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 rates derivatives markets.
The SABR model describes a single forward , such as a LIBOR forward rate, a forward swap rate, or a forward stock price. The volatility of the forward is described by a parameter . SABR is a dynamic model in which both and 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 and . Here, and are two correlated Wiener processes with correlation coefficient
where, for clarity, we have set . The value denotes a conveniently chosen midpoint between and (such as the geometric average or the arithmetic average ). We have also set
The function entering the formula above is given by: