Random Walk of N Correlated Assets

Question

I am trying to value an option on N assets, say $S^1, S^2,..., S^N$ that expires in $\Delta T$ years using Monte Carlo simulation. I have read many sources that state I should use the following formula for each asset:

$S_T^i = S_0^i exp( (\mu_i - \sigma_i^2/2)\Delta T + \alpha_i\sigma_i\sqrt{\Delta T})$

Where:

• The $i$'s are used to differentiate the different assets.
• $S_t^i$ denotes the price of asset $S^i$ at time t.
• $(\alpha_1,...,\alpha_N)$ are derived by taking the Cholesky decomposition $LL^*$ of the "correlation matrix" and then applying it to N iid standard normal random variables $(\epsilon_i,...,\epsilon_N)$.

My questions are:

1. Does the "correlation matrix" represent the correlations between the Assets or of the Asset returns?
2. Does the Cholesky method simply accomplish drawing from multivariate normal distribution with mean $(0,...,0)$ and variance-covariance matrix of the answer to my first question?

Thank you in advance.

Answer 1

The correlation matrix refers to the correlations between the asset returns. In fact, it can be seen as follows. Each asset follows a geometric Brownian motion, i.e., $$ \frac{{\rm d}S_t^i}{S_t^i}=\mu_i{\rm d}t+\sigma_i{\rm d}W_t^i, $$ where the correlation between $W_t^i$ and $W_t^j$ is supposed to be $$ \text{Corr}\left(W_t^i,W_t^j\right)=\rho_{ij}. $$ Therefore, the correlation matrix referes to the correlations between the asset returns.

The Cholesky decomposition helps to transform $N$ independent normal random variables into $N$ correlated normal random variables, with the correlation matrix $\rho_{ij}$ as above. This can be seen as follows. Solve the SDE for each asset, and $$ S_t^i=S_0^i\exp\left[\left(\mu_i-\frac{1}{2}\sigma_i^2\right)t+\sigma_iW_t^i\right]. $$ As we are only interested in the sampling of $S_T^i$, the above formula yields $$ S_T^i=S_0^i\exp\left[\left(\mu_i-\frac{1}{2}\sigma_i^2\right)T+\sigma_iW_T^i\right]. $$ Here each $W_T^i\sim\mathcal{N}(0,T)$. By contrast, when all $W_T^i$'s are viewed as a whole, i.e., a vector of $N$ random variables, we have $$ \mathbf{W}\sim\mathcal{N}(\mathbf{0},T\Sigma), $$ where the $\left(i,j\right)$-th entry of the square matrix $\Sigma$ reads $$ \Sigma_{ij}=\rho_{ij}, $$ because the components of $\mathbf{W}$ are correlated. Now, suppose we have another vector of $N$ random variables, denoted by $\mathbb{Z}$, that follows $$ \mathbf{Z}\sim\mathcal{N}(\mathbf{0},I_N), $$ meaning that the components of $\mathbf{Z}$ are independent and identically distributed as standard normal. This is what we could generate numerically. Our target is to make use of this $\mathbf{Z}$ to get $\mathbf{W}$. This could be implemented by $$ \sqrt{T}L\mathbf{Z}, $$ where $L$ satisfies $$ \Sigma=LL^{\top}. $$ Since $$ \sqrt{T}L\mathbf{Z}\sim\mathcal{N}(\mathbf{0},T\Sigma), $$ it may provide sampling of $\mathbf{W}$.