Abstract
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.
Highlights
To understand, describe, and predict phenomena, scientists have come up with the scientific method of reasoning
Entropic inference is an inductive inference framework designed with proper tools to cope with situations where incomplete information is at our disposal [1,2,3]
Prior information about the subject and the scale invariance symmetry lead us to choose the logarithm of the exchange rate; we want to model the dynamics of the logarithm of the exchange rate
Summary
Describe, and predict phenomena, scientists have come up with the scientific method of reasoning. The relative entropy is designed such that it can incorporate the new information and update the state of partial knowledge, the probability distribution [4,5]. In our formalism, the dynamical models are derived by maximizing the relative entropy. The relative entropy is designed as a tool of inference to update the state of partial belief when new information is available. In order to have our formalism be manifestly scale invariant, we wish to formulate our formalism such that the probability densities are scalar functions This choice will lead to choosing the logarithm of the exchange rate as the proper variable to model. In Entropic Dynamics, the information about the subject matter takes the form of a constraint equation on the probability density function. We derive the dynamics of European Options, which is the Black–Scholes–Merton partial differential equation
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