Role of scaling in the statistical modeling of finance

TL;DR

This paper proposes a new stochastic model for financial index evolution based on the scaling properties of return distributions, capturing volatility clustering and multi-scaling features.

Contribution

It introduces a heteroskedastic, non-Markovian martingale process constructed through scaling principles inspired by statistical mechanics and renormalization group methods.

Findings

Model reproduces volatility clustering.

Captures multi-scaling of return distributions.

Aligns with empirical statistical properties of financial data.

Abstract

Modeling the evolution of a financial index as a stochastic process is a problem awaiting a full, satisfactory solution since it was first formulated by Bachelier in 1900. Here it is shown that the scaling with time of the return probability density function sampled from the historical series suggests a successful model. The resulting stochastic process is a heteroskedastic, non-Markovian martingale, which can be used to simulate index evolution on the basis of an auto-regressive strategy. Results are fully consistent with volatility clustering and with the multi-scaling properties of the return distribution. The idea of basing the process construction on scaling, and the construction itself, are closely inspired by the probabilistic renormalization group approach of statistical mechanics and by a recent formulation of the central limit theorem for sums of strongly correlated random variables.