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Unformatted text preview: 4 Interest Rate Derivative Securities 251 2 + ( + ) 2 / e ( T t ) ( + ) ( + ) e ( T t ) (  ) / ! ( + ) / = 2 ( + ) e ( T t ) (  ) (  ) / 2 e ( T t ) ( + ) e ( T t ) (  ) ( + ) / = 2 ( + ) e ( T t ) (  ) 2 / e ( + )( T t ) / = 2 e ( + )( T t ) / 2 ( + ) e ( T t ) (  ) 2 / , the two expressions are identical. 10. Describe a way to determine the market price of risk for the spot interest rate. Solution : Let dr = u ( r, t ) dt + w ( r, t ) dX, where r is the spot interest rate. Suppose that the price of any interest rate derivative satisfies V t + 1 2 w 2 2 V r 2 + ( u ( r, t ) w ) V r rV = 0 , where ( r, t ) is the market price of risk for the spot interest rate. Because ( r, t ) is unknown, in order to use this equation to price derivatives, we need to find . Suppose that is a function of t , i.e., = ( t ). Then this function, as the solution of the following inverse problem, can be determined by the term structure of interest rates or, equivalently, by the zerocoupon bond price curve. Suppose that t = 0 corresponds to today and todays spot interest rate is r * . Let V ( r, t ; T * ) be the solutions of the problem...
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 Spring '10
 Zhu

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