Exercise 24.6.4 Consider the symmetricrandom walk for modeling the short rate, r (t +1) = +r

Exercise 24.6.4 Consider the symmetricrandom walk for modeling the short rate, r (t +1) = α +ρr (t)±σ. Let V denote the current value of an interest rate derivative, Vu its value at the next period if rates rise, and Vd its value at the next period if rates fall. Define u ≡ eα+ρr (t)+σ /P(t, t +2) and d ≡ eα+ρr (t)−σ /P(t, t +2), so they are the gross one-period returns on the two-period zero-coupon bond when rates go up and down, respectively. (1) Show that a portfolio consisting of $B worth of one-period bonds and two-period zero-coupon bonds with face value $ to match the value of the derivative requires

= Vu −Vd

(u−d) P(t, t +2)

, B= uVd −dVu

(u−d) er (t) .

(2) Prove that V = pVu +(1− p) Vd er (t) , where p ≡ (er (t) −d)/(u−d).