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# Black model

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

The derivation of 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.

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

$c = e^{-rT}(FN(d_1) - KN(d_2))$
where
$r$ is the risk-free interest rate
$F$ is the current forward price of the underlying for the option maturity
$d_1 = \frac{log(\frac{F}{K}) + \frac{\sigma^2t}{2}}{\sigma\sqrt t}$
$d_2 = d_1 - \sigma\sqrt t$
$\sigma$ is the volatility of the forward price.
and $N(.)$ is the standard cumulative Normal distribution function.

The put price is

$p = e^{-rT}(KN(-d_1) - FN(-d_2))$

• 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.

All Wikipedia text is available under the terms of the GNU Free Documentation License

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