Lera Sandro Claudio, Sornette Didier
ETH Zurich, Singapore-ETH Centre, 1 CREATE Way, No. 06-01 CREATE Tower, 138602 Singapore.
ETH Zurich, Department of Management, Technology, and Economics, Scheuchzerstrasse 7, 8092 Zürich, Switzerland.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Dec;92(6):062828. doi: 10.1103/PhysRevE.92.062828. Epub 2015 Dec 28.
We study the performance of the euro-Swiss franc exchange rate in the extraordinary period from September 6, 2011 to 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 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. Nonparametric empirical estimates of drift and volatility show that the predicted first-order analogy between economics and physics is incorrect. The clue is to realize that the random-walk nature of financial prices results from the continuous anticipation of traders about future opportunities, whose aggregate actions translate into an approximate efficient market with almost no arbitrage opportunities. With the Swiss National Bank's stated commitment to enforce the barrier, traders' 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, Q. J. Econ. 106, 669 (1991)]. We present 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 economic 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.
我们研究了2011年9月6日至2015年1月15日这一特殊时期欧元兑瑞士法郎汇率的表现，在此期间瑞士国家银行实施了1欧元兑1.20瑞士法郎的最低汇率。在基于几何布朗运动构建的一般框架内，并基于金融中的布朗运动与物理学之间的类比，这种空间约束的一阶效应将以与穿越被禁止障碍的路径相关的排斥熵力的形式先验地进入。漂移和波动率的非参数经验估计表明，经济学与物理学之间预测的一阶类比是不正确的。关键在于认识到金融价格的随机游走性质源于交易员对未来机会的持续预期，他们的总体行动转化为一个几乎没有套利机会的近似有效市场。由于瑞士国家银行明确承诺实施该障碍，交易员对这一行动的预期导致漂移消失，同时汇率波动率取决于与障碍的距离。这种效应由克鲁格曼模型描述[P.R.克鲁格曼，《目标区与汇率动态》，《经济学季刊》106，669（1991）]。我们提供了直接的定量经验证据，表明克鲁格曼的理论模型对欧元兑瑞士法郎目标区提供了准确描述。受经济模型见解的启发，我们修正了最初的经济学 - 物理学类比，并表明，在受阻扩散的背景下，这两个系统终究可以用相同的数学来描述。使用最近提出的一种扩展类比，即嵌入与基础订单簿相关的分子流体中的胶体布朗粒子，我们推导得出，在接近限制边界时，两个系统的动力学都由一个具有非常小的恒定漂移和线性扩散系数的随机微分方程描述。作为一个附带结果，我们给出了布朗粒子靠近壁时线性流体动力学扩散系数的简化推导。
