# hw10-3 - 4 Interest Rate Derivative Securities Problems 1 a...

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4 Interest Rate Derivative Securities Problems 1. a) Suppose the spot interest rate is a known function r ( t ). Consider a bond with a face value Z and assume that it pays a coupon with a coupon rate k ( t ), that is, during a time interval [ t, t + dt ], the coupon payment is Zk ( t ) dt . Show that the value of the bond is V ( t ) = Ze - R T t r ( τ ) " 1 + Z T t k τ ) e R T ¯ τ r ( τ ) d ¯ τ # . b) Suppose that r ( t ) and k ( t ) are equal to constants r and k , respectively. Show that in this case, V ( t ) = Z e - r ( T - t ) [1 + k (e r ( T - t ) - 1) /r ] . c) Suppose that the bond pays coupon payments at two specified dates T 1 and T 2 before the maturity date T and the payments are Zk 1 and Zk 2 , respectively. According to the formula given in a), and assuming T 1 < T 2 , find the values of the bond for t [0 , T 1 ) , t ( T 1 , T 2 ), and t ( T 2 , T ), respectively, and give a financial interpretation of these expressions. Solution : a) Because during the time interval [ t, t + dt ], the holder of this bond will earn dV + Zk ( t ) dt and it is risk free, we have dV + Zk ( t ) dt = r ( t ) V dt, i.e., dV - r ( t ) V dt = - Zk ( t ) dt. Multiplying both sides of the equation by e R T t r ( τ ) yields

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238 4 Interest Rate Derivative Securities e R T t r ( τ ) ( dV - r ( t ) V dt ) = - Zk ( t ) e R T t r ( τ ) dt, i.e., d ( e R T t r ( τ ) V ) = - Zk ( t ) e R T t r ( τ ) dt. Integrating the equation from t to T , we arrive at Z T t d ( e R T t r ( τ ) V ) = V ( T ) - e R T t r ( τ ) V ( t ) = - Z Z T t k τ ) e R T ¯ τ r ( τ ) d ¯ τ or V ( t ) = V ( T ) e - R T t r ( τ ) + ˆ Z Z T t k τ ) e R T ¯ τ r ( τ ) d ¯ τ ! · e - R T t r ( τ ) = Ze - R T t r ( τ ) ˆ 1 + Z T t k τ ) e R T ¯ τ r ( τ ) d ¯ τ ! . b) Based on the result in a), in this case we have V ( t ) = Ze - r ( T - t ) ˆ 1 + k Z T t e r ( T - ¯ τ ) d ¯ τ ! = Ze - r ( T - t ) h 1 + k ( e r ( T - t ) - 1) /r i . c) Because k ( t ) = k 1 δ ( t - T 1 ) + k 2 δ ( t - T 2 ) , we have V ( t ) = Ze - R T t r ( τ ) " 1 + Z T t ( k 1 δ τ - T 1 ) + k 2 δ τ - T 2 )) e R T ¯ τ r ( τ ) d ¯ τ # = Ze - R T t r ( τ ) 1 + k 1 e R T T 1 r ( τ ) + k 2 e R T T 2 r ( τ ) · for t (0 , T 1 ) , Ze - R T t r ( τ ) 1 + k 2 e R T T 2 r ( τ ) · for t ( T 1 , T 2 ) , Ze - R T t r ( τ ) for t ( T 2 , T ) = Ze - R T t r ( τ ) + k 1 Ze - R T 1 t r ( τ ) + k 2 Ze - R T 2 t r ( τ ) for t (0 , T 1 ) , Ze - R T t r ( τ )
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