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 $\theta = -1$?
