Consider a one-factor HJM model where the initial forward curve is given as 6% for one year and 7% between one and two years. The evolution of continuously-compounded one-year forward rates beginning at time $T$ , is given by the following binomial process: $f ( t + 1 , T ) = f ( t , T ) + \alpha \pm 0.02$ , where the up and down movements are equiprobable. What is the price of a one-year call option on a two-year 6.5% coupon bond, with a strike price of $100 ex-coupon?

Question

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The price of the bond at time 0 is 100. The price of the bond at time 1 if the rate goes up is 100 * exp(0.065) * exp(0.07 + 0.02) = 108.20. The price of the bond at time 1 if the rate goes down is 100 * exp(0.065) * exp(0.07 - 0.02) = 101.80. The probability of the rate going up is 0.5, and the probability of the rate going down is also 0.5. Therefore, the expected price of the bond at time 1 is (108.20 * 0.5) + (101.80 * 0.5) = 105.00. The strike price of the call option is 100, so the payoff of the call option at time 1 is max(0, 105.00 - 100) = 5.00. The risk-neutral probability of the rate going up is 0.5, and the risk-neutral probability of the rate going down is also 0.5. Therefore, the price of the call option at time 0 is 5.00 * 0.5 = 2.50.

Consider a One-Factor HJM Model Where the Initial Forward Curve TTT

Consider a One-Factor HJM Model Where the Initial Forward Curve $T$