AUTHOR=Contreras G. Mauricio , Ortiz H. Roberto 

TITLE=Three little arbitrage theorems

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=Volume 9 - 2023

YEAR=2023

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2023.1138663

DOI=10.3389/fams.2023.1138663

ISSN=2297-4687

ABSTRACT=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 $\tau = T -t$.