hw11-1 - 4 Interest Rate Derivative Securities 243 with A(T...

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4 Interest Rate Derivative Securities 243 with A ( T ) = B ( T ) = 0 and determine the system of ordinary diFerential equations the functions A ( t ) and B ( t ) should satisfy. Solution : Substituting V ( r, t ) = e A ( t ) - rB ( t ) into the equation yields dA dt - r dB dt + ( a 0 + a 1 r ) B 2 - ( b 0 + b 1 r ) B - r = 0 . Thus dA dt + a 0 B 2 - b 0 B = 0 , dB dt - a 1 B 2 + b 1 B + 1 = 0 . This system of ordinary diFerential equations with the conditions A ( T ) = B ( T ) = 0 has a unique solution. Therefore the original equation has a solution in the form V ( r, t ) = e A ( t ) - rB ( t ) . Because A ( T ) = B ( T ) = 0, we have V ( r, T ) = 1 . Consequently, the solution of the system of ordinary diFerential equations with conditions A ( T ) = B ( T ) = 0 gives the solution of the bond problem. 6. In the Vasicek model, the spot interest rate is assumed to satisfy dr = (¯ μ - γr ) dt + p - βdX, β < 0 , γ > 0 , where ¯ μ, γ and β are constants and dX is a Wiener process. Let the market price of risk λ ( r, t ) = ¯ λ - β . Then the price V ( r, t ; T ) of a zero- coupon bond maturing at time T with a face value
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hw11-1 - 4 Interest Rate Derivative Securities 243 with A(T...

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