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Using the Updated Heston Variance in the Next Price Step

Article Quant Q&A · Author: AQT
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Summary

The document explains how the variance path enters a discretized Heston simulation. In the asset-price update, the current step uses the current variance, floored at zero under the fully truncated scheme. The correlated price shock combines the variance shock with an independent normal shock, so the variance innovation can affect the asset return through the correlation parameter.

After updating variance to the next time point, that new value is used in the following asset-price step. The accepted answer clarifies this time ordering: the simulated variance is needed for the next period, and the process repeats through the option’s maturity. The document gives no numerical example or discussion of time-step bias, alternative truncation schemes, or calibration; its contribution is a concise clarification of how to sequence the updates.

Key ideas

  • The asset update for a given interval uses the variance at the start of that interval.
  • The fully truncated scheme replaces negative current variance with zero in its drift and diffusion terms.
  • The price shock is correlated with the variance shock through the specified correlation parameter.
  • The newly simulated variance is used when advancing the asset price over the following interval.

Tags

Derivatives pricingVolatilityStatistics
Full text
# How the variance process in discretised form influence the asset price in the Heston model


# How the variance process in discretised form influence the asset price in the Heston model












I'm trying to do Monte Carlo simulation paths of an asset price with time step $\Delta t$ via the discretised Euler scheme. My main question is how does the variance process influence the asset price simulation on the next time step, $(t+\Delta t)$ from the equations below?

First off, let the discretised asset price process be: \begin{eqnarray} S_{t+\Delta t} = S_t \exp \left( \left(\mu - \frac{1}{2} v_t^{+} \right) \Delta t + \sqrt{v_t^{+} \Delta t} Z_S \right) \end{eqnarray}

The fully truncated scheme of variance process is as follows: \begin{eqnarray} \ v_{t+\Delta t} &=& \ v_t + \kappa (\theta - \ v_t^{+}) \Delta t + \sigma \sqrt{\ v_t^{+} \Delta t} Z_V, \end{eqnarray}

with correlation equation $Z_S = \rho Z_V + \sqrt{1-\rho^2}Z_2$. Where $Z_V$ and $Z_2$ are independent standard normal variables. Next, substituting the correlation equation into the asset price process gives us: \begin{eqnarray} S_{t+\Delta t} = S_t \exp \left( \left(\mu - \frac{1}{2} v_t^{+} \right) \Delta t + \sqrt{v_t^{+} \Delta t}* \left( \rho Z_V + \sqrt{1-\rho^2}Z_2 \right) \right) \end{eqnarray}

## Answer by KaiSqDist (score 1, accepted)

https://quant.stackexchange.com/a/79425

I am not quite sure what is your confusion here.

If I understood what you meant, you use $v_{t+\Delta t}$ in $S_{t+2\Delta t}$ simulation. This repeats until the maturity of the option. The only reason you simulate the variance rate is because it is to be used for the next period.

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