tssolution3 - IEOR E4710 Term Structure Models Spring 2005...

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IEOR E4710 Term Structure Models: Spring 2005 Columbia University Instructor: Martin Haugh Solutions to Assignment 3 Question 1 (a)Solution . Since W t is a Gaussian distribution for any fixed t [ 0 , T ] , X T : = R T 0 W t dt is also a Gaussian distribution. Obviously, E [ X T ] = 0. And var ( X T ) = E [ X 2 T ] = E [( Z T 0 W t dt ) 2 ] = 2 Z T 0 Z t 0 cov ( W u , W t ) dudt = 2 Z T 0 Z t 0 ududt = T 3 3 . Thus, X T is a normal random variable with mean 0 and variance T 3 3 . (b)Solution . For a normal random variable X N ( μ , σ 2 ) , we have E [ exp ( X )] = exp ( μ + σ 2 2 ) , so E Q [ exp ( - X T )] = exp ( T 3 6 ) . (c)Solution . Since r u = r t + R u t θ ( s ) ds + σ [ W u - W t ] for u [ t , T ] , Z T t = E Q t h e - R T t r u du i = e - r t ( T - t ) - R T t R u t θ ( s ) dsdu E Q t h e - σ R T t W u - W t du i = e - r t ( T - t ) - R T t R u 0 θ ( s ) dsdu + σ 2 ( T - t ) 3 6 . (d)Solution . By (c) , P ( 0 , T ) = Z T 0 = exp ( - r 0 T - Z T 0 Z u 0 θ ( s ) dsdu + σ 2 T 3 6 ) . We can also write P ( 0 , T ) = exp ( - R T 0 f ( 0 , t ) dt ) . Equating the two expressions for P ( 0 , T ) and taking logarithms, we find that r 0 T + Z T 0 Z u 0 θ ( s ) dsdu - σ 2 T 3 6 = Z T 0 f ( 0 , t ) dt . Differentiating twice with respect to the maturity argument T , we find that θ ( t ) = T f ( 0 , T ) | T = t + σ 2 t = t f ( 0 , t ) + σ 2 t . Thus, bond prices produced by the Ho-Lee model will match a given set of bond prices P ( 0 , T ) if the function θ is tied to the initial forward curve f ( 0 , t ) in this way. (e)Solution . By (c) , Z T t can be written as Z T t = exp ( A ( t , T ) - B ( t , T ) r t ) , where B ( t , T ) = T - t . Notice that the Ho-Lee model is also a Gaussian model, the exactly same argument as in Example 1 of the Continuous- Time Short Rate Models lecture notes will work. Then the time t price C t of a European call option on a zero- coupon bond that expires at time T with payoff ( Z U T - K ) + has an expression C t = Z T t E P T t [( Y U T - K ) + ] , where Y U t : = Z U t / Z T t and dY U t = Y U t [ S ( t , U ) - S ( t , T )] dW P T t = Y U t [ σ B ( t , U ) - σ B ( t , T )] dW P T t = σ ( U - T ) Y U t dW P T t , which indicates Y U T is a log-normal random variable and ln ( Y U T ) N ln ( Z U t / Z T t ) - 1 2 σ 2 ( U - T ) 2 ( T - t ) , σ 2 ( U - T ) 2 ( T - t ) , thus C t = Z T t E P T t [( Y U T - K ) + ] = Z U t Φ ( h ) - KZ T t Φ ( h - b ) where h = 1 b ln Z U t KZ T t + 1 2 b and b 2 = σ 2 ( U - T ) 2 ( T - t ) . / 1
Question 2 Solution . The spot rate r ( t , T ) in the period [ t , T ] is defined as r ( t , T ) = - ln ( Z T t ) T - t = B ( t , T ) r t - A ( t , T ) T - t , which is an affine function in r t , for both Vasicek and CIR models. The price of the caplet with maturity