Random processes for long-term market simulations

ArXiv ID: 2511.18125 “View on arXiv”

Authors: Gilles Zumbach

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 plannings, for example in the financial component of social insurances 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 details a multivariate process that incorporate 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’ 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 by a probabilistic forecast allows us to analyze the impact of the drift values, and then to incorporate this uncertainty in the Monte Carlo simulations.

Keywords: Strategic Asset Allocation (SAA), Monte Carlo simulations, Multivariate time series modeling, Heteroskedasticity, Probabilistic forecasting, Equities (Asset Allocation)

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail
• Why: The paper introduces advanced mathematical concepts like multivariate ARCH processes and non-central Student distributions, indicating high complexity. It also presents empirical analyses comparing indices with Monte Carlo simulations and discusses parameter estimation challenges, showing solid empirical grounding.

  flowchart TD
    A["Research Goal<br>Improve Long-Term Portfolio Simulations<br>by Modeling Asset Return Dynamics"] --> B{"Key Methodology Steps"}
    B --> C1["Identify Limitations<br>of Standard Model<br>Constant Drift/Normal"]
    B --> C2["Develop Multivariate Model<br>incorporating: Heteroskedasticity,<br>Fat Tails, Negative Correlations"]
    B --> C3["Replace Point Forecast<br>with Probabilistic Forecast<br>for Drift/Parameters"]
    
    C1 & C2 & C3 --> D["Computational Process<br>Monte Carlo Simulations"]
    
    D --> E{"Data & Inputs Used"}
    E --> F["Historical Financial Time Series<br>(Returns, Volatilities)"]
    E --> G["Asset Allocation Strategy<br>Strategic Asset Allocation SAA"]
    
    F & G --> D
    
    D --> H["Key Findings & Outcomes<br>1. Probability Density of Portfolio Values<br>2. Quantified Impact of Parameter Uncertainty<br>3. Risk Assessment for Long-Term Planning"]