hw11-2 - 4 Interest Rate Derivative Securities 251 2 + ( +...

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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 zero-coupon 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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hw11-2 - 4 Interest Rate Derivative Securities 251 2 + ( +...

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