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I was asked to simulate the following geometric Brownian motion to get paths for the SPX stock price. the process follows a Log-Linear stochastic volatility.

$dS_t = \mu S_tdt+e^VS_tdW_1 $

where the volatility follows the process:

$dV_t=\alpha(\theta-V_t)dt+\gamma dW_2 $

where $dW_2=\rho dW_1+\sqrt{1-\rho^2}dZ$

My initial price is 30.2 ($S_0$) and I need 1 year, so T=252. I was asked to use $dt=\dfrac{0.5}{252}$ and was given $\theta = -1$, $\rho = -0.4$ and two regimes for the mean reversion parameters: $\alpha = 1 and 150$ and $\gamma = 0.3 and 8$

I'm using $\mu = 0.02$ right now

In my original time series for SPX, my prices range from 29 to 50, which is a quite big variance

When using Euler discretization, starting with $V_0 = 1$ , a never get prices over 40, so I'm wondering whether the problem here is just in the mean reversion regimes I was asked to use or whether I'm not choosing a reasonable value for $V_0$. Also, does it make any sense to have $\theta = -1$?

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    $\begingroup$ I can add my R code here if it helps... $\endgroup$ – Amy Zhang Mar 12 at 21:07
  • $\begingroup$ Did you try with $\gamma=0$ ? And with $V_0=-1$ ? You should try simple special cases first to assess if there are bugs in your implementation so going back to constant or determinisitic vol is a good first check. You should recover lognormal model. $\endgroup$ – Ezy Mar 13 at 13:00

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