Vol Ref
Vol Ref is a direct subtype of Financial
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TYPE INCLUSION RELATIONSHIPS
AVAILABLE FUNCTIONS
AVAILABLE CREATE FUNCTION KEYS
EXAMPLE OF OBJECTS OF TYPE Vol Ref
This type represents a collection of specification parameters being used by Vol objects in defining the exact type of volatility that these objects represent.
A formal description follows:
In order to properly define a volatility of some stochastic process against some risk factor, two separate stochastic processes must be first defined.
The first is the process where the volatility applies. It is supplied as an object of type Quotable by the key Ref Quotable
The second is the reference process defining the risk factor. It is supplied as an object of also type Quotable by the key Risk Factor
It is assumed that the Quotable Value x1 of the Quotable in Ref Quotable follows the jump-diffusion process:
dx1 = μ1dt + σ1dw1 + dN1
where μ1 = drift, t = time, σ1 = overall volatility, w1 = wiener process, N1 = pure jump process.
It is similarly assumed that the value x2 of the Quotable in Risk Factor follows the jump-diffusion process:
dx2 = μ2dt + σ2dw2 + dN2
We can always decompose as σ1dw1 = σ1'dw2 + σ1''dw3 where dw3 is orthogonal to dw2, so that
dx1 = μ1dt + σ1'dw2 + σ1''dw3 + dN1
This final equation describes how the value x1 of Ref Quotable is evolved against the non-jump random increment dw2 of the value x2 of Risk Factor
The quantity σ1' is exactly the volatility described here.