When Risks and Uncertainties Collide: Quantum Mechanical Formulation of Mathematical Finance for Arbitrage Markets

51 Pages Posted: 20 Jun 2019 Last revised: 8 Jul 2019

Date Written: June 28, 2019

Abstract

Geometric Arbitrage Theory reformulates a generic asset model possibly allowing for arbitrage by packaging all assets and their forwards 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 explicitly the Schrödinger equations by means of spectral decomposition of the Hamilton operator, coincides with those obtained by solving the stochastic Euler Lagrange equations derived by a variational principle and providing therefore consistency. Arbitrage bubbles are computed.

Keywords: Geometric Arbitrage, Arbitrage Bubbles, Minimal Arbitrage Dynamics

JEL Classification: C02, D58, D90, G11, G12

Suggested Citation

Farinelli, Simone and Takada, Hideyuki, When Risks and Uncertainties Collide: Quantum Mechanical Formulation of Mathematical Finance for Arbitrage Markets (June 28, 2019). Available at SSRN: https://ssrn.com/abstract=3404437 or http://dx.doi.org/10.2139/ssrn.3404437

Simone Farinelli (Contact Author)

Core Dynamics GmbH ( email )

Scheuzerstrasse 43
Zurich, 8006
Switzerland

Hideyuki Takada

Toho University ( email )

Room 4421
Miyama 2-2-1
Funabashi, Chiba 274-8510
Japan
(+81)-47-472-1856 (Phone)

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