Swaption CubeSubtype of
This type is exclusively used to describe the volatility of forward interest rate swap rates.
Web blog example here
It thus only makes sense if the entry defined within Vol Spec relates to a .
The fundamental assumption is that for a fixed expiry T1 and underlying swap maturity T2, the forward rate F(T1,T2) is a martingale diffused as dF = σ(F^β)dw according to the stochastic volatility model
Note that we do not assume that the same SABR parameters apply to all different combinations of T1 and T2.
In other words, for each pair T1, T2, we allow the respective forward rate F(T1,T2) to diffuse as dF = σ(F^β)dw, but with parameters α, β, ν, ρ that depend on T1, T2.
All this gives rise to a non-flat Black volatility structure in the following sense:
For any pair T1, T2, the respective European swaption with strike K would have a theoretical price P(K,T1,T2).
In general, there must exist a Black vol σB that would lead to the exact same price P when the lognormal Black diffusion dF = (σB)Fdw had been assumed instead.
This defines the vol σB as a function of K, T1, T2, of which the graphical display requires 3 axes for the independent parameters.
The exact same argument applies with regard to the vol σN and vol σL associated with a normal and shifted lognormal diffusion of the forward rate F respectively.
We could perhaps specify the SABR vol structure through an array of quartets α, β, ν, ρ, where each quartet corresponds to a pair T1, T2 and specifies the stochastic vol diffusion of the respective forward rate F(T1,T2).
A far simpler method adopted by Deriscope is to specify instead the 3-dimensional grid of market vols as defined above.
Such an input grid is referred as "vol cube" and may consist by the σB, σN or σL.
In addition, certain SABR model switches, such as initial guesses of SABR parameters, may be specified in an optional input object of type
The 3-dimensional grid is specified by a object containing volatilities for various combinations of strike spread, expiry and swap tenor.
Typically the first cube dimension spans the strike spreads (differences from the atm rate level), the second the option expiries and the third the swap tenors.
The strike spread coordinate must include the number 0 and the corresponding 2-dimensioal sub-table must contain the at-the-money swaption vols.
The 2-dimensioal sub-tables associated with the non-zero strike spread coordinates should not contain the absolute vol levels but rather the vol spreads, defined as the differences between the vol levels for the respective strike and the atm vol levels.
Bilinear interpolation is assumed for any missing entries in the supplied tables.
The QuantLib implementation is the SwaptionVolCube1.