5.22.2005

Q&A: Calculate the price of the interest rate put option using the Black model

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%. The notional principal is $10 million.

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

Elsewhere on BlogSpot

A Point So Often Missed: The Presence of an Administered Rate
INCITE - ... of all sorts of financial risk is what is known as the Risk Free Rate (RFR). What the RFR intends to capture is the sum of the Real Rate of Risk (which ...

Deconstructing VIX
RANDOM ROGER S BIG PICTURE - The risk free rate of return an investor can get from a Treasury Bill goes into every type of options pricing model that I have ever heard of...

Angry Bear
ECONOMIST S VIEW - Of course, this inverse relationship between the exogenous growth rate and the ... v. bonds so as to achieve the optimal; mix of expected return and risk...

Revisiting the Lottery
POLITICAL CALCULATIONS - Not that long ago, I looked at the cost of risk versus the value of the benefit ... This higher level will be directly related to the marginal tax rate that ...

October 2004
WISBLAWG - FROM THE UW LAW LIBRARY - Free Webinar Convincing Your Boss or Client! That It s Not All Free on ... Web searching have a new privacy risk to worry about: Google s free new tool ...

1 comment:

  1. Half of difficulty following your calculation is figuring out why you use * in some places, and x in other places to denote multiplication. That's multiplication, right?

    ReplyDelete