Three little arbitrage theorems

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 $\tilde{K} = K e^{-A_N(T)}$.