Sheldon Ross,

University of Southern California

Gambler ruin problems and pricing a barrier option under a jump diffusion model


Suppose there are r gamblers, with gambler i initially having a fortune n(i).  In our first model we suppose that at each stage two of the gamblers are chosen to play a game, equally likely to be won by either player, with the winner of the game receiving 1 from the loser.  Any gambler whose fortune becomes 0 leaves, and this continues until there is only a single gambler left.  We are interested in the mean number of players that involve both players i and j.  In our second model we suppose that all remaining players contribute 1 to a pot, which is equally likely to be won by each of them.  The problem here is to determine the expected number of games played until one player has all the funds.

If time permits, we will also discuss how to efficiently simulate the expected return from an up and in (or up and out) barrier call option under the assumption that the price of the security follows a geometric Brownian motion with random jumps.