Black model

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The Black model (sometimes known as the Black-76 model) is a variant the Black-Scholes option pricing model. It is widely used in the futures market and interest rate market for pricing bond options. It was first presented in a paper written by Fischer Black in 1976.

The main problem with the Black model is that it does not easily deal with price correlation of multiple options. Each option is considered to be priced independently of other options and the linkages between the prices of different options is not easily incorporated into the model.

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

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The Black formula

The Black formula is similar to the Black-Scholes formula for valuating 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 struck 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>
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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 follows such a process. 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.

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See also

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External links

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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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