Random processes for long-term market simulations

Gilles Zumbach

Quantitative Finance, 2026, vol. 26, issue 2, 299-324

Abstract: For long-term investments, model portfolios are defined at the level of indexes, a setup known as Strategic Asset Allocation (SAA). The possible outcomes at a scale of a few decades can be obtained by Monte Carlo simulations, resulting in a probability density for the possible portfolio values at the investment horizon. Such studies are critical for long-term wealth planning, for example, in the financial component of social insurance or in accumulated capital for retirement. The quality of the results depends on two inputs: the process used for the simulations and its parameters. The base model is a constant drift, a constant covariance and normal innovations, as pioneered by Bachelier. Beyond this model, this document presents in detail a multivariate process that incorporates the most recent advances in the models for financial time series. This includes the negative correlations of the returns at a scale of a few years, the heteroskedasticity (i.e. the volatility's dynamics), and the fat tails and asymmetry for the distributions of returns. For the parameters, the quantitative outcomes depend critically on the estimate for the drift, because this is a non-random contribution acting at each time step. Replacing the point forecast with a probabilistic forecast allows us to analyze the impact of the drift values and then to incorporate this uncertainty in the Monte Carlo simulations. The main change introduced by the negative return correlations is the partial decoupling between the volatility (along the time direction) and the standard deviation of the terminal values. The definition for the process is supplemented by graphs comparing empirical results obtained from major indices with the values computed with Monte Carlo simulations. Finally, the main statistics for the wealth at increasing time are presented, showing the key features added by the components beyond the basic normal random walk.

Date: 2026
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DOI: 10.1080/14697688.2025.2604768

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