The Excel Derivatives Periscope



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!