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?


  • $\begingroup$ 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. $\endgroup$ Commented Dec 11, 2018 at 19:05
  • 1
    $\begingroup$ As Heston has analytic characteristic function I calculated this same conditional expectation using Fourier-based methods ala Oosterlee & Ruijter. $\endgroup$ Commented Dec 11, 2018 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 $\endgroup$ Commented Dec 12, 2018 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/… $\endgroup$ Commented Dec 12, 2018 at 20:18

1 Answer 1


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.

  • $\begingroup$ 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. $\endgroup$ Commented Dec 12, 2018 at 10:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.