# FxPaul

Math in finance or vice versa

## SABR model calibration – attempt 2

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. It was developed by Patrick Hagan, Deep Kumar, Andrew Lesniewski, and Diana Woodward.

The SABR model describes a single forward F, such as a LIBOR forward rate, a forward swap rate, or a forward stock price. The volatility of the forward F is described by a parameter σ. SABR is a dynamic model in which both F and σ are represented by stochastic state variables whose time evolution is given by the following system of stochastic differential equations:
$dF_t = \sigma_t F_t^{\beta} dW_t$
$d\sigma_t = \alpha\sigma_t dZ_t$
Constant parameters should satisfy the condition $0 \leq \beta \leq 1, \alpha \geq 0$
Here, $W_t$ and $Z_t$ are two correlated Wiener processes with correlation coefficient $-1\leq\rho\leq 1$. For simplicity sake, we assume that $\beta = 1$, therefore, we put $dZ_t dW_t = \rho dt$:
$dF_t = \sigma_t F_t dW_t$
$d\sigma_t = \alpha\sigma_t dZ_t$

Written by fxpaul

November 1, 2011 at 13:50

3. Correlation is bound in the interval $[-1, 1]$