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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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This note was uploaded on 09/12/2009 for the course MATH 45240 taught by Professor Saari during the Spring '09 term at UC Irvine.
- Spring '09