FINM345A09TakeHomeFinal10 - FINM345/STAT390 Stochastic...

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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 ) 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 ) 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 ) 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 ) 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
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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 t 0 ( σ ( s, T ) dW ( s ) - 0 . 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 ) - 0 . 5 σ 2 ( t, t ) dt + dt · t 0 ∂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 out of the course or from book questions. Also, from calculus or advanced calculus/analysis, you will need to know how to handle the derivative of a doubly time-dependent integral like t 0 f ( s, t ) ds with respect to t or be able to derive it in dt -precision. } { An extended hint: For sufficiently small Δ t , or just using dt with dt - precision ( = { dt } ), you want to consider the increment t + dt 0 f ( s, t + dt ) ds - t 0 f ( s, t ) ds = { dt } ( f ( t, t ) + t 0 f T ( s, t ) ds ) · dt , where f T ( t, T ) = ( ∂f/∂T )( t, T ) . and the integral with dW ( s ) can be handled similarly since the
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