# Entropic Dynamics of Exchange Rates and Options

## TL;DR

This paper introduces an entropic dynamics framework for modeling exchange rates and options, providing an alternative to traditional models by using entropic inference and scale invariance principles.

## Contribution

It develops a novel entropic inference-based approach to model exchange rate dynamics and derives the Garman-Kohlhagen model within this framework.

## Key findings

- Derivation of scale-invariant geometric Brownian motion for exchange rates.
- Development of a risk-neutral measure for FX options valuation.
- Formalism naturally leads to the Garman-Kohlhagen model.

## 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.