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 $\alpha$ value for $f ( 1,1 )$ ?

Question

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 $\alpha$ value for $f ( 1,1 )$ ?

Quizplus · Accepted Answer

The $\alpha$ value for $f(1,1)$ is calculated based on ensuring the expected rate at time $t+1$ matches the initial forward rate for that period. Given the initial forward rate for the period between one and two years is 7% (or 0.07 in decimal form), and the process for the rate evolution includes an up or down movement of $\pm 0.02$, the expected rate at $t+1$ should equal the initial forward rate. The formula for the expected rate is $f(t,T) + \alpha + 0.5(0.02) - 0.5(0.02) = f(t,T) + \alpha$. Since $f(t,T)$ for $t=0, T=1$ is 6% (or 0.06), and we want the expected rate to be 7%, we set $0.06 + \alpha = 0.07$, solving for $\alpha$ gives $\alpha = 0.01$ or $+0.0002$ when expressed in the format consistent with the options provided.

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

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