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Unformatted text preview: Suggested Answers for Math of Finance, Test # 2; March 10, 2009 1. a. (3) State the Efficient Market Hypothesis. b. (2) Give an example where the EMH holds, and where it does not. Answer: This answer can be found in the notes. 2. (10) Find a change of variables to convert V t + S 2 2 V S 2 + S V S- V = 0 into an equation with constant coefficients. Answer: By use of the chain rule, we have that V S = V x dx dS To eliminate the bad portions of the problem, let dx dS = 1 S . To compute the second derivative, 2 V S 2 = S ( V S ) = S [ 1 S V x ] =- 1 S 2 V x + 1 S S [ V x ] . But by using the chain rule one more time, S [ V x ] = x [ V x ] dx dS = 1 S 2 V x 2 . Substituting this term into the above, and then all of these terms into the original equation, we have the much simpler expression V t + 2 V x 2- V = 0 . 3. (10) a. Prove (i.e., give the details) that if t < T , then a European Put P ( S,t ) must go below the curve max ( E- S, 0) . State the conditions when this happens. Answer: Using the Put-Call Parity, solving for P , and then adding and subtracting E yields P ( S,t ) = C ( S,t ) + Ee- r ( T- t )- S = E- S + [ E ( e- r ( T- t )- 1) + C ( S,t )] ....
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- Spring '09