Three little arbitrage theorems

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 A N . The first theorem states that if A N = 0, then the solution at matu...

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Bibliographic Details
Published inFrontiers in applied mathematics and statistics Vol. 9
Main Authors Contreras G., Mauricio, Ortiz H., Roberto
Format Journal Article
LanguageEnglish
Published Frontiers Media S.A 21.04.2023
Subjects
arbitrage
Black–Scholes model
derivatives
option pricing
quantum mechanics
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Summary: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 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 of r . The second theorem states that if A N ≠ 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 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 of r (τ) = r + (1/τ) A N (τ), where τ = T − t .
ISSN:2297-4687
2297-4687
DOI:10.3389/fams.2023.1138663