Central Tendency (CT)

The model is

\[\begin{split}dp_{t}&=\left(r+\left(\lambda-\frac{1}{2}\right) \sigma_{t}^{2}\right)dt+\sigma_{t}dW_{t}^{r},\\ d\sigma_{t}^{2}&=\kappa_{\sigma}\left(v_{t}^{2}-\sigma_{t}^{2}\right)dt +\eta_{\sigma}\sigma_{t}dW_{t}^{\sigma},\\ dv_{t}^{2}&=\kappa_{v}\left(\mu-v_{t}^{2}\right)dt+\eta_{v}v_{t}dW_{t}^{v},\end{split}\]

with \(p_{t}=\log S_{t}\), and \(Corr\left[dW_{s}^{r},dW_{s}^{\sigma}\right]=\rho\), or in other words \(W_{t}^{\sigma}=\rho W_{t}^{r}+\sqrt{1-\rho^{2}}W_{t}^{v}\). Also let \(R\left(Y_{t}\right)=r\).