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?
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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 ($d=2$).
$\hat{\mu_i} = \frac{1}{n}\sum\limits_{k = 1}^n X_{m-k + 1, i}, \quad i = 1,\dots, d$
$\hat{R_{ij}} = \frac{1}{n-1}\sum\limits_{k=1}^n\left(X_{m-k +1, i}-\hat{\mu_i} \right) \left(X_{m-k +1, j}-\hat{\mu_j} \right), \quad i,j = 1,\dots,d$
