Geometric Brownian Motion Price Processes in high Dimensions

This is my first post so I am open for an suggestions in formating improvement. For some reason I can not find suitable literature for the following problem

What I want to do is calculate the option value of a basket defined through $$d$$ different assets of the stock market defined through geometric brownian motions $$S = S_1(t), \dots, S_d(t)$$, that satisfy the following SDE:

$$dS_i(t) = \mu_i S_i(t) dt + \sum_{j=1}^{d} \sigma_{ij} S_i(t) dW_j(t), \quad t \in [0,T], i = 1,\dots, d$$

where $$\mu_i$$ are drift parameters, $$(\sigma_{ij})_{ij}\in \mathbb{R}^{d\times d}$$ is the defining a volatility matrix and $$W_i(t)$$ are independent brownian motions on a probability space. Furthermore we have that $$E(dW_i) = 0$$ and $$E(dW_i^2) = dt$$ and furthermore $$E(dW_i dW_j)= \rho_{ij}dt$$.

In the literature I found that "its easy" to show that through integration the system can be transformed for $$i=2$$ to: $$S_1(t) = S_1(0) \exp\left[ \left( \mu_i - \dfrac{\sigma_{11}^{2}+\sigma_{12}^{2} }{2} \right) t + \sigma_{11}W_1(t) + \sigma_{12} W_2(t)\right]$$

$$S_2(t) = S_2(0) \exp \left[\left(\mu_i - \dfrac{\sigma_{21}^{2}+\sigma_{22}^{2} }{2} \right) + \sigma_{21}W_1(t) + \sigma_{22} W_2(t) \right]$$

Question: What are the values of the system of integrated geometric brownian motions in the general case, (or e.g. $$i = 10$$).

I found in a script of the KIT that its possible to model a European basket through the geometric brownian Motion

$$dS(t) = \mu S dt + diag(\sigma) diag(S) L dW(t)$$ with $$S(t_*) = S_*$$ where $$\sigma = (\sigma_1, \dots, \sigma_d)$$, $$\mu > 0$$, $$S = (S_1, \dots, S_d)$$ , $$dW = (dW_1, \dots, dW_d)$$ and $$L$$ is the lower trianagle matrix which can be derived through the colesky decomposition of $$\Sigma = L L^T$$, while

$$\Sigma := \left(\begin{matrix}1&\rho_{12}&\cdots&\rho_{1d}\\\rho_{21}&1&\ddots&\vdots\\\vdots&\ddots&1&\rho_{d-1d}\\\rho_{d1}&\cdots&\rho_{dd-1}&1\end{matrix}\right)$$.

Question: Can someone help me to derive that system of geometric BM please?

• Do you know how to do the "easy" computation in the $i=2$ case? The solution there can be written in matrix notation. This will tell you what the formula for the general case will look like. If you use matrix/vector notation the proof for the $i=2$ case and the general case are essentially identical. – quarague Feb 11 at 10:49
• To be really honest with you I do not really know the "easy" computation of the $i=2$ case. In the literature I found it said that the computation is "easily done"... Thanks though @quarague – Christian.gruener Feb 11 at 16:38
• Hey I do not know if this pushes my topic to the top but I am still searching for help with my problem. – Christian.gruener Feb 19 at 12:15