An interest rate put option based on a 90-day underlying rate has an exercise rate of 5.5% and expires in 150 days. The forward rate is 5.25% and the volatility is 0.08. The continuously compounded risk-free rate is 4%. Calculate the price of the interest rate put option using the Black model. The notional principal is $10 million.

Correct Answer: $6,625

The time to maturity is T = 150/365 = 0.4110.
f₀(T) = 0.0525
X = 0.055
d₁ = [ln(f₀(T)/X) + (σ²/2)/T] / (σ T^1/2) = [ln(0.0525/0.055) + 0.08²/2 x 0.4110] / (0.08 x 0.4110^1/2 = -0.8815
d₂ = d₁ - σ T^1/2 = -0.8815 - 0.08 x 0.4110^1/2 = -0.9327
N(d₁) = N(-0.8815) = 1 - N(0.8815) = 0.1894
N(d₂) = N(-0.9327) = 1 - N(0.9327) = 0.1762

p = e^{-0.04 x 0.4110} [0.055 x (1 - 0.1762) - 0.0525 x (1 - 0.0.1894)] = 0.002708

Two adjustments:
1. Use the forward rate to discount the result back from day 240 to day 150: 0.002708 x e ^{-0.0525 x (90/365)} = 0.00265.
2. Convert the result to a periodic rate based on a 90-day rate, using the customary 360-day year: 0.00265 x (90/360) = 0.0006625.

Therefore, the price is 10,000,000 x 0.0006625 = $6,625.