# 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 $dZ_t dW_t = \rho dt$:
$dF_t = \sigma_t F_t dW_t$
$d\sigma_t = \alpha\sigma_t dZ_t$

## SABR solution

In this case it is possible to integrate equation and find an exact solution:
$d(\ln F_t) = -\frac{\sigma_t^2}{2} dt + \sigma_t dW_t$
$d(\ln \sigma_t) = -\frac{\alpha^2}{2} dt + \alpha dZ_t$
Therefore:
$F_t = F_0 \exp\left[-\frac{\sigma_t^2}{2} t + \sigma_t W_t \right]$
$\sigma_t = \sigma_0 \exp\left[-\frac{\alpha^2}{2}t + \alpha Z_t\right]$

Therefore:
$F_t = F_0 \exp\left[-\frac{\sigma_t^2}{2} t + \sigma_t W_t \right]$
$\sigma_t = \sigma_0 \exp\left[-\frac{\alpha^2}{2}t + \alpha Z_t\right]$

## Calibration by simple linear regression

Let’s introduce new variables for convenience:
$f_t = \ln F_t$
$s_t = \ln \sigma_t$

By naive approach, in finite differences the SABR model looks:
$f_t - f_{t-1} = -\frac{1}{2} dt \exp{2s_t} + dW_{t-1} \exp{s_t}$
$s_t - s_{t-1} = -\frac{\alpha^2}{2} dt + \alpha dZ_{t-1}$

Assumption of correlation gives
$f_t - f_{t-1} = -\frac{1}{2} \exp{2s_t} \Delta t + \exp{s_t} \sqrt{dt} \Delta N_{t-1}$
$s_t - s_{t-1} = -\frac{\alpha^2}{2} \Delta t + \alpha \sqrt{dt} \left(\rho \Delta N_{t-1} + \sqrt{1-\rho^2} \Delta M_{t-1}\right)$
where $\Delta N$ and $\Delta M$ are independent normal distributed random variables.

I don’t know what to do further with it. So, I’ve to fallback to brute force Maximum Likelihood Estimation with Monte Carlo sampling to obtain conditional probability.

Written by fxpaul

November 1, 2011 at 13:50