# Stochastic Long-Run Mean Instantaneous Variance in Heston Model (and extensions)?

I'm working on my dissertation in Financial Economics, focusing on the topic of Stochastic Volatility Jump Diffusion models; and I'm playing around with some ideas for model extensions. In particular, I am quite interested in the idea of an extension in which the Heston long-run mean instantaneous variance parameter is also stochastic and mean-reverting; i.e. with a (risk-neutral) SDE system of: $$dS_{t}=(r_{t}-q_{t}-\lambda \bar{k})S_{t}dt+\sqrt{\nu_{t}}S_{t}dW_{t}^{S}+kS_{t}dN_{t}$$ $$d\nu_{t}=\kappa_{\nu}(\theta_{t}-\nu_{t})dt+\sigma_{\nu}\sqrt{\nu_{t}}dW_{t}^{\nu}$$ $$d\theta_{t}=\kappa_{\theta}(\theta_{\infty}-\theta_{t})dt+\sigma_{\theta}\sqrt{\theta_{t}}dW_{t}^{\theta}$$ $$d\left=\rho dt$$ $$N_{t}\sim Pois(\lambda t)$$ $$k=\exp(J^{S})-1$$ $$\bar{k}=\mathbb{E}^{Q}\left[k\right],$$

such that $$\theta_{\infty}$$, $$\kappa_{\theta}$$, $$\sigma_{\theta}$$ are the analogous paramaters of the long-run mean instantaneous variance DE to those of the DE for the spot instantaneous variance $$\nu_{t}$$. I leave open the questions of the jump size distribution (from which the general solution approach is independent assuming independent jumps) as to whether or not the long-run mean instantaneous variance is correlated with any of the other parameters in the model. (I see no reason why it should have to be.)

My question to you guys: is anyone aware of any literature that deals with such a model specification, either with or without jumps included? I ask because I have tried deriving the European Option price Heston/Bates style that would require a $$\theta_{t}$$ coefficient in the Characteristic Function, and I end up with a rather nasty Riccati DE for that coefficient that is stretching me beyond the limit of my talent I fear. (Or which may simply have no closed-form solution.)

Alternatively, if any of you are feeling lucky today and would like to help a simple mind like my mine try to derive a solution, that would also be greatly appreciated; let me know, I'll type out the DE for you to take a stab at.

• The approach of modelling the long-term mean level of the variance process as stochastic is quite common. One of the earliest references that I am aware of is faculty.baruch.cuny.edu/jgatheral/Bachelier2008.pdf - see the slide titled "double CEV dynamics". – LocalVolatility Mar 3 at 20:25
• Hi Local Volatility, my apologies for the late response. Thank you so much for the link! It was really helpful. – pmms12585 Apr 11 at 8:02