Exponential OU Stochastic Volatility and Related Models
Summary
The document identifies a stochastic volatility specification in which the asset follows geometric Brownian motion while volatility is the exponential of an Ornstein–Uhlenbeck process. The log volatility mean-reverts, and its driving Brownian motion may be correlated with the asset’s return shock. The accepted answer says this combination has no single special name, though it belongs to the broader family of stochastic volatility models.
The answer compares it with familiar alternatives: Heston volatility uses a square-root process for variance, while the Hull–White stochastic volatility formulation is described using geometric Brownian motion for variance. Other responses connect related formulations to work by Scott and Heston and to the Barndorff-Nielsen–Shephard model. These labels can vary across sources, and the discussion offers no evidence about prevalence or model performance. It recommends evaluating such specifications against chosen targets or as benchmarks, and notes that changing the underlying dynamics leads to additional models such as SABR and stochastic local volatility.
Key ideas
- Exponential OU volatility means the logarithm of volatility follows a mean-reverting OU process.
- The asset return shock can be correlated with the shock driving volatility.
- The combination with geometric Brownian motion does not have one universally accepted special name.
- Heston and Hull–White formulations provide related stochastic volatility comparisons.
- Model usefulness depends on the application and calibration targets.
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# Stochastic volatility model with exponential OU volatility # 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? ## Answer by ir7 (score 6, accepted) https://quant.stackexchange.com/a/9842 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 $d[B,W]_t = \rho dt$, $\rho\in[-1,1].$ - CIR (Cox-Ingersoll-Ross) for $\sigma_t^2$ (when combined with GBM-like underlying dynamics, it is the popular Heston SV model) $$d\sigma_t^2 = \kappa(\theta - \sigma_t^2)dt + \zeta \sigma_tdB_t$$ - GBM for $\sigma_t^2$ and $\rho=0$ (when combined with GBM-like underlying dynamics, it is the Hull-White SV model) $$d\sigma_t^2 = -\kappa\sigma_t^2dt + \zeta \sigma_t^2dB_t$$ - (Yours) Exponential OU for $\sigma_t$ (when combined with GBM-like underlying dynamics, it has no special name) $$d\ln \sigma_t = \kappa(\theta - \ln \sigma_t)dt + \zeta dB_t$$ - 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. ## Answer by Richi Wa (score 4) https://quant.stackexchange.com/a/9826 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). ## Answer by sigma (score 1) https://quant.stackexchange.com/a/15405 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 )
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