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SABR_ModelSABR Model is a child type of Model[Quotable] that represents the SABR Stochastic Alpha Beta Rho volatility model (2002) whereby a single forward F - such as a forward swap rate with a given maturity and tenor or a forward stock price with a given maturity - is modelled as a two-factor diffusion process that follows the SDE:
dF = Γ³(F^ΓΆ)dw
where w is a Wiener process, ΓΆ is the Beta constant and Γ³ is the forward's stochastic volatility, which itself follows the SDE:
dΓ³ = ΓΓ³dΓΉ
where v (Nu) is constant and ΓΉ is another Wiener process having correlation Γ± (Rho) with w.Web reference available here
We refer to the initial value of Γ³ as Γ΅ (Alpha), i.e. Γ΅ = Γ³(0)
Typically one uses the SABR model for the simultaneous description of the evolution of a collection of forward rates, such as the forward swap rates spanned by several combinations of swap start dates and tenors.
In the latter case, each forward swap rate is assigned its own quartet of SABR parameters.
Deriscope enables the user to specify a particular SABR model by optionally supplying initial guess values for the constant parameters Γ΅, ΓΆ, Γ, Γ± for each forward.
Alternatively a flat guess value may be defined for each parameter that will apply to all forwards.
Then Deriscope will generate the optimal set of parameters that causes a given input set of instruments to have SABR-implied prices that match the supplied market prices.
A good example is the so called swaption volatility cube, where for each forward swap rate at least 3 swaptions referencing that rate but having different strikes are defined.
Under a 3-dimensional coordinate system where the axis x measures the swaption expiry, the axis y measures the tenor of the underlying swap and the axis z measures the swaption strike, a single point represents a swaption, so that a given collection of swaptions corresponds to a 3-dimensional grid of points.
To each such point (i.e. to each swaption) a certain market price is assigned, typically quoted as either Black or normal vol.
The collection of all these points together with their assigned market vols is known as market volatility cube.
Assuming the four SABR parameters associated with each point are known, a SABR-implied swaption price can be calculated for each point.
The collection of all these points together with their SABR-implied prices is known as SABR-implied volatility cube.
The optimization routine implemented in Deriscope effectivelly calculates the four SABR parameters per point so that the two volatility cubes match as much as possible.
The user can dictate whether some of the guess parameter values should be kept fixed during the optimization.