Stocks

From (45) we get the Euler-approximation

$$\ln S_{j,m} - \ln S_{j,m-1} = \mu_j\cdot\Delta + \sigma_j\cdot\sqrt{\Delta}\cdot N_{j+2,m} \quad (1 \leq j \leq d)$$ (50)

where the $N_{j+2,m}$ (fixed $j$) are i.i.d. $N(0,1)$. This implies for $M\in\mathbb{N}^+$

$$S_{j,M} = S_{j,0}\exp\left(\mu_jM\Delta + \sigma_j\sum_{m=1}^{M} \sqrt{\Delta}N_{j+2,m}\right) .$$ (51)