hw6solutions

# hw6solutions - minimize u s.t u 1-A T y ≥ 1 T y = 1 y ≥...

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ORF 307 Homework 6 Solutions Exercise 1. We solve this problem by optimizing on the expected value of the asset to be priced. The constraints are that the discounted expected value of the known payoﬀs is equal to the prices of the assets for which we know the price of. The expected values are taken with respect to the risk neutral probabilities which are the variables of our problem. Let y be these risk neutral probabilities and the maximum price problem is: max Σ 10 i =1 y i ( x i - 50) + s.t. Σ 10 i =1 y i ( x i - 40) + = 21 Σ 10 i =1 y i ( x i - 60) + = 10 Σ 10 i =1 y i x i = 50 Σ 10 i =1 y i = 1 y 0 Note that x is a parameter to this linear program. Refer to the Matlab code for how to implement this. The minimum price is the same program with the objective multiplied by -1. The result is that the price of the asset is in the interval [14.667, 15.5]. Exercise 2. The payoﬀ matrix for a two-player, zero-sum game is given by A = - 1 - 2 - 3 - 1 1 1 5 - 2 1 The row player’s optimal strategy y is attained by the following min-max problem: min y max x y T Ax where x is the column player’s strategy. This problem can be formulated as

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Unformatted text preview: minimize u s.t. u 1-A T y ≥ 1 T y = 1 y ≥ 1 Equivalently, minimize u s.t. u + y 1 + y 2-5 y 3 ≥ u-2 y 2-y 2 + 2 y 3 ≥ u + 3 y 1-y 2-y 3 ≥ y 1 + y 2 + y 3 = 1 y 1 , y 2 , y 3 ≥ The column player’s strategy y could be solved by the following LP: maximize v s.t. Ax-v 1 ≥ 1 T x = 1 x ≥ which is equivalent to maximize v s.t. v 1-Ax ≤ 1 T x = 1 x ≥ or explicitly, maximize v s.t. v + x 1-2 x 2 + 3 x 3 ≥ v + x 2-x 2-x 3 ≥ v-5 x 1 + 2 x 2-x 3 ≥ x 1 + x 2 + x 3 = 1 x 1 , x 2 , x 3 ≥ If use MATLAB to solve either of the programs we ﬁnd the the row player’s optimal strategy is y T = { . 1429 , . 6190 , . 2381 } and the column player’s optimal strategy is x T = { . 2857 , . 5714 , . 1429 } The return value is-. 4286. This implies that the row player has an advantage: he is expected to win . 4286 dollars. 2...
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## This note was uploaded on 04/30/2008 for the course ORF 307 taught by Professor Alexandrew.d'aspremont during the Spring '08 term at Princeton.

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hw6solutions - minimize u s.t u 1-A T y ≥ 1 T y = 1 y ≥...

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