# Heston model computations

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!

• 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. – Daneel Olivaw Dec 11 '18 at 19:05
• As Heston has analytic characteristic function I calculated this same conditional expectation using Fourier-based methods ala Oosterlee & Ruijter. – James Spencer-Lavan Dec 11 '18 at 19:21
• @ 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 – noob-mathematician Dec 12 '18 at 10:44
• This is an ok reference on 2d COS inc Heston bivariate characteristic function open.uct.ac.za/bitstream/handle/11427/8520/… – James Spencer-Lavan Dec 12 '18 at 20:18

## 1 Answer

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.

• 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. – noob-mathematician Dec 12 '18 at 10:44