**CppQuant Answer**

- To price European options on futures we can use the Black model: p = e^(-r*T)*{X*[1 - N(d2)] - f(0,T)*[1 - N(d1)]}
- The time to maturity is T = 150/365 = 0.4110.
- f(0, T) = 0.0525, X = 0.055
- d1 = [ln( f(0T)/X) + (s^2/2)/T] / (s T^0.5) = [ln(0.0525/0.055) + 0.082/2 * 0.4110] / (0.08 x 0.41101/2 = -0.8815
- d2 = d1 - s T^0.5 = -0.8815 - 0.08 x 0.41101/2 = -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.

- The answer is given under assumption that the option payoff occurs at the option expiration. However, this interest rate option expires in 90 days and pays off 90 days that that. Therefore, we need to 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
- As the underlying rate and exercise rate are expressed as annual rates, the answer is an annual rate. However, interest rate option prices are often quoted as periodic rates. We need to convert the result to periodic rate based on a 90-day rate and using the customary 360-day year: 0.00265 x (90/365) = 0.0006625.
- the price is 10,000,000 x 0.0006625 = $6,625.

*Bonus Points*

- c = e^(-r*T)*[f(0,T)*N(d1) - X*N(d2)]
- As with the Black-Scholes-Merton formula, this model applies to European options only. We can also use the model for American options on forwards as they are never exercised early.
- volatility refers to the volatility of the continuously compounded change in the futures price.

*Category: C++ Quant > Derivatives > Options*

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