Can you please check if the logic or way the previous Expert solved problem 4 is correct? because the calculations do not seem to be correct.
Problem 4
Let's denote the 2-year and 6-year continuously compounded rates as r2 and r6, respectively. The payoff of the derivative in 4 years is given by (r6 - r2) * $10 million.
Using the Black-76 model, we can calculate the value of the derivative as:
V=e−⋊T×[$10m×(FN(d1)−K×FN(d2))]
where:
T = 4 years
K = 1 (since we're looking at the difference between two rates)
sigma = 20%
F_N(d1) and F_N(d2) are the probabilities that a standard normal variable is greater than or equal to d1 and d2, respectively. These can be calculated using the Black-76 formula.
To calculate d1 and d2, we need to use the following equations:
d1=ln(FK)+0.5σ2TσT
d2=d1−σT
where F is the forward rate between years 2 and 6, which can be calculated using the formula:
F=er66er22
Plugging in the values, we get:
F=e0.046e0.042=1.1892
d1=ln(1.18921)+0.50.2240.24=1.6036
d2=1.6036−0.24=1.2036
F_N(d1) = 0.9332
F_N(d2) = 0.8850
Therefore, the value of the derivative is:
V=e−0.04×4×[$10m×(0.9332−1×0.8850)]
=$1,587,007.12
So the value of the derivative is approximately $1,587,007.12.