Estimating Correlated Spot and Futures Returns for VaR Simulation

Summary

The document explains which correlation information to use when simulating one-day spot and futures price paths with a Cholesky decomposition. It frames the simulation in terms of changes in the risk factors, specifically the log returns of spot and futures, and assumes these changes follow a bivariate normal distribution. The historical relationship between spot and futures is relevant when estimated from these return changes, rather than from the price levels themselves.

The method estimates the mean vector and covariance matrix from aligned historical observations of the two return series. Cholesky decomposition factors that covariance matrix into a matrix used to transform independent standard normal draws into correlated simulated returns. Those returns can then be used to produce future price scenarios for risk calculations such as VaR. The answer outlines the variance-covariance approach but does not address whether the normality assumption is appropriate, how to select a historical window, or how to validate the VaR estimate. Its results therefore depend on the chosen data and distribution assumptions.

Key ideas

• Estimate dependence from aligned historical spot and futures return changes, not simply from their price levels.
• The proposed model assumes the two risk factor returns are jointly normal.
• Estimate the mean vector and covariance matrix from historical return observations.
• Use the Cholesky factor to convert independent normal draws into correlated simulated returns.

Tags

# Cholesky correlation


# 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?

## Answer by SmurfAcco (score 1)

https://quant.stackexchange.com/a/53186

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

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.