## Swaption Cube

Subtype of Vol InputThis 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 Ref Quotable defined within

**relates to a Swap Rate.**

*Vol Spec*The fundamental assumption is that for a fixed expiry

**and underlying swap maturity**

*T1***, the forward rate**

*T2***is a martingale diffused as**

*F(T1,T2)***according to the stochastic volatility model SABR Model**

*dF = σ(F^β)dw*Note that we do not assume that the same SABR parameters apply to all different combinations of

**and**

*T1***.**

*T2*In other words, for each pair

**, we allow the respective forward rate**

*T1, T2***to diffuse as**

*F(T1,T2)***, but with parameters**

*dF = σ(F^β)dw***that depend on**

*α, β, ν, ρ***.**

*T1, T2*All this gives rise to a non-flat Black volatility structure in the following sense:

For any pair

**, the respective European swaption with strike**

*T1, T2***would have a theoretical price**

*K***.**

*P(K,T1,T2)*In general, there must exist a Black vol

**that would lead to the exact same price**

*σB***when the lognormal Black diffusion**

*P***had been assumed instead.**

*dF = (σB)Fdw*This defines the vol

**as a function of**

*σB***, of which the graphical display requires 3 axes for the independent parameters.**

*K, T1, T2*The exact same argument applies with regard to the Normal vol

**and Shifted Lognormal vol**

*σN***associated with a normal and shifted lognormal diffusion of the forward rate**

*σL***respectively.**

*F*We could perhaps specify the SABR vol structure through an array of quartets

**, where each quartet corresponds to a pair**

*α, β, ν, ρ***and specifies the stochastic vol diffusion of the respective forward rate**

*T1, T2***.**

*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

**or**

*σB, σN***.**

*σL*In addition, certain SABR model switches, such as initial guesses of SABR parameters, may be specified in an optional input object of type SABR Model

The 3-dimensional grid is specified by a HyperTable 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*