# Negative theta in Log-linear stochastic volatility model

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 $$\theta = -1$$?

• I can add my R code here if it helps... – Amy Zhang Mar 12 '19 at 21:07
• 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. – Ezy Mar 13 '19 at 13:00