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In the Heston model the dynamics of a single-asset $S$ are given by:

$dS_t = rS_tdt+S_t \sqrt{V_t}dW^S$

where $W^s$ is a brownian-motion $W^S$ and the square root variance process $V$ is given by the SDE:

$dV_t = a(\bar{V}- V_t)dt + \eta \sqrt{V_t}dW_t^V$,

with $a,\bar{V}, \eta$ constants and $W^V$ has fixed correlation to $W^S$ equal to $\rho$.

I want to compute the conditional expectation, $E[ V_t | S_t ] $. By solving the SDE for the Variance-process the problem reduces to computing a stochastic integral,

$E [ \int_{0}^{t}...dW_s^V | S_t]$. Anyone has any idea how to compute that?

Thanks!

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  • $\begingroup$ As far as I know there is no closed-form solution to the SDE. The square-root variance process is a Cox-Ingersoll-Ross (CIR) process thus its distribution is non-central Chi-squared. The process can be characterized as a sum of Ornstein-Uhlenbeck processes. $\endgroup$ – Daneel Olivaw Dec 11 '18 at 19:05
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    $\begingroup$ As Heston has analytic characteristic function I calculated this same conditional expectation using Fourier-based methods ala Oosterlee & Ruijter. $\endgroup$ – James Spencer-Lavan Dec 11 '18 at 19:21
  • $\begingroup$ @ James Spencer-Lavan - Do you have any paper in mind that I should look at? I found a ton of bibliography out there but many of the papers are irrelevant. Thank you $\endgroup$ – noob-mathematician Dec 12 '18 at 10:44
  • $\begingroup$ This is an ok reference on 2d COS inc Heston bivariate characteristic function open.uct.ac.za/bitstream/handle/11427/8520/… $\endgroup$ – James Spencer-Lavan Dec 12 '18 at 20:18
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A quick trick:

from "Dupire, A Unified Theory of Volatility" we have $$E[V_t | S_t] = \sigma_{\text{loc}}(S_t, t)^2$$ where $\sigma_{\text{loc}}(S, t)$ is the local volatility. We also have from the Dupire formula that $$ \sigma_{\text{loc}}(K, T)^2 = \frac{\frac{\partial C}{\partial T}}{\frac{1}{2}K^2\frac{\partial^2 C}{\partial K^2}} $$ (in the case where drift and interest rate are zero, otherwise there are additional terms), and finally there are semi-analytic formulas for the call price $C(K,T)$, based on Fourier transform as mentionned by @ James Spencer-Lavan, for which you will find plenty of implementations. So you have all the ingredients readily available for your calculation.

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  • $\begingroup$ I see your point and I fully agree with that. In particular I have something more general to compute, namely $E[V_t^1 | S_t^1, S_t^2 ]$, where the brownian of the variance of asset 1 is also correlated to the brownian of asset 2. $\endgroup$ – noob-mathematician Dec 12 '18 at 10:44

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