# FxPaul

Math in finance or vice versa

## SABR model calibration

Don’t use this!

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 $\rho = 1$, therefore, we put $Z_t = W_t$:
$dF_t = \sigma_t F_t^{\beta} dW_t$
$d\sigma_t = \alpha\sigma_t dW_t$

Written by fxpaul

June 17, 2011 at 09:27

Posted in trading math

## Is EUR/USD mean reverting?

Nassim Taleb in his book “Dynamic Hedging” provides an empirical rule to validate if the market has mean reversion tendency. It states that if volatility lowers if the longer time frame is used, then the mean reversion takes place.
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Written by fxpaul

June 16, 2011 at 20:52

Posted in trading math

## Exponential Ornstein-Uhlenbeck process

$dS_t = \theta(\mu - S_t) dt + \sigma dW_t$
$P_t= \exp S_t$