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In a two-player zero sum game, the matrix below shows the value to Player 1. Player I's strategies are labelled I to VI, where Player 2's strategies are labelled A to F. A B C D E F 13 29 8 12 16 23 II 18 22 21 22 29 31 III 18 22 31 31 27 37 IV 11 22 12 21 21 26 V 18 16 19 14 19 28 VI 23 22 19 23 30 34 (i) Show, by eliminating dominated strategies, that the game can be reduced to the following 3 x 3 matrix. [3] b C 13 29 8 B 18 22 31 Y 23 22 19 (ii) Explain whether or not this new 3 x 3 matrix has any saddle points. [2] Now consider a randomised strategy for Player 2, denoted X, whereby strategy "a" is chosen with probability p and strategy 'c' is chosen with probability 1 -p, 0

0 either from above (down-and-out) or from below (up-and-out). The down-and-out call has the following payoff at time T: max(ST - K,0) if min S, 2b, OSIST 0 otherwise. Assume this special option is written on the given stock, has the same strike price and maturity as the European call option described in part (ii) and the barrier b is fixed at 48. (iii) Calculate the price of this contract using the binomial tree model and risk- neutral valuation. [3] (iv) Determine the price of the down-and-out contract when b = 40, without performing any further calculations. [21