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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(d1) - KN(d2))

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]
σ is the volatility of the forward price.
and N(.) is the standard cumulative Normal distribution function.

The put price is

p = e - rT(KN( - d1) - FN( - d2))