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.

Highlights

  • In pursuit of understanding and describing phenomena, scholars often encounter situations in which information about the subject of interest is limited

  • Bachelier modeled the dynamics of stock price using the stochastic process and he came up with what is known as Geometric Brownian Motion

  • We laid down an entropic framework to model the dynamics of stocks and European options

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Summary

Introduction

In pursuit of understanding and describing phenomena, scholars often encounter situations in which information about the subject of interest is limited. Entropic inference is an inductive inference framework equipped with proper tools to handle situations where incomplete information is available [1,2,3] Such tools are probability theory, relative entropy, and information geometry. The relative entropy is designed to update the state of partial knowledge, namely the probability distribution, whenever a new piece of information is available. Bachelier modeled the dynamics of stock price using the stochastic process and he came up with what is known as Geometric Brownian Motion. The dynamics of stocks and pricing of Options were further developed by Merton to include jumps [18]. While the main focus of other articles on the subject are to assume different stochastic models for the dynamics of stock price, we derive the stochastic dynamic model from our formalism. The celebrated Black–Scholes-Merton, BSM, differential equation [18] is derived by taking the time derivative of the expected payoff at maturity

Entropic Stock Model
Scale Invariant Measure
Statistical Model
The Prior
Volatility and Entropic Clock
The Constraints
Entropic Instant and Probability Dynamics
European Option Pricing
Black–Scholes Model
BSM Differential Equation
Summary and Discussion
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