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 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(τ) = r + (1/τ) A_N(τ), where τ = T -t.