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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.
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  • $\begingroup$ Your question is quite confusing, what exactly are steps 1 and 2? $\endgroup$
    – KaiSqDist
    Commented May 18 at 21:42
  • $\begingroup$ Apologies for the confusion. Steps 1 and 2 are both merely descriptions of what I've done and understood to simulate $S_{t+ \Delta t}$ asset price in question, not the time steps if that's what you thought I was referring to. The question is does the value of the variance process used in the asset price process of the next time step? <br/> For example, do we use the value $v_{t+ \Delta t}$ calculated at time 1 for the asset price $S_{t+ \Delta t}$ at time 2? $\endgroup$
    – AQT
    Commented May 19 at 11:18
  • $\begingroup$ Yes that's correct, which is exactly what I explained in my answer below. $\endgroup$
    – KaiSqDist
    Commented May 19 at 13:10

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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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  • $\begingroup$ Thank you very much for the confirmation. However, I have 1 more question. Do we use the same value used in $Z_V$ in $v_{t+ \Delta t}$ for the calculation of $S_{t+2 \Delta t}$? For example, if $Z_V=1.1$ in $v_{t+ \Delta t}$, then we use the same value $Z_V=1.1$ for $Z_V$ in the correlation equation of $S_{t+2 \Delta t}$? $\endgroup$
    – AQT
    Commented Jun 10 at 13:56
  • $\begingroup$ Why do you have to care about the correlation equation though? The correlation part is already included in the stochastic process for $S_{t+2\Delta t}$ $\endgroup$
    – KaiSqDist
    Commented Jun 10 at 14:33
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    $\begingroup$ Apologies for any confusion. Yes I know that the correlation equation is part of the price process calculation. What I meant was if the value of the variable $Z_V$ in both $v_{t+ \Delta t}$ and $S_{t+2 \Delta t}$ must strictly be the same to simulate the asset price of the next time step. Example, if an arbitrary number $Z_V = 1.1$ is used, then both the variance and asset price process must use $Z_V = 1.1$. The reason why I need to know exactly is because I need to be completely sure that I’m simulating each time step correctly. $\endgroup$
    – AQT
    Commented Jun 10 at 14:44
  • $\begingroup$ Oh I see. No, the $Z_V$ should be different ($Z_V$ should be the same period as $S_{t+2\Delta t}$). As the $Z_S$ "belongs" to the stock price simulation, it should be the same period, but for the variance rate it should lag. $\endgroup$
    – KaiSqDist
    Commented Jun 10 at 15:03
  • $\begingroup$ I've implemented a fix to my calculation sheet with regards to $Z_V$ in both $v_{t+ \Delta t}$ and $S_{t+2 \Delta t}$ so that both processes use different values. However, I noticed that having $\rho = -0.9$ or $\rho = 0.9$ does not influence the expected value (or mean) of the Monte Carlo simulations. Is this an intended effect? Previously, I fixated $Z_V$ for both $v_{t+ \Delta t}$ and $S_{t+2 \Delta t}$ to be the same values and the result is having a mean that increases/decreases (changes) with positive/negative $\rho$ respectively. I've since fixed it as mentioned though. $\endgroup$
    – AQT
    Commented Jun 10 at 16:18

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