Black model

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, also referred to as LIBOR Market Model.

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 F.

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

:$c\; =\; e^\{-rT\}*\; [FN(d\_1)\; -\; KN(d\_2)]$

The put price is

:$p\; =\; e^\{-rT\}*\; [KN(-d\_2)\; -\; FN(-d\_1)]$

where

:$d\_1\; =\; frac\{ln(F/K)\; +\; (sigma^2/2)T\}\{sigmasqrt\{T$

:$d\_2\; =\; frac\{ln(F/K)\; -\; (sigma^2/2)T\}\{sigmasqrt\{T\; =\; d\_1\; -\; sigmasqrt\{T\}.$

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 undiscounted expected future value.

ee also

*Financial mathematics
*Black-Scholes

External links

*Options on Futures: [http://www.quantnotes.com/fundamentals/futures/optionsonfutures.htm quantnotes.com]
* [http://www.cba.ua.edu/~rpascala/greeks2/GFOPMForm.php 'Greeks' Calculator using the Black model] , Razvan Pascalau, Univ. of Alabama

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.