# Stochastic volatility model with exponential OU volatility

I have a friend in the industry who said they are interested in the model I gave in the title. Whether they use it, idk.

$dS_t= S_t(rdt+ \sigma_t dW_t)$

And $\sigma_t$ is the exponential of an OU process. The brownian motions are negatively correlated.

Does this thing have name? Is it widely used? What is it called?

Let $dS_t = \mu_tS_tdt + \sigma_tS_tdW_t$ be the underlying GBM (Geometric Brownian Motion)-like dynamics as in the question.

Let $B_t$ a Brownian motion such that $d[B,W]_t = \rho dt$, $\rho\in[-1,1].$

1. CIR (Cox-Ingersoll-Ross) for $\sigma_t^2$ (when combined with GBM-like underlying dynamics, it is the popular Heston SV model) $$d\sigma_t^2 = \kappa(\theta - \sigma_t^2)dt + \zeta \sigma_tdB_t$$

2. GBM for $\sigma_t^2$ and $\rho=0$ (when combined with GBM-like underlying dynamics, it is the Hull-White SV model) $$d\sigma_t^2 = -\kappa\sigma_t^2dt + \zeta \sigma_t^2dB_t$$

3. (Yours) Exponential OU for $\sigma_t$ (when combined with GBM-like underlying dynamics, it has no special name) $$d\ln \sigma_t = \kappa(\theta - \ln \sigma_t)dt + \zeta dB_t$$

4. Lognormal for $\sigma_t$ (when combined with GBM-like underlying dynamics, it has no special name) $$d\sigma_t = \kappa(\theta - \sigma_t)dt + \zeta \sigma_tdB_t$$

They can all be useful, depending on what you want or, simply, as benchmark of one against the other (when calibrated to the same targets). Note that if you are willing to revisit the underlying dynamics itself, you get other SV (stochastic volatility) models. For example, look up SABR or SLV (stochastic local volatility) models.

The model is similar to the Barndorﬀ-Nielsen - Shephard model. But this model is much more general.

On the other hand in this paper by Heston it is exactly your form that is used.

Already Scott in 1987 considered a model of your form (see this)

Finally in this thesis you find the names Hull-White model (of course there is the interest rate model too) and Heston model for this kind of model (sometimes only in the case that the variance is modelled as square-root process, sometimes in OU-case too).

• I am actually looking at the BNS model. this is helpful. thank you. – Lost1 Dec 30 '13 at 12:40
• I wouldn't say this model is more general? BNS uses a jump process, this is a BM. different approaches to dress the leverage effect. – Lost1 Dec 30 '13 at 12:41
• Also $\sigma_t$ here (or square root of it) is modelled as the OU not the exponential of OU? – Lost1 Dec 30 '13 at 12:44
• I would say that modelling with jumps is more general. At least it is tehnically more difficult and if you can do the jump case then the other case will be easy. A jump process is a natural model for vol in my opinion. Something like a Hawkes [process](fiquant.mas.ecp.fr/ioane_files/HawkesCourseSlides.pdf) looks really good. – Ric Dec 30 '13 at 12:59
• And yes - as I can see it: Heston is with a CIR model for the variance. And you need some non-linear transformation if you model vol as OU-process. The Heston with CIR-vol seems to be the most common. Have you ever looked at Heston-Nandi? – Ric Dec 30 '13 at 13:01

You can use the log-normal model directly: Affine Approximation for Moment Generating Function of Log-Normal Stochastic Volatility Model ( http://ssrn.com/abstract=2522425 )