Editing Geometric Brownian motion (section)

==Multivariate version==

GBM can be extended to the case where there are multiple correlated price paths.<ref name="musielarutkowski">Musiela, M., and Rutkowski, M. (2004), Martingale Methods in Financial Modelling, 2nd Edition, Springer Verlag, Berlin.</ref>

Each price path follows the underlying process

:<math>dS_t^i = \mu_i S_t^i\,dt + \sigma_i S_t^i\,dW_t^i,</math>

where the Wiener processes are correlated such that <math> \operatorname{E}(dW_{t}^i \,dW_{t}^j) = \rho_{i,j} \, dt</math> where <math>\rho_{i,i} = 1</math>.

For the multivariate case, this implies that
:<math>\operatorname{Cov}(S_t^i, S_t^j) = S_0^i S_0^j e^{(\mu_i + \mu_j) t }\left(e^{\rho_{i,j} \sigma_i \sigma_j t}-1\right).</math>

A multivariate formulation that maintains the driving Brownian motions <math>W_t^i</math> independent is 

:<math>dS_t^i = \mu_i S_t^i\,dt + \sum_{j=1}^d \sigma_{i,j} S_t^i\,dW_t^j,</math>

where the correlation between <math>S_t^i</math> and <math>S_t^j</math> is now expressed through the <math>\sigma_{i,j} = \rho_{i,j}\, \sigma_i\, \sigma_j</math> terms.