# hmw4 - Exercise 3 a Solve Exercise 4.10 from [GMS + 06]. b...

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Numerical Methods for SDEs, Fall 2006. Course Instructor: Ra´ul Tempone. Homework Set 4, due Thursday Sept 28. Last revised, Aug 30, 2006. Exercise 1 Assume that S ( t ) is the price of a single stock. Derive a Monte Carlo and a PDE method to determine the price of a contingent claim with with the (path dependent) payoﬀ Z T 0 h ( t, S ( t )) dt, for a given function h . Exercise 2 Derive the Black-Scholes equation for a general system of stocks S ( t ) R d of the form dS i ( t ) = a i ( t, S ( t )) dt + d X j =1 b ij ( t, S ( t )) dW j ( t ) , i = 1 , . . . , d and the European option with ﬁnal payoﬀ f ( T, S ( T )) = g ( S ( T )) . Here g : R d R is a given function e.g. g ( s ) = max ± P d i =1 s i d - K, 0 ² . Hint: Generalize the classroom derivation considering a self ﬁnancing portfolio with all stocks and the option.
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Unformatted text preview: Exercise 3 a Solve Exercise 4.10 from [GMS + 06]. b Optional for extra credit. Consider the SDE dX ( t ) =-aX ( t ) dt + b ( X ( t )) dW ( t ) , where a &amp;gt; . Is it possible to determine the function b such that X has a prescribed limit distribution with density p ( x ) as t ? If the answer is yes, carry out explicit computations for the uniform case p ( x ) = 1 , if-. 5 x . 5 , , otherwise. References [GMS + 06] J. Goodman, K.S. Moon, A. Szepessy, R. Tempone, and Z. Zouraris. Stochastic and Partial Dierential Equations with Adapted Numerics. Lecture Notes , 2006. 1...
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## This note was uploaded on 12/14/2011 for the course MAD 5932 taught by Professor Gallivan during the Fall '06 term at FSU.

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