SABR Model
SABR Model is a direct subtype of Model[Quotable]
aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa
TYPE INCLUSION RELATIONSHIPS
AVAILABLE FUNCTIONS
AVAILABLE CREATE FUNCTION KEYS
TYPICAL OBJECTS OF TYPE SABR Model
This type represents the SABRStochastic 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:
dFₜ = αₜFₜᵝdw
where w is a Wiener process, β is the Beta constant and αₜ is the forward's stochastic volatility, which itself follows the SDE:
dαₜ = ναₜdu
where v (Nu) is constant and u is another Wiener process having correlation ρ (Rho) with w. Web reference available here
We refer to the initial value of αₜ as α (Alpha), i.e. α = αₜ(0)
More details on the model definition and and the calibration of its parameters (also described below) are available at here
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 underlying swap tenors.
In the latter case, each forward swap rate with a given fixed swap start date and underlying swap tenor is assigned its own quartet of SABR parameters α, β, ν, ρ.
Below is a schematic diagram of the time evolution of three such forward swap rates:
Deriscope enables the user to specify a particular SABR model by optionally supplying initial guess values for all applicable parameter quartets.
Alternatively a flat guess value may be defined for each parameter that will apply to all forward swap rates.
Then Deriscope will generate the optimal set of parameters that causes a given input set of instruments to have SABR-implied prices that match - as much as possible - the supplied market prices.
In practice, the optimal set of parameters is found by matching - within tolerance - the given market volatilities rather than the market prices, which is much easier to do because there exist approximate closed form formulas that return the vols implied by any given set of SABR parameters.
Specifically, the following formula from Hagan's paper gives the Black vol:
with the following reduced form in the case for At-The-Money options:
The following formula gives the normal vol:
Note, the shifted lognormal vols cannot be produced by SABR parameters calibrated to vols quoted in either Black or normal form.
They require SABR parameters calibrated to calibrated to vols quoted in shifted lognormal form with the exact same shift amount.
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 swaption price is assigned, typically quoted as a volatility in Black (i.e. lognormal), normal or shifted lognormal terms.
The collection of all these points together with their assigned market vols is known as market volatility cube, schematically depicted below:
Assuming the four SABR parameters associated with each point are known - note that points on a vertical line, i.e. a line paraller to the strike axis, share the same SABR parameters -, 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, schematically depicted below:
The optimization routine implemented in Deriscope effectivelly calculates the four SABR parameters per vertical line 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.