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.

Contents

1 The Black formula 2 Derivation and assumptions 3 See also 4 External links 5 References

The Black formula

The Black formula for a call option on an underlying struck at K, expiring T years in the future is

c = e ^{- rT}(FN(d₁) - KN(d₂))

where

r is the risk-free interest rate

F is the current forward price of the underlying for the option maturity

σ is the volatility of the forward price.

and N(.) is the standard cumulative Normal distribution function.

The put price is

p = e ^{- rT}(KN( - d₂) - FN( - d₁)).

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.