Black model

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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.

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[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

<math> c = e^{-rT}*[FN(d_1) - KN(d_2)]</math>

where

<math>r</math> is the risk-free interest rate
<math>F</math> is the current forward price of the underlying for the option maturity
<math>d_1 = \frac{ln(\frac{F}{K}) + \frac{\sigma^2T}{2}}{\sigma\sqrt T}</math>
<math>d_2 = d_1 - \sigma\sqrt T</math>
<math>\sigma</math> is the volatility of the forward price.
and <math>N(.)</math> is the standard cumulative Normal distribution function.

The put price is

<math> p = e^{-rT}*[KN(-d_2) - FN(-d_1)]</math>

[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.it:Formula di Black
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