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Vol Type refers to List of definitions of what "volatility" exactly (mathematically) means in the context of a given stochastic process, a given risk factor and some fixed future time.
Available Vol Type types:
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!
Variation of the
Vol Spec::Vol Type::Black whereby dF = σdw that results in F(T) being normally distributed.
E{ max(F(T)-K,0) } can be then calculated through a formula known as Bachelier formula.
Note that the normal vol determines the absolute size of the fluctuation around the starting value, in contrast to the Black vol that measures the relative deflection.
Shifted Lognormal
Variation of the
Vol Spec::Vol Type::Black whereby the logmormal distribution of F(T) is shifted by a certain amount θ to the right.
When interest rates are negative, a positive θ is used so that the lognormal distribution is shifted to the left towards the negative rates regime.
The shifted lognormal terminal distribution arises from the dynamics d(F+θ) = σ(F+θ)dw, which treats d(F+θ) as being always positive.
It follows that a positive θ results to an F at time T of which the lognormal distribution is shifted to the left by an amount equal to θ.
Note this volatility convention reduses to the Black convention when θ = 0.