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Stream: theory: applied category theory

Topic: finance in commutative diagrams

Jules Hedges (Nov 20 2020 at 17:20):

I just stumbled across a quite nice blog post: "finance explained in commutative diagrams"
https://www.thenewflesh.net/2016/04/04/finance-explained-in-commutative-diagrams.html

Fawzi Hreiki (Nov 20 2020 at 17:34):

This reminds me of a paper I once saw which uses linear logic and fibrations to model deontic logic (the logic of obligations, permissions, etc) to supposedly talk about contract law and the like

Fawzi Hreiki (Nov 20 2020 at 17:35):

Found it: https://www.sciencedirect.com/science/article/pii/S1570868314000573

Jules Hedges (Nov 20 2020 at 17:36):

Cool!

Fawzi Hreiki (Nov 20 2020 at 17:37):

It actually seems pretty interesting and who knows, maybe one day this stuff could be used to automate large scale mediation of automated contracts

Matteo Capucci (he/him) (Nov 20 2020 at 18:44):

This is very interesting, especially the closing remarks

Matteo Capucci (he/him) (Nov 20 2020 at 18:44):

And well-written, I've got to say

Eric Forgy (Dec 13 2020 at 17:39):

I'm just seeing this :blush:

Finance is pretty fun. After my doctorate from UIUC and a stint at MIT Lincoln Lab, I switched careers to finance back in 2005 and have been there since.

I've maintained a math finance blog since 2007: Phorgy Phynance. The "Ph" is a homage to my days in physics and linking physics to finance :nerd:

The "no arbitrage" principle as a commuting diagram is not new. Physicists have been tinkering with finance for decades. No arbitrage is occassionally presented as parallel transport so that lack of arbitrage is related to "curvature", i.e. nontrivial parallel transport around a loop. Taking this seriously has led to an entire field of "gauge theory of finance" :sweat_smile:

Some fun things:

• The Nobel prize winning Black-Scholes equation is essentially the heat equation. Stock prices are modeled via Brownian motion.
• The "time value of money" can be interpretted as a gauge transformation.
• Arbitrage can be interpretted as curvature.
• Stochastic calculus can be viewed in the context of noncommutative geometry.

Some of my fun/informal articles:

• Einstein meets Markowitz: Relativity Theory of Risk-Return: Looks at risk-return in the context of a Minkowski spacetime with Brownian motion representing space.
• Discrete Black-Scholes Model: Uses a form of discrete stochastic calculus derived from principles of noncommutative geometry.
• Black-Scholes and Schrodinger: Demonstrates that Black-Scholes can be written as Schrodinger's equation complete with noncommutative space and momentum operators.
• Gauge Transforming Black-Scholes: Makes the link between Black-Scholes and the heat equation explicit and shows it can be interpretted as a gauge transformation.
• Currency exchange rates, directed graphs, and quivers: Shows how modelling currencies is naturally done on a directed graph and uses the free construction of a quiver given a directed graph to complete an "FX matrix".
• Categorified Option Pricing Theory: Just some brief high-level thoughts on a potentially interesting area of future research.

Fun stuff :nerd: