Currency target-zone modeling: An interplay between physics and economics

Abstract

We study the performance of the euro/Swiss franc exchange rate in the extraordinary period from September 6, 2011 and January 15, 2015 when the Swiss National Bank enforced a minimum exchange rate of 1.20 Swiss francs per euro. Within the general framework built on geometric Brownian motions (GBM) and based on the analogy between Brownian motion in finance and physics, the first-order effect of such a steric constraint would enter a priori in the form of a repulsive entropic force associated with the paths crossing the barrier that are forbidden. Non-parametric empirical estimates of drift and volatility show that the predicted first-order analogy between economics and physics are incorrect. The clue is to realise that the random walk nature of financial prices results from the continuous anticipations of traders about future opportunities, whose aggregate actions translate into an approximate efficient market with almost no arbitrage opportunities. With the Swiss National Bank stated commitment to enforce the barrier, traders's anticipation of this action leads to a vanishing drift together with a volatility of the exchange rate that depends on the distance to the barrier. This effect is described by Krugman's model [P.R. Krugman. Target zones and exchange rate dynamics. The Quarterly Journal of Economics, 106(3): [669][670][671][672][673][674][675][676][677][678][679][680][681][682] 1991]. We give direct quantitative empirical evidence that Krugman's theoretical model provides an accurate description of the euro/Swiss franc target zone. Motivated by the insights from the economical model, we revise the initial economics-physics analogy and show that, within the context of hindered diffusion, the two systems can be described with the same mathematics after all. Using a recently proposed extended analogy in terms of a colloidal Brownian particle embedded in a fluid of molecules associated with the underlying order book, we derive that, close to the restricting boundary, the dynamics of both systems is described by a stochastic differential equation with a very small constant drift and a linear diffusion coefficient. As a side result, we present a simplified derivation of the linear hydrodynamic diffusion coefficient of a Brownian particle close to a wall.

Modeling of long-range memory processes with inverse cubic distributions by the nonlinear stochastic differential equations

“…Therefore, these remote deviations can influence the spectrum but they are hardly observable and barely explorable directly. Dierent investigations result in dierent asymptotics of the heavy-tailed distributions, including log-normal, stretched-exponential, Weibull, incomplete Gamma and other distributions [17,48,[56][57][58][59]. Therefore, the distribution ( ) ∼ P x x 2 in some finite interval of the relatively large fluctuations as in our model, yielding β < 1 and long-range memory, does not contradict the known investigation outcomes.…”

“…Equation 10and other similar random walk models are used for analysis of the Euro/ Swiss franc exchange rate [48]. phenomena, while finite time studies reveal the main magnitudes of the observable (e.g.…”

“…Equation ( 10) and other similar random walk models are used for analysis of the Euro/ Swiss franc exchange rate [48].…”

Understanding the Nature of the Long-Range Memory Phenomenon in Socioeconomic Systems

“…Equation ( 8 ) and similar random walk models have been used to model the EUR/CHF exchange rate [ 73 ]. It has also lead to numerous modifications by our group, which we discuss in detail in the following subsections.…”

“…Notably, a few more recent works refer to or use some of our results and are more application-minded. In [ 73 ] a non-linear SDE was derived, providing both physical and economic arguments, to study the performance of EUR/CHF exchange rate. The derived SDE belongs to the class described by ( 8 ).…”