Three little arbitrage theorems

TL;DR

This paper establishes three theorems characterizing solutions to a generalized Black-Scholes equation with arbitrage bubbles, revealing conditions under which solutions match or differ from the standard model and linking arbitrage effects to variable interest rates.

Contribution

It introduces three new theorems that describe the behavior of solutions to a generalized Black-Scholes equation incorporating arbitrage bubbles, connecting arbitrage measures to solution forms.

Findings

Solutions with zero arbitrage number match standard Black-Scholes solutions.

Solutions with non-zero arbitrage number involve higher derivatives of standard solutions.

For any arbitrage number, solutions correspond to a Black-Scholes model with a variable interest rate.

Abstract

We prove three theorems about the exact solutions of a generalized or interacting Black-Scholes equation that explicitly includes arbitrage bubbles. These arbitrage bubbles can be characterized by an arbitrage number $A_N(T)$. The first theorem states that if $A_N(T) = 0$, then the solution at the maturity of the interacting equation is identical to the solution of the free Black-Scholes equation with the same initial interest rate $r$. The second theorem states that if $A_N(T) \ne 0$, the solution can be expressed in terms of all higher derivatives of solutions to the free Black-Scholes equation with the initial interest rate $r$. The third theorem states that whatever the arbitrage number is, the solution is a solution to the free Black-Scholes equation with a variable interest rate $r(\tau) = r + (1/\tau) A_N(\tau)$. Also, we show, by using the Feynman-Kac theorem, that for the special case of a Call contract, the exact solution for a Call with strike price $K$ is equivalent to the usual Call solution to the Black-Scholes equation with strike price $\tilde{K} = K e^{-A_N(T)}$.