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.
are represented by stochastic state variables whose time evolution is given by the following system of stochastic differential equations: with the prescribed time zero (currently observed) values
controls the height of the ATM implied volatility level.
The above dynamics is a stochastic version of the CEV model with the skewness parameter
The general case can be solved approximately by means of an asymptotic expansion in the parameter
Under typical market conditions, this parameter is small and the approximate solution is actually quite accurate.
Also significantly, this solution has a rather simple functional form, is very easy to implement in computer code, and lends itself well to risk management of large portfolios of options in real time.
It is convenient to express the solution in terms of the implied volatility
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
entering the formula above is given by Alternatively, one can express the SABR price in terms of the Bachelier's model.
The approximation accuracy and the degree of arbitrage can be further improved if the equivalent volatility under the CEV model with the same
[2] A SABR model extension for negative interest rates that has gained popularity in recent years is the shifted SABR model, where the shifted forward rate is assumed to follow a SABR process for some positive shift
Since shifts are included in a market quotes, and there is an intuitive soft boundary for how negative rates can become, shifted SABR has become market best practice to accommodate negative rates.
The SABR model can also be modified to cover negative interest rates by: for
Its exact solution for the zero correlation as well as an efficient approximation for a general case are available.
[3] An obvious drawback of this approach is the a priori assumption of potential highly negative interest rates via the free boundary.
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.
[4] 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.
[5] The SABR model can be extended by assuming its parameters to be time-dependent.
An advanced calibration method of the time-dependent SABR model is based on so-called "effective parameters".
[6] Alternatively, Guerrero and Orlando[7] show that a time-dependent local stochastic volatility (SLV) model can be reduced to a system of autonomous PDEs that can be solved using the heat kernel, by means of the Wei-Norman factorization method and Lie algebraic techniques.
Explicit solutions obtained by said techniques are comparable to traditional Monte Carlo simulations allowing for shorter time in numerical computations.
As the stochastic volatility process follows a geometric Brownian motion, its exact simulation is straightforward.
However, the simulation of the forward asset process is not a trivial task.
Recently, novel methods have been proposed for the almost exact Monte Carlo simulation of the SABR model.
[8] Extensive studies for SABR model have recently been considered.