Black model

From Wikipedia, the free encyclopedia

The Black model (sometimes known as the Black-76 model) is a variant of the Black-Scholes option pricing model. Its primary applications are for pricing bond options, interest rate caps / floors, and swaptions. It was first presented in a paper written by Fischer Black in 1976.

Black's model can be generalized into a class of models known as log-normal forward models.

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

[edit] The Black formula

The Black formula is similar to the Black-Scholes formula for valuing stock options except that the spot price of the underlying is replaced by the forward price.

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

c = e - r T (F N (d 1) - K N (d 2))

where

r

is the risk-free interest rate

F

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

$d_1 = \frac{ln(\frac{F}{K}) + \frac{\sigma^2T}{2}}{\sigma\sqrt T}$

$d_2 = d_1 - \sigma\sqrt T$

σ

is the volatility of the forward price.

and

N (.)

is the standard cumulative Normal distribution function.

The put price is

p = e - r T (K N ( - d 2) - F N ( - d 1)).

[edit] 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 at maturity of the option is log-normally distributed. 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.

[edit] See also

[edit] External links

Options on Futures: quantnotes.com or riskglossary.com
Foreign exchange options: riskglossary.com
Online real-time pricing models: Black, Black & Scholes, Garman & Kohlhagen calculator, sitmo.com

[edit] 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.