Black formula
The Black formula is a function that maps 4 positive numbers f,K,T,σ and a special integer φ = ±1 to a value BLACK(f,K,T,σ,φ) according to the formula:
(f,K,T,σ,φ) -> BLACK(f,K,T,σ,φ) := Dφ[fN(φd₊) - KN(φd₋)]
where
D is the discount factor for a maturity equal to T.
N(x) is the cumulative distribution function (CDF) of the of the standard normal distribution
d± = [ln(f/K) ± ½σ²T]/(σT¹ᐟ²)
While the above formula is just a definition, one can prove that if V(0) represents the price (i.e. present value) as of time t = 0 of a European call (if φ = 1) or put (if φ = -1) with expiry T (in annual units) and strike K on any tradable asset of which the forward price with maturity T observed at t = 0 is denoted as f(0;T), then V(0) is given by the formula:
V(0) = BLACK(f(0;T),K,T,σ,φ)
provided f(t;T), i.e. the forward price observed at time t, is diffused over t as df(t;T) = σf(t;T)dw with respect to the forward martingale measure.
The latter provision is sufficient, but by no means necessary, since all that is needed to prove the stated relationship is that by the time T, ln(f(T;T)) is distributed according to a normal distribution with standard deviation σT¹ᐟ² and mean ln(f(0;T)) - ½σ²T
The Black formula is often used to quote the price of options in volatility terms even when the underlying forward price is not diffused lognormally.
For example, consider a stock option with strike K and expiry T and also assume the underlying forward stock price f(t;T) is diffused in a complicated way so that its distribution at time T is not lognormal.
If that stock option were traded at time t = 0 at a price V(0), the traders would not quote its price V(0), but rather the number σ that solved the Black formula:
V(0) = BLACK(f(0;T),K,T,σ,φ)
and refer to σ as the option's Black volatility at time t = 0.
Note, this is a pure quotation choice that makes no assumption on the underlying's stochastic evolutuion.
For example, even if the forward stock price were assumed to evolve in a wat that f(T;T) were a constant C, then the Black volatility for any pair K,T such that C > K would be the number σ satisfying D(C-K) = Black(C,K,T,σ,φ).