Stochastic Asset Flow Equations: Interdependence of Trend and Volatility

45 Pages Posted: 27 Feb 2021

See all articles by Carey Caginalp

Carey Caginalp

University of Pittsburgh; Chapman University

Gunduz Caginalp

University of Pittsburgh - Department of Mathematics

David Swigon

University of Pittsburgh - Department of Mathematics

Date Written: December 30, 2020

Abstract

The interaction between the volatility and price dynamics is explored. We model stochastic asset prices using the asset flow model with randomness arising directly from supply and demand. We show that the volatility is smallest at the extrema of the price. Linearizing the stochastic differential equation (SDE) about equilibrium, we obtain an exact relation for the autocovariance function, and relate it to the (3 by 3) Jacobian of the linearized SDE. In particular, we find the conditions under which one has a pair of complex conjugate eigenvalues of the Jacobian resulting in oscillations. The frequency of the oscillations depends only on the imaginary part of the complex pair, while the decay rate depends only on the real eigenvalue and the real parts of the complex pair. For the deterministic system, oscillations typically decay rapidly. However, randomness induces oscillations to continue indefinitely with a frequency that depends on the parameters of the deterministic system. The computations and analytical results presented here demonstrate that volatility increases when traders place greater emphasis on trend, confirming a generally held belief among practitioners.

Keywords: autocovariance, random supply, demand, stochastic differential equations, behavioral effects, price oscillations

JEL Classification: G00, G02

Suggested Citation

Caginalp, Carey and Caginalp, Gunduz and Swigon, David, Stochastic Asset Flow Equations: Interdependence of Trend and Volatility (December 30, 2020). Available at SSRN: https://ssrn.com/abstract=3757759 or http://dx.doi.org/10.2139/ssrn.3757759

Carey Caginalp

University of Pittsburgh

Pittsburgh, PA 15260
United States

Chapman University ( email )

One University Dr.
Orange, CA 92866
United States

Gunduz Caginalp (Contact Author)

University of Pittsburgh - Department of Mathematics ( email )

507 Thackeray Hall
Pittsburgh, PA 15260
United States
412-624-8339 (Phone)
412-624-8397 (Fax)

David Swigon

University of Pittsburgh - Department of Mathematics ( email )

507 Thackeray Hall
Pittsburgh, PA 15260
United States

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