# FxPaul

Math in finance or vice versa

## Exponential Ornstein-Uhlenbeck process

Let’s consider parameter estimation for the following modification of Ornstein-Uhlenbeck process:
$dS_t = \theta(\mu - S_t) dt + \sigma dW_t$
$P_t= \exp S_t$
This model is simplification of Schwarz Model 1, one of Short-rate models

To find the explicit formula of the process, let’s apply Ito lemma to the following function
$f(t,x) = \exp x$
Calculating partial derivatives:
$\frac{\partial f}{\partial t} = 0,\, \frac{\partial f}{\partial x} = f,\, \frac{\partial^2 f}{\partial x^2} = f$
and finally by Ito lemma:
$dP_t = P_t\left(\theta(\mu - S_t) dt + \sigma dW_t\right) + \frac{1}{2} P_t \left(dS_t\right)^2$
As $(dW_t)^2 = dt$ and $dW_t dt = dt dW_t = (dt)^2 = 0$:
$(dS_t)^2 = \sigma^2 dt$
Therefore:
$dP_t = \left(\theta(\mu - \ln P_t) + \frac{1}{2} \sigma^2 \right) P_t dt + \sigma P_t dW_t$

Thus this process might be used in Monte Carlo simulations as follows:
$P_t - P_{t-1} = \left(\theta(\mu - \ln P_{t-1}) + \frac{1}{2} \sigma^2 \right) P_{t-1} dt + \sigma P_{t-1} \sqrt{\Delta t} dW_t$
And process parameters can be estimated on $\ln P_t$ as it is done for standard Ornstein-Uhlenbeck process here.

Written by fxpaul

June 8, 2011 at 08:28

### 3 Responses

1. Hi Paul,

Thank you for this great article.

I wonder what would be an ideal solution for the calibration of this process? I tried the method from your other link but got stuck at deciding the volatility sigma. As can be seen from the above equation, we have 0.5*sigma^2*dt and another one which is sigma*sqrt(dt). If we use OLS, would the constant (intercept) = 0.5*sigma^2 and sde = sigma*sqrt(dt)? Hence which one we use for estimating the sigma of the process?

Many thanks.
Kind regards,
Julie

Julie Wong

November 15, 2013 at 10:36

• Hi Julie,

Sorry for the delay but the end of the year is always filled with a lot of task. I wrote a separate article about the calibration: http://wp.me/p1gkpt-cf

Hope it helps and it is not so late.

Regards,
Pavel

fxpaul

November 20, 2013 at 13:06

2. […] the previous article about process calibration we derived the following updating formula: Now we can rearrange equation’s parts as follows: […]