Can You Hear the Shape of a Market? Geometric Arbitrage and Spectral Theory
Education / Math
Today we look at very interesting paper written by Simone Farinelli & Hideyuki Takada. We would like to thank Alexander Lipton for bringing this work to our attention.
Farinelli & Takada hypothesize that utilizing gauge symmetries, the Geometric Arbitrage Theory reformulates any asset model.
The “paper develops a conceptual structure—called Geometric Arbitrage Theory(GAT)—to link arbitrage modeling in generic markets with spectral theory.”
Furthermore, “The main idea of the GAT approach consists of modeling markets made of basic financial instruments together with their term structures as principal fibre bundles. The financial features of this market—such as no-arbitrage and equilibrium—are then characterized in terms of standard differential geometric constructions—such as curvature—associated with a natural connection in this fibre bundle.
Mathematically ideal drums with membranes of these two different shapes (but otherwise identical) would sound the same, because the eigenfrequencies are all equal, so the timbral spectra would contain the same overtones. This example was constructed by Gordon, Webb and Wolpert. Notice that both polygons have the same area and perimeter.
As a result, allowing for arbitrage by means of a stochastic principal fibre bundle with a connection whose curvature measures the “instantaneous arbitrage capability”. The cash flow bundle is the associated vector bundle.
Furthermore, the zero eigenspace of its connection Laplacian parameterizes all risk-neutral measures equivalent to the statistical one. A market satisfies the No-Free-Lunch-with-Vanishing-Risk (NFLVR) condition if and only if 0 is in the discrete spectrum of the Laplacian.
The Jarrow–Protter–Shimbo theory of asset bubbles and their classification and decomposition extend to markets not satisfying the NFLVR. Euler’s characteristic of the asset nominal space and non-vanishing of the homology group of the cash flow bundle are both topological obstructions to NFLVR.
Download The Full Paper: (PDF) Can You Hear the Shape of a Market? Geometric Arbitrage and Spectral Theory (researchgate.net)
Written by Simone Farinelli
Quantitative Asset Liability Management in the banking, life & non life, first & re- insurance, pension fund and asset management/hedge fund industries, as well as in the energy sector.
As advisory board member of Core Dynamics GmbH: Core Dynamics is a quantitative software and consulting startup located in Zurich, Switzerland.
Moreover, the mission of Core Dynamics is to provide the next generation of integrated quantitative software applications in combination with high-quality consulting services in quantitative fields.
The software solutions are embedded into a powerful workflow engine enabling customers to implement quantitative related processes end-to-end in one system. That is to say, our product offerings enable data feedings, quantitative assessments, reporting as well as governance, risk and compliance controls in one full-fledged solution. As a result, this allows our clients to control and more efficiently operate their businesses.
Furthermore, by considerably reducing operational risks and by anticipating increasing regulation in this area. In addition, interfaces to industry standard tools (Matlab and R) allow customers to leverage their existing infrastructure.
Publications to be found online on
https://www.researchgate.net/profile/Simone_Farinelli
Written by Hideyuki Takada
Author Page for Hideyuki Takada :: SSRN
Hideyuki Takada – Google Scholar
Education / Math
Can You Hear the Shape of a Market? Geometric Arbitrage and Spectral Theory
