Physics Maths Engineering
Three little arbitrage theorems
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Abstract
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 AN. The first theorem states that if AN = 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 AN ≠ 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)AN(τ), where τ = T − t.
Key Questions about 'Three Little Arbitrage Theorems'
The article "Three Little Arbitrage Theorems" by Mauricio Contreras G. and Roberto Ortiz H., published in Frontiers in Applied Mathematics and Statistics in April 2023, explores the exact solutions of a generalized Black–Scholes equation that explicitly incorporates arbitrage bubbles. Source
1. How does the inclusion of arbitrage bubbles affect the Black–Scholes equation?
The study demonstrates that incorporating arbitrage bubbles into the Black–Scholes framework leads to solutions that differ from those of the standard model, providing a more accurate representation of market dynamics where arbitrage opportunities exist. Source
2. What are the implications of the arbitrage number (Aₙ) on the solution of the Black–Scholes equation?
The authors establish that when the arbitrage number Aₙ equals zero, the solution at maturity aligns with that of the free Black–Scholes equation with the same initial interest rate. Conversely, if Aₙ is non-zero, the interacting solution can be expressed in terms of higher derivatives of the free Black–Scholes equation's solutions. Source
3. How does the variable interest rate r(τ) = r + (1/τ)Aₙ(τ) influence the Black–Scholes equation?
The study reveals that for a given arbitrage number, the interacting solution satisfies the free Black–Scholes equation with a variable interest rate defined by r(τ) = r + (1/τ)Aₙ(τ), where τ = T − t. This formulation allows for a dynamic adjustment of the interest rate in response to arbitrage conditions. Source
