Simulations with exact means and covariances

To perform risk and portfolio management, we must represent the distribution of the risk factors that affect the market. The most flexible approach is in terms of scenarios and their probabilities, which includes historical scenarios, pure Monte Carlo and importance sampling (see Glasserman, 2004).

Here, we present a simple method to generate scenarios from elliptical distributions with given sample means and covariances. This is very important in applications such as mean-variance portfolio optimisation, which are heavily affected by incorrect representations of the first two moments.

The same problem has been tackled by, among others, Wedderburn (1975), Cheng (1985), Li (1992) and Alexander, Ledermann & Ledermann (2008). However, these approaches require handling matrices or loops of the same size as the number of scenarios. This quickly becomes intractable for large Monte Carlo simulations. Instead, our method is a multivariate generalisation of the intuitive shift/rescaling that appears in, for example, Boyle, Broadie & Glasserman (1995). This method amounts to solving a matrix Riccati equation independent of the number of scenarios.

[image] - Simulations with exact means and covariances (PDF, 242KB)

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