Abedi Mohammad, Bartolomeo Daniel
Department of Physics, University at Albany-SUNY, Albany, NY 12222, USA.
Entropy (Basel). 2019 Jun 13;21(6):586. doi: 10.3390/e21060586.
An Entropic Dynamics of exchange rates is laid down to model the dynamics of foreign exchange rates, FX, and European Options on FX. The main objective is to represent an alternative framework to model dynamics. Entropic inference is an inductive inference framework equipped with proper tools to handle situations where incomplete information is available. Entropic Dynamics is an application of entropic inference, which is equipped with the entropic notion of time to model dynamics. The scale invariance is a symmetry of the dynamics of exchange rates, which is manifested in our formalism. To make the formalism manifestly invariant under this symmetry, we arrive at choosing the logarithm of the exchange rate as the proper variable to model. By taking into account the relevant information about the exchange rates, we derive the Geometric Brownian Motion, GBM, of the exchange rate, which is manifestly invariant under the scale transformation. Securities should be valued such that there is no arbitrage opportunity. To this end, we derive a risk-neutral measure to value European Options on FX. The resulting model is the celebrated Garman-Kohlhagen model.
构建了汇率的熵动力学模型来模拟外汇汇率（FX）的动态变化以及外汇的欧式期权。主要目标是提供一个用于模拟动态变化的替代框架。熵推断是一个归纳推断框架，配备了处理信息不完整情况的适当工具。熵动力学是熵推断的一种应用，它配备了时间的熵概念来模拟动态变化。尺度不变性是汇率动态变化的一种对称性，在我们的形式体系中有所体现。为使形式体系在这种对称性下明显不变，我们选择汇率的对数作为建模的合适变量。通过考虑有关汇率的相关信息，我们推导出汇率的几何布朗运动（GBM），它在尺度变换下明显不变。证券的估值应确保不存在套利机会。为此，我们推导了一个风险中性测度来对外汇欧式期权进行估值。得到的模型就是著名的加曼 - 科尔哈根模型。
