# Mean Reverting Heston Model?

Is there a name for a variation on the Heston Stochastic Process Model where not only the underlying volatility but the asset price itself is mean-reverting? I'm looking to model long term equity index returns, which I would argue show both volatility mean-reverting and asset-price mean reverting behavior. Clearly, where such an equity index could be thought to (at least approximately) mean-revert to some fixed long-term volatility, it also mean-reverts to the asset price point expected by annualized compounding returns (about 7% in real terms for the SP500).

As far as I know, the Heston Model does nothing to account for the second behavior. How would one modify the model to accommodate for that?

Edit: I tried to consider the following model. Any thoughts on it?

$$dS_t = \sqrt{v_t}S_tdB_t^{(1)} + a_1(\mu_t - S_t)$$

Where $$u_t := \mathbb{E}[S_t]$$ is the expected equilibrium at time t as per expected annualized compounding, $$a_1$$ is speed of mean reversion for price, $$B_t^{(1)}$$ is one dimensional Brownian Motion, and $$v_t$$ is the variance process $$\{v_t, t\geq 0\}$$ as defined with:

$$dv_t = \sigma\sqrt{v_t}dB_t^{(2)} + a_2(v_t - \nu)$$

Where, in turn, $$\sigma$$ is the constant vol of vol, $$B_t^{(2)}$$ is one dimensional Brownian Motion correlated to $$B_t^{(1)}$$ by $$Cov(B_t^{(1)},B_t^{(2)})=\rho$$, $$a_2$$ is speed of mean reversion for volatility, and $$\nu$$ is the long run average for volatility.

Edit 2: I meant to mean reverting returns, not a fixed price level.

• Hi, this (look may get you started. You may also start from ‚first principles‘ with Duffie/Pan/Singleton‘s Transform Methods paper . Oct 9 '20 at 16:42
• I saw this paper, actually. Or, well, an earlier version of it, anyway. I understand why the authors define $S_t = exp(X_t)$ but fail to see why they define $dX_t = [\theta(t) - \kappa X_t - \frac{v_t}{2}]dt + \sqrt{v_t}dW_t$. The second term makes enough sense to me -- it's merely its equivalent as in the Heston Model -- but why in the world are we subtracting variance from equilibrium mean at time t? And shouldn't we have $\kappa (\theta(t)-X_t)$ for the mean reversion? Oct 9 '20 at 17:43
• @TheMathBoi The factor $-\frac{1}{2}v_t$ is normally relates to Ito's Lemma when you take logs, so don't worry about it. Whether you have $\theta-\kappa X_t$ or $\kappa(\theta-X_t)$ doesn't matter either because you can just rescale $\theta$ Oct 9 '20 at 19:17

Let us start with the classical Heston model with underlying price $$S_t$$ and variance $$v_t$$,

\begin{align} \frac{dS}{S}&=\mu dt+\sqrt{v_t}dW_1\\ dv_t&=\kappa(\theta-v_t)dt+\sigma\sqrt{v_t}dW_2 \end{align} and $$E(dW_1dW_2)=\rho dt$$

From here on, if you want to introduce a mean reverting price level, I might suggest the following adjustment of your asset process:

$$dS/S=\kappa_S(\theta_S-lnS_t) dt + \sqrt{v_t}dW_1$$

Loosely speaking, $$e^{\theta_s+g(\theta,\kappa,\sigma,\rho)}$$ is the long run price level, with $$g$$ some correction term for the steady state variance.

We can now either directly simulate this setup, or we make use of the Fourier transform machinery for linear jump diffusion processes, knowing that - under the transformation $$y=lnS$$ - the system

\begin{align} dy&=\kappa_S(\theta_S-y-0.5v_t) dt + \sqrt{v_t}dW_1\\ dv_t&=\kappa(\theta-v_t)dt+\sigma\sqrt{v_t}dW_2 \end{align}

is clearly in the class of linear processes. The next step would be to perform the analysis in DPS2000.

EDIT

Under the physical measure, if you want to model your returns to be mean-reverting, you should be able to work with the following:

\begin{align} dy&=(\mu_t-0.5v_t) dt + \sqrt{v_t}dW_1\\ dv_t&=\kappa(\theta-v_t)dt+\sigma\sqrt{v_t}dW_2\\ d\mu&=\kappa_{\mu}(\theta_{\mu}-\mu_t)dt+\sigma_{\mu}dW_3 \end{align}

and you should even be able to specify correlations between $$dW_1,dW_3$$ and $$dW_2,dW_3$$.

Under the risk neutral measure, you should be able to introduce a mean-reverting risk-free-rate-of-return process, though. Again, see source 1 above.

• If, on the other hand, you want to introduce mean reverting returns, then we need to introduce another process for $\mu$. Please let me know if that's what you truly want. Oct 9 '20 at 19:10
• I'm sorry, I might have been a bit inexact in my language. I was talking about mean reverting returns. In the formulas I have in my edit, I consider an equilibrium price at time t, $\mu_t=\mathbb{E}[S_t]=S_0*e^{rt}$. Thank you so much for your help and advice, by the way. I'm an undergrad with only very minimal exposure to stochastic calc, so this doesn't come naturally to me. Oct 10 '20 at 15:54
• I am wondering whether, in the risk neutral world, the specification of the mean process is of any use (i.e. in an option pricing application), see for example this thread and the reference within: quant.stackexchange.com/questions/22189/…. I will nevertheless add a line to my answer. Oct 10 '20 at 20:03