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

• Financial mathematics
• Black-Scholes

External links

• Options on Futures: quantnotes.com or riskglossary.com
• Foreign exchange options: riskglossary.com
• Generalized Black-Scholes Calculator by Espen Gaarder Haug (himself)

References

• Black, Fischer (1976). The pricing of commodity contracts, Journal of Financial Economics, 3, 167–179.
• Garman, Mark B. and Steven W. Kohlhagen (1983). Foreign currency option values, Journal of International Money and Finance, 2, 231–237.