FINM345A09TakeHomeFinal10 - FINM345/STAT390 Stochastic...

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Unformatted text preview: FINM345/STAT390 Stochastic Calculus – Hanson – Autumn 2009 Take-Home Final Examination: Due by 6:30pm CST Monday 07 December 2009 (7:30pm EST at UBS; 7:30am Tuesday 08 Dec. in Singapore) in Chalk FINM345 Assignment Submenu • You must show your work, code and/or worksheet for full credit. The work must also be your own and points will be deducted for likely copies. • There are multiple points per question and grades depend on both the correct answer and the quality as well as quantity of the justification. • Points will be deducted for late submission by a point per hour per problem to begin, not to exceed one half the total of earned points. • The exam is open Lecture notes and course textbook, but any other refer- ences used must be cited in a scholarly fashion. • In your exam submission, include a copy or reasonable facsimle of this signed statement: On my honor this take-home exam is my own work, except for any citation to resources that I have used. Signed: . (10 points) Corrections or emphasis are in Red as are comments, December 3, 2009 0. Innovations beyond what the problem asks for. (variable points) 1. JD SDE Transformations: { Comment: Note that for sufficiently small Δ t , and dt , you can use the zero-one jump law, but for general values you need the jump form below. } (a) Given dS ( t ) = S ( t ) · ( μ ( t ) dt + σ ( t ) dW ( t )) + dP ( t ) X j =1 S ( T- j ) ν ( T- j , Q j ) , (1) show that d 1 S ( t ) = 1 S ( t ) · ( f ( S ( t ) ,t ) dt + g ( S ( t ) ,t ) dW ( t ))+ dP ( t ) X j =1 h ( S ( T- j ) ,T- j ,ν ( T- j , Q j )) , (1 . 5) by finding the functions f ( s,t ), g ( s,t ) and h ( s,t,ν ( q )) explicitly. (10 points) (b) Given dY ( t ) = μ ( Y ( t ) ,t ) dt + σ ( Y ( t ) ,t ) dW ( t ) + dP ( t ) X j =1 ν ( Y ( T- j ) ,T- j , Q j ) , (2) show that d (exp( Y ( t )) = F ( Y ( t ) ,t ) dt + G ( Y ( t ) ,t ) dW ( t ) + dP ( t ) X j =1 H ( Y ( T- j ) ,T- j ,ν ( Y ( T- j ) ,T- j , Q j )) , (3) by finding the functions F ( y,t ), G ( y,t ) and H ( y,t,ν ( y,t,q )) explicitly. (15 points) 1 2. Stochastic Calculus Example from Forward Contracts * : Consider the price of a forward contract for energy at time t expiring at T , satisfying dF ( t,T ) = F ( t,T ) σ ( t,T ) dW ( t ) . (4) (a) Show that F ( t,T )= F (0 ,T ) exp Z t ( σ ( s, T ) dW ( s )- . 5 σ 2 ( s, T ) ds ) , (5) by stochastic calculus. (15 points) (b) If the spot price is S ( t ) = F ( t,t ) at t and if Y ( t ) = ln( S ( t )), show that dY ( t )= ∂ ln( F ) ∂T (0 ,t ) · dt + σ ( t,t ) dW ( t )- . 5 σ 2 ( t,t ) dt + dt · Z t ∂σ ∂T ( s,t ) · ( dW ( s )- σ ( s,t ) ds ) , (6) where the partial with respect to T is the partial with respect to the second argument of F or σ . (20 points) (c) Show that dS ( t ) = S ( t ) ( dY ( t ) + 0 . 5 σ 2 ( t,t ) dt ) . (7) (10 points) { * Background note: You do not have to know anything about forward contracts for this problem, but ONLY about stochastic calculus. This problem and problem 1(a) arose outproblem, but ONLY about stochastic calculus....
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This note was uploaded on 04/19/2010 for the course FIN 390 taught by Professor Hansen during the Fall '09 term at University of Illinois, Urbana Champaign.

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FINM345A09TakeHomeFinal10 - FINM345/STAT390 Stochastic...

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