Model of continuous random cascade processes in financial markets

TL;DR

This paper introduces a continuous stochastic cascade model for financial market volatility, using differential equations and Brownian motions, which successfully reproduces empirical volatility distributions and multifractality.

Contribution

The paper develops a novel continuous cascade model with stochastic differential equations for volatility, incorporating two Brownian motions, and demonstrates its effectiveness on real stock data.

Findings

Model reproduces empirical volatility pdfs

Captures multifractality of financial time series

Aligns with observed empirical facts

Abstract

This article present a continuous cascade model of volatility formulated as a stochastic differential equation. Two independent Brownian motions are introduced as random sources triggering the volatility cascade. One multiplicatively combines with volatility; the other does so additively. Assuming that the latter acts perturbatively on the system, then the model parameters are estimated by application to an actual stock price time series. Numerical calculation of the Fokker--Planck equation derived from the stochastic differential equation is conducted using the estimated values of parameters. The results reproduce the pdf of the empirical volatility, the multifractality of the time series, and other empirical facts.