M503(hw1)

M503(hw1) - MATH-503 Homework-3 Solution Note: In this...

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MATH-503 Homework-3 Solution Note: In this solution, a strategy h = ( x,y 1 , ··· ,y n ), where x is the riskless asset, y i is the number of shares of the risky asset. 1) a) i. ” ”. If 1 + R > u , consider the strategy h = ( S 0 , - 1). Clearly, V h 0 = 0, and V h 1 = ± S 0 (1 + R ) - uS 0 , w.p. P u S 0 (1 + R ) - dS 0 , w.p. P d Since 1 + R > u , we have V h 1 > 0, which implies V is dominant. This contradicts with the no dominant strategy condition, so 1 + R u . If 1 + R < d , consider the strategy h = ( - S 0 , 1). Again, we have V h 0 = 0, and V h 1 = ± uS 0 - S 0 (1 + R ) , w.p. P u dS 0 - S 0 (1 + R ) , w.p. P d Since 1+ R < d , we have V h 1 > 0, which leads to the contradiction. Thus, 1+ R d . Finally, we have d 1 + R u . ii. ” ”. Suppose there is a dominant strategy H = ( x,y ). Then we must have x = - yS 0 and V H 1 = ± yS 0 [ u - (1 + R )] , w.p. P u yS 0 [ d - (1 + R )] , w.p. P d H is dominant, so we must have ± d > 1 + R, if y > 0 u < 1 + R, if y < 0 Clearly, it’s a contradiction. So there is no dominant strategy.
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M503(hw1) - MATH-503 Homework-3 Solution Note: In this...

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