Vol_Spec__Vol_Type__Black

Appropriate when the value *x(T)* of the Quotable in Key Vol Spec::Ref Quotable at a given time *T* with respect to the given risk factor is distributed in any fashion.

For conceteness, *x(T)* could be the forward price of some stock or the forward (Libor, swap or FX) rate with forward maturity *T* as observed at some earlier variable time *t*.

Then the *Black* volatility surface of *x* with respect to maturity *T* and strike *K* is the function *σ(K,T)* defined by requiring that the equality *E{ max(x(T)-K,0) } = Black(K,T,σ)* holds for all appropriate values *T* and *K*.

Here *E* denotes the Expectation operator with respect to a measure where *x* behaves as martingale and *Black(K,T,σ)* is defined to equal *E{ max(F(T)-K,0) }* for some *F* diffused as *dF = σFdw*.

For any fixed *T*, *F(T)* is lognormally distributed and *E{ max(F(T)-K,0) }* can be easily calculated through the well known Black formula.

Note that the Black volatility of a process *x* for a given pair *T* and *K* has nothing to do with the true volatility of *x*.

For example, if *x* is a constant *C*, then its Black volatility for any pair *K,T* such that *C > K* would be the number *σ* satisfying *C-K = Black(K,T,σ)*, even though *x* is a non-volatile, constant process!