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.

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

$d_{1}={\frac {\ln(S/K)+(r+\sigma ^{2}/2)T}{\sigma {\sqrt {T}}}}$

$d_{2}={\frac {\ln(S/K)+(r-\sigma ^{2}/2)T}{\sigma {\sqrt {T}}}}=d_{1}-\sigma {\sqrt {T}}.$

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.

See also

• Financial mathematics
• Black-Scholes

External links

• Options on Futures: quantnotes.com
• '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.