# Entropic Dynamics of Stocks and European Options

## TL;DR

This paper introduces an entropic inference framework for modeling stock and European option dynamics, deriving the GBM and Black-Scholes models through scale invariance and no-arbitrage principles.

## Contribution

It presents a novel entropic approach to derive classical financial models like GBM and Black-Scholes from fundamental symmetry and information constraints.

## Key findings

- Derives the Geometric Brownian Motion model using entropic inference.
- Develops the Fokker-Planck equation for stock price density.
- Reproduces the Black-Scholes model and its differential equation.

## Abstract

We develop an entropic framework to model the dynamics of stocks and European Options. Entropic inference is an inductive inference framework equipped with proper tools to handle situations where incomplete information is available. The objective of the paper is to lay down an alternative framework for modeling dynamics. An important information about the dynamics of a stock's price is scale invariance. By imposing the scale invariant symmetry, we arrive at choosing the logarithm of the stock's price as the proper variable to model. The dynamics of stock log price is derived using two pieces of information, the continuity of motion and the directionality constraint. The resulting model is the same as the Geometric Brownian Motion, GBM, of the stock price which is manifestly scale invariant. Furthermore, we come up with the dynamics of probability density function, which is a Fokker--Planck equation. Next, we extend the model to value the European Options on a stock. Derivative securities ought to be prices such that there is no arbitrage. To ensure the no-arbitrage pricing, we derive the risk-neutral measure by incorporating the risk-neutral information. Consequently, the Black--Scholes model and the Black--Scholes-Merton differential equation are derived.