Endogenous Stochastic Arbitrage Bubbles and the Black--Scholes model

TL;DR

This paper introduces a stochastic arbitrage bubble model within the Black--Scholes framework, deriving a generalized equation that captures stochastic arbitrage effects and connects to classical models in certain limits.

Contribution

It develops a stochastic arbitrage bubble model that generalizes the Black--Scholes equation without undetermined parameters and explores its asymptotic behaviors.

Findings

The model reduces to Black--Scholes in deterministic and certain stochastic limits.

Explicit formulas are provided for Gaussian and lognormal bubbles.

The Black--Scholes model is a low energy limit of the proposed stochastic model.

Abstract

This paper develops a model that incorporates the presence of stochastic arbitrage explicitly in the Black--Scholes equation. Here, the arbitrage is generated by a stochastic bubble, which generalizes the deterministic arbitrage model obtained in the literature. It is considered to be a generic stochastic dynamic for the arbitrage bubble, and a generalized Black--Scholes equation is then derived. The resulting equation is similar to that of the stochastic volatility models, but there are no undetermined parameters as the market price of risk. The proposed theory has asymptotic behaviors that are associated with the weak and strong arbitrage bubble limits. For the case where the arbitrage bubble's volatility is zero (deterministic bubble), the weak limit corresponds to the usual Black-Scholes model. The strong limit case also give a Black--Scholes model, but the underlying asset's mean value replaces the interest rate. When the bubble is stochastic, the theory also has weak and strong asymptotic limits that give rise to option price dynamics that are similar to the Black--Scholes model. Explicit formulas are derived for Gaussian and lognormal stochastic bubbles. Consequently, the Black--Scholes model can be considered to be a "low energy" limit of a more general stochastic model.