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}
In simple terms, to get the next simulated price $S_{t+\Delta t}$, I've first:
1. Find the value of the discretised variance process $v_{t+\Delta t}$.
2. Find the value of the asset price of $S_{t+\Delta t}$ using the same independent variable $Z_V$ found in step 1; $v_{t+\Delta t}$.
THE PROBLEM
Now, I am not sure what to do with the value of $v_{t+\Delta t}$ since variance $v_{t}$ in step 2 are all using the previous period (1 time step back) values. Do I substitute the value found in step 1 into $v_t$ found in $S_{t+\Delta t}$?
However, I know that $v_t$ in $S_{t+\Delta t}$ only corresponds to the instantaneous variance value at time $t$ (or the previous time step so to speak) so it wouldn't make sense to use $v_{t+\Delta t}$ as the 'instantaneous' value.
Just a brief info, I'm only using Excel for this problem since I'm merely trying to understand the mechanics of how the Heston model works in 'real-time' for pricing an asset (not options but instead non-derivative assets like stocks for instance).
Any enlightenment and education would be very much appreciated.
