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, also referred to as LIBOR Market Model.

Contents

[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 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(d1) − KN(d2)]

The put price is

p = e − rT * [KN( − d2) − FN( − d1)]

where

d_1 = \frac{\ln(F/K) + (\sigma^2/2)T}{\sigma\sqrt{T}}
d_2 = \frac{\ln(F/K) - (\sigma^2/2)T}{\sigma\sqrt{T}} = d_1 - \sigma\sqrt{T}.

[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 undiscounted expected future value.

[edit] See also

[edit] External links

[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.
v • d • e
Derivatives market
Derivative (finance)
Options
Swaps
Other derivatives
Languages