Multiplicative random cascades with additional stochastic process in financial markets

Abstract

Multiplicative random cascade model naturally reproduces the intermittency or multifractality, which is frequently shown among hierarchical complex systems such as turbulence and financial markets. As described herein, we investigate the validity of a multiplicative hierarchical random cascade model through an empirical study using financial data. Although the intermittency and multifractality of the time series are verified, random multiplicative factors linking successive hierarchical layers show strongly negative correlation. We extend the multiplicative model to incorporate an additional stochastic term. Results show that the proposed model is consistent with all the empirical results presented here.

Model of continuous random cascade processes in financial markets

“…) is consistent with the constraint condition (18) with accuracy. If one imposes the other constraint (19), then the parameters have the following functional form of…”

“…Results show that the discrete cascade model using the random multiplication process did not reproduce the statistical property of the multiplier factors. Therefore, as an alternative model, a discrete random multiplicative cascade process with additional additive stochastic process [16,17,18], or a model formulated as the Fokker-Planck equation considering the cascade process as a continuous Markov process [19,20,21,22,23] was proposed. Those models have been applied to stock market or foreign exchange market data, yielding empirical results including the statistics of multipliers.…”

Model of Continuous Random Cascade Processes in Financial Markets

“…shows that deviation of the quadratic curve from the origin results from the parameter b A (λ), as demonstrated from an empirical study in [18].…”

“…Moreover, those models have been applied to data analysis of the stock market and the foreign exchange market, where they have been successful. Herein, we propose a continuous cascade model formulated as a stochastic differential equation of volatility including two independent modes of Brownian motion: one has multiplicative coupling with volatility; the other has additive coupling, as in an improved discrete cascade model for the stock market, with effectiveness clarified by results of earlier research [18]. The model parameters were estimated by application to a stock price time series.…”

“…Results show that the discrete cascade model using the random multiplication process did not reproduce the statistical property of the multiplier factors. Therefore, as an alternative model, a discrete random multiplicative cascade process with additional additive stochastic processes [16][17][18], or a model formulated as the Fokker-Planck equation considering the cascade process as a continuous Markov process [19][20][21][22][23] was proposed. Those models have been applied to stock market or foreign exchange market data, yielding empirical results including the statistics of multipliers.…”