Recovering Conditional Heston Variance from Local Volatility

Summary

The note asks how to calculate the conditional expectation of variance given the asset price in the Heston stochastic volatility model. Its proposed shortcut uses the connection between this conditional variance and the square of local volatility, allowing the quantity of interest to be inferred without directly evaluating the conditional stochastic integral from the variance process.

Local volatility can in turn be obtained from call prices and their derivatives with respect to maturity and strike through the Dupire relation, with additional terms required when rates or drift are nonzero. The note points out that semi-analytic Heston call pricing formulas based on Fourier methods can supply those prices. This is a computational route, rather than a worked derivation: it gives no numerical example or implementation details, and readers must account for the model assumptions and derivative estimation needed in practice.

Key ideas

• The Heston model couples asset returns to a mean-reverting stochastic variance process.
• The response relates conditional variance given the asset price to squared local volatility.
• The Dupire relation expresses local variance using maturity and strike derivatives of call prices.
• Fourier-based call pricing formulas can provide inputs for the calculation.
• Nonzero rates or drift require corresponding terms in the local-volatility formula.

Tags

# Heston model computations


# 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!

## Answer by Antoine Conze (score 2)

https://quant.stackexchange.com/a/43027

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.