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Unformatted text preview: Suggested Answers for Final Exam, Math of Finance, March 19, 2009 1. Today, shortly after you leave this exam, California will be playing Maryland in the NCAA Basketball tournament. Sam believes Maryland will win; he is giving 3:2 odds. Jane supports California with 3:1 odds. a. (10) You have $ 100 to bet, and you want to make the same amount of money no matter which team wins. How should you bet and how much will you win? Answer: Bet x with Sam and 100 x with Jane. If Maryland wins, you win $3(100 x ) and you lose $ x . Thus your profit is 300 3 x x , or 300 4 x, so you want 75 > x . If California wins, then your profit is (100 x ) + 3 2 x = 5 2 x 100 , or you want x > 40. For a fixed income, set the equations equal to obtain 300 4 x = 5 2 x 100 , or 400 = 13 2 x. So, x = 800 13 and the fixed winnings are 300 4 800 13 . That is the answer, but if you had a calculator, you would find that it is about $53.85. b. (10) Sam believes that his bet is a fair one in that he is willing to take either side. Compute what Sam believes is the likelihood of California winning. Answer: A fair bet is where E ( X ) = 0. If p C is the likelihood of California winning, then 1 p C is the likelihood of Maryland winning. For each dollar bet with Sam, we have 0 = E ( X ) = 1(1 p C ) 3 2 p C = 1 5 2 p C . Solving, we have that p C = 2 5 . 2. a. (10) Find the value of c so that f ( x ) = cx 2 on [0 , 1] , and zero elsewhere, is a pdf. Answer: 1 = R 1 cx 2 dx = c x 3 3  1 = c 3 . Thus, c = 3. b. ( 10) Using (a), if X ( x ) = x , compute E ( X ) . Answer: E ( X ) = R 1 X ( x ) f ( x ) dx = 3 R 1 x 3 dx = 3 4 x 4  1 = 3 4 ....
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 Spring '09
 Saari
 Math, Exponential Function, Derivative, Differential Calculus, Mathematical finance

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