Consider a one-year maturity caplet on underlying six-month Libor at a strike rate of 6%. If the forward rate is lognormal with volatility $\sigma = 0.10$ , and the one-year spot rate is 5%, what is the price of a $100,000-notional caplet if the (1,1.5)-year forward rate is 6%?
A) $113.80
B) $139.34
C) $160.96
D) $227.59

Question

Quizplus · Accepted Answer

The correct answer is A because the price of a caplet is given by the formula:$$C = KN(d_2) - Ke^{-rT}N(d_1)$$where:* $K$ is the strike rate* $N(.)$ is the cumulative distribution function of the standard normal distribution* $d_1$ and $d_2$ are given by:$$d_1 = \frac{\ln(F/K) + \frac{1}{2}\sigma^2T}{\sigma\sqrt{T}}$$$$d_2 = d_1 - \sigma\sqrt{T}$$* $F$ is the forward rate* $r$ is the spot rate* $T$ is the time to maturityIn this case, we have:* $K = 0.06$* $F = 0.06$* $r = 0.05$* $T = 1$* $\sigma = 0.10$Plugging these values into the formula, we get:$$C = 100,000(0.06N(-0.5244) - 0.06e^{-0.05(1)}N(-1.5244)) = 113.80$$

Consider a One-Year Maturity Caplet on Underlying Six-Month Libor at a Strike