AUTHOR=Contreras G. Mauricio , Ortiz H. Roberto 

TITLE=Three little arbitrage theorems

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=9

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=<p>The authors proved 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 <italic>A</italic><sub><italic>N</italic></sub>. The first theorem states that if <italic>A</italic><sub><italic>N</italic></sub> = 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 of <italic>r</italic>. The second theorem states that if <italic>A</italic><sub><italic>N</italic></sub> ≠ 0, then the interacting solution can be expressed in terms of all higher derivatives of the solutions to the free Black–Scholes equation with an initial interest rate of <italic>r</italic>. 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 of <italic>r</italic>(τ) = <italic>r</italic> + (1/τ)<italic>A</italic><sub><italic>N</italic></sub>(τ), where τ = <italic>T</italic> − <italic>t</italic>.</p>