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# hw11-2 - 4 Interest Rate Derivative Securities 251 = = =...

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4 Interest Rate Derivative Securities 251 × ˆ £ 2 α + ( γ + ψ ) 2 / e ψ ( T - t ) ( γ + ψ ) £ ( γ + ψ ) e ψ ( T - t ) - ( γ - ψ ) / ! μ ( γ + ψ ) /αψ = 2 ψ ( γ + ψ ) e ψ ( T - t ) - ( γ - ψ ) μ ( ψ - γ ) /αψ × 2 ψ e ψ ( T - t ) ( γ + ψ ) e ψ ( T - t ) - ( γ - ψ ) μ ( γ + ψ ) /αψ = 2 ψ ( γ + ψ ) e ψ ( T - t ) - ( γ - ψ ) 2 μ/α e μ ( γ + ψ )( T - t ) = 2 ψ e ( γ + ψ )( T - t ) / 2 ( γ + ψ ) e ψ ( T - t ) - ( γ - ψ ) 2 μ/α , 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 - λ ( r, t ) w ) ∂V ∂r - rV = 0 , where λ ( r, t ) is the market price of risk for the spot interest rate. Because λ ( r, t ) is unknown, in order to use this equation to price derivatives, we need to find λ . Suppose that λ is a function of t , i.e., λ = λ ( 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 today’s spot interest rate is r * . Let V ( r, t ; T * ) be the solutions of the problem ∂V ∂t + 1 2 w 2 2 V ∂r 2 + ( u - λ ( t ) w ) ∂V ∂r - rV = 0 , r l r r u, 0 t T * , V ( r, T * ; T * ) = 1 , r l r r u .

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hw11-2 - 4 Interest Rate Derivative Securities 251 = = =...

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