In mathematical finance, the SABR model is a stochastic volatility model, which attempts to capture the volatility smile in derivatives markets. The name stands for "stochastic alpha, beta, rho", referring to the parameters of the model. The SABR model is widely used by practitioners in the financial industry, especially in the interest rate derivative markets. It was developed by Patrick S. Hagan, Deep Kumar, Andrew Lesniewski, and Diana Woodward.
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Dynamics
The SABR model describes a single forward
with the prescribed time zero (currently observed) values
The constant parameters
The above dynamics is a stochastic version of the CEV model with the skewness parameter
Asymptotic solution
We consider a European option (say, a call) on the forward
Except for the special cases of
It is convenient to express the solution in terms of the implied volatility of the option. Namely, we force the SABR model price of the option into the form of the Black model valuation formula. Then the implied volatility, which is the value of the lognormal volatility parameter in Black's model that forces it to match the SABR price, is approximately given by:
where, for clarity, we have set
and
The function
Alternatively, one can express the SABR price in terms of the normal Black's model. Then the implied normal volatility can be asymptotically computed by means of the following expression:
It is worth noting that the normal SABR implied volatility is generally somewhat more accurate than the lognormal implied volatility.
SABR for the negative rates
A SABR model extension for Negative interest rates that has gained popularity in the recent years is the shifted SABR model, where shifted forward rate is assumed to follow a SABR process
for some positive shift
The SABR model can also be modified to cover Negative interest rates by:
for
An obvious drawback of this approach is the a priori assumption of potential highly negative interest rates via the free boundary.
Arbitrage problem in the implied volatility formula
Although the asymptotic solution is very easy to implement, the density implied by the approximation is not always arbitrage-free, especially not for very low strikes (it becomes negative or the density does not integrate to one).
One possibility to "fix" the formula is use the stochastic collocation method and to project the corresponding implied, ill-posed, model on a polynomial of an arbitrage-free variables, e.g. normal. This will guarantee equality in probability at the collocation points while the generated density is arbitrage-free . Using the projection method analytic European option prices are available and the implied volatilities stay very close to those initially obtained by the asymptotic formula.
Another possibility is to rely on a fast and robust PDE solver on an equivalent expansion of the forward PDE, that preserves numerically the zero-th and first moment, thus guaranteeing the absence of arbitrage.
Extensions
The SABR model can be extended by assuming its parameters to be time-dependent. This however complicates the calibration procedure. An advanced calibration method of the time-dependent SABR model is based on so-called "effective parameters".