Constrained Random Walk Models for Euro/Swiss Franc Exchange Rates: Theory and Empirics

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), 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. It turns out that this naive theory is proved empirically to be completely mistaken. 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, trader's anticipation of this action leads to a volatility of the exchange rate that depends on the distance to the barrier. This effect described by Krugman's model [P.R. Krugman. Target zones and exchange rate dynamics. The Quarterly Journal of Economics, 106(3):669-682, 1991] is supported by non-parametric measurements of the conditional drift and volatility from the data. To the best of our knowledge, our results are the first to provide empirical support for Krugman's model, likely due to the exceptional pressure on the euro/Swiss franc exchange rate that made the barrier effect particularly strong. Despite the obvious differences between "brainless" physical Brownian motions and complex financial Brownian motions resulting from the aggregated investments of anticipating agents, we show that 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.