Abstract
This article presents 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, the model parameters are estimated by the 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 probability density function of the empirical volatility, the multifractality of the time series, and other empirical facts.
Highlights
In financial time series, past coarse-grained measures of volatility correlate better to future fine-scale volatility than the reverse process
The original model, in which the stochastic process that connects each layer of the spatial scale is an independent random multiplication process, contradicts results obtained through empirical research
An improved discrete random multiplicative cascade model with additional additive stochastic processes was proposed along with a model formulated as a Fokker–Planck equation by considering cascade processes as a continuous Markov process
Summary
Past coarse-grained measures of volatility correlate better to future fine-scale volatility than the reverse process. Gashghaie et al investigated details of the self-similar transformation rule of the probability density function of price fluctuations and the nonlinear scaling law of the structure function (nth moment of fluctuations), signifying the multifractality of the time series, in their study of the time series of foreign exchange. They pointed out the similarity of price changes in the financial time series to the velocity difference between two spatial points in turbulence [10, 11]. Intermittency at each time scale produces a characteristic hierarchical structure designated as multifractality [8, 9]
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