# Cholesky correlation

I have historic time series for spot and futures and I want to now simulate future price paths for 1 day to get the distribution and from there compute the value at risk. My question is now since i am generating correlated random numbers, what correlation am i supposed to input into the cholesky decomposition. Is it supposed to be the correlation between the spot and future of the historical series?

For Monte Carlo simulation it is necessary to suggest a distribution function $$F$$. You want to simulate based on the observations $$(S_{1}, \dots , S_{t}, F_{1}, \dots, F_{t})$$. Here, you assume your risk factor changes $$(X_{S,t+1} = \log(S_{t+1}/S_t), X_{F,t+1}= \log(F_{t+1}/F_t))$$ to be bivariate normal. Now, you want to estimate the parameters $$\mu$$ amd $$R$$?
If you assume $$Y_1, \dots, Y_d \sim N(0,1)$$ iid, then $$\mu + A\textbf{Y} \sim N_d(\mu, R)$$
You have to find the Cholesky decomposition $$A$$ of $$R:$$ $$R = AA^T$$.
Variance-covariance method to estimate parameters ($$d=2$$).
$$\hat{\mu_i} = \frac{1}{n}\sum\limits_{k = 1}^n X_{m-k + 1, i}, \quad i = 1,\dots, d$$ $$\hat{R_{ij}} = \frac{1}{n-1}\sum\limits_{k=1}^n\left(X_{m-k +1, i}-\hat{\mu_i} \right) \left(X_{m-k +1, j}-\hat{\mu_j} \right), \quad i,j = 1,\dots,d$$