Three little arbitrage theorems - Contreras G., Mauricio; Ortiz H., Roberto (Frontiers in Applied Mathematics and Statistics, 2023)
We prove three theorems about the exact solutions of a generalised or interacting Black--Scholes equation that explicitly includes arbitrage bubbles. These arbitrage bubbles can be characterised by an arbitrage number $A_N$. The first theorem states that if $A_N = 0$, then the solution at 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 $A_N \ne 0$, the interacting solution can be expressed in terms of all the higher derivatives of the solutions to the free Black--Scholes equation with an initial interest rate $r$. The third theorem states that for a given arbitrage number, the interacting solution is a solution to the free Black--Scholes equation but with a variable interest rate $r(\tau) = r + (1/\tau) A_N(\tau)$, where $\tau = T -t$.