Joshi et a. "Quant Job Interview Questions and Answers". The book also contains a solution. – LocalVolatility Sep 10 '16 at 22:15
• I checked the solution: they have a similar approach. the pb is I don't understand the last line (when he states "thus the result is 100*42/32-100)...have you a clue on this? :) – phacoo Sep 11 '16 at 2:37