Black model
The Black model (sometimes known as the Black-76 model) is a variant (and more general form) of the Black-Scholes option pricing model. It is widely used in the futures market and interest rate market for pricing options. It was first presented in a paper written by Fischer Black in 1976.
The Black formula
The Black formula for a call option on an underlying struck at K, expiring T years in the future is
- <math> c = e^{-rT}(FN(d_1) - KN(d_2))<math>
where
- <math>r<math> is the risk-free interest rate
- <math>F<math> is the current forward price of the underlying for the option maturity
- <math>d_1 = \frac{log(\frac{F}{K}) + \frac{\sigma^2t}{2}}{\sigma\sqrt t}<math>
- <math>d_2 = d_1 - \sigma\sqrt t<math>
- <math>\sigma<math> is the volatility of the forward price.
- and <math>N(.)<math> is the standard cumulative Normal distribution function.
The put price is
- <math> p = e^{-rT}(KN(-d_2) - FN(-d_1))<math>
Derivation and assumptions
The derivation of the pricing formulas in the model follows that of the Black-Scholes model almost exactly. The assumption that the spot price follows a log-normal process is replaced by the assumption that the forward price follows such a process. From there the derivation is identical and so the final formula is the same except that the spot price is replaced by the forward - the forward price represents the expected future value discounted at the risk free rate.
See also
