# negative values in geometric brownian motion

A GBM

$$\frac{dx}{x} = \mu dx + \sigma dW$$

solves to

$$x_t = x_o e^{(\mu - \sigma^2)t + \sigma W_t}$$

From the solution, it is clear that $$x_t$$ cannot become negative. However, it is not so clear from the SDE. In fact, if I do simulations using the SDE, x very frequently becomes negative for certain parameter combinations.

Is the non-negativity of $$x_t$$ valid only in case $$dt->0$$

Here is one simulation:

• Your discretization appears wrong. For example, E3 = E2+ E2 * D3 = E2 * (1+D3). – Gordon Jan 28 at 19:26
• Thanks @Gordon Edited. Issue is still the same – dayum Jan 28 at 19:35
• You formula is still incorrect: E2 is $x_t$ whereas D3 gives you $\Delta_t=\text{d}x_t/x_t$ thus in E3 you should have E2+E2*D3 ($x_t+x_t\Delta_t$) instead of D2+E2*D2 ($\Delta_t+x_t\Delta_t$). – Daneel Olivaw Jan 28 at 20:00
• Why not simulate d log x? That's more stable and I think avoids the issues you're having. – ilovevolatility Jan 29 at 2:13