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 $100 notional one-year cap on the one-year forward rate at a strike rate of 7%.

Question

Quizplus · Accepted Answer

The price of the cap can be calculated using the given binomial process for the evolution of forward rates. The initial forward rate for the period between one and two years is 7%. According to the process, the rate can either go up to 7.02% or down to 6.98%. Since the cap strike rate is 7%, only the upward movement results in a payoff. The expected payoff is therefore $0.5 \times \max(7.02\% - 7\%, 0) = 0.5 \times 0.02\% = 0.01\%$. Discounting this back to the present value at the initial one-year forward rate of 6% gives $0.01\% / (1 + 6\%) \approx 0.0094\%$. On a $100 notional, this is approximately $0.0094, or when rounded and presented in the correct format, $0.87.

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

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