# When Risks and Uncertainties Collide: Mathematical Finance for Arbitrage Markets in a Quantum Mechanical View

## TL;DR

This paper introduces a geometric and quantum mechanical framework for modeling arbitrage markets, linking stochastic asset dynamics with quantum theory to analyze market risks and arbitrage opportunities.

## Contribution

It develops a novel geometric and quantum approach to model arbitrage markets, unifying stochastic asset dynamics with quantum mechanics.

## Key findings

- Quantum models match classical stochastic results
- Curvature measures market arbitrage potential
- Schrödinger equation solutions align with variational principles

## Abstract

Geometric arbitrage theory reformulates a generic asset model possibly allowing for arbitrage by packaging all asset and their forward dynamics into a stochastic principal fibre bundle, with a connection whose parallel transport encodes discounting and portfolio rebalancing, and whose curvature measures, in this geometric language, the instantaneous arbitrage capability generated by the market itself. The asset and market portfolio dynamics have a quantum mechanical description, which is constructed by quantizing the deterministic version of the stochastic Lagrangian system describing a market allowing for arbitrage. Results, obtained by solving the Schroedinger equation, coincide with those obtained by solving the stochastic Euler Lagrange equations derived by a variational principle and providing therefore consistency.