ma503b-10final

# ma503b-10final - MATH503b/506 (SPRING 2010) FINAL PROJECT...

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MATH503b/506 (SPRING 2010) FINAL PROJECT Due Friday, May 7, 2010 1) Consider a bond markdet in which the short rate is modeled by the SDE: dr ( t ) = μ ( t,r ( t )) dt + σ ( t,r ( t )) d ¯ W ( t ) , (1) where ¯ W is a Brownian motion under the objective measure P . a) Explain how a “martingale measure” Q should be deﬁned in the bond market? b) What should the Q -dynamics of the short rate look like? c) Suppose that X = Φ( r ( T )) is a contingent claim on the short rate, and assume that its price takes the form Π( t ; X ) = F ( t,r ( t )), t [0 ,T ]. Derive in details the Q -dynamics of Π. (You should make appropriate assumptions on the way, but you do not have to prove the Term Structure Equation.) d) Using the Q -dynamics of Π to derive the “risk neutral valuation” formula for Π( t ; X ). e) Argue that the forward rate and short rate have the following relation: f ( t,T ) = E Q t,r ( t ) h r ( T )exp n - R T t r ( s ) ds oi E Q t,r ( t ) h exp n - R T t r ( s ) ds oi

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## This note was uploaded on 10/08/2010 for the course MA 503 taught by Professor Majin during the Fall '09 term at USC.

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ma503b-10final - MATH503b/506 (SPRING 2010) FINAL PROJECT...

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