5 thoughts on “Finance explained in commutative diagrams

  1. I feel like what you’re actually saying is that the collection of all possible commodities forms some sort of a setoid: all morphisms are invertible (which you seem to be assuming implicitly), and there is at most one isomorphism between any two objects (the no-arbitrage principle).

    The assignment of exchange rates forms a functor from the commodity category to the multiplicative group of positive reals (or rationals?), viewed as a category with one object. (That is, it’s a commutative diagram labeled by commodities with numbers for arrows).

    Then you describe some natural transformations e.g. the wait natural transformation from the current prices functor to the future prices functor, or the exchange rate natural transformation from the US prices functor to the Chinese prices functor. The diagrams you draw are the ones making those natural (or failing to do so, in the later examples).

    Finally, your terminal object discussion is a bit off. The fact that there is a unique map from every object to the terminal object doesn’t mean those morphisms can’t be invertible; especially since you seem to be implicitly assuming that all morphisms in your commodity category are in fact invertible. Rather, what you have is that either any two commodities can be exchanged (in which case there is a unique map between any two objects, and *every* commodity is terminal) or there are a pair of commodities which no commodity can have maps to both of (in which case there is no terminal commodity). The moral here is that terminal objects of groupoids are not interesting.

    1. Thanks and agreed. The no-arbitrage principle is the assumption that the space of contracts forms a setoid. I deliberately avoided this language in the post, but it’s definitely how I’d have liked to say it. Once we free ourselves from the no-arbitrage principle, things get a bit hazy for me, and I haven’t taken the time to pin it down. Should we view arbitrage contracts as reversible arrows or irreversible? If you can sign such a contract, does this not pre-supposed that one can issue one. However, it should be much harder to issue such a contract. This gives arbitrage contracts an asymmetry that is not present in an arbitrage-free contract. Maybe the easiest thing to do is to set the objects as quantities of commodities rather than raw commodities. Then the arbitrage contracts are induced by a partial (probably total) ordering.

      With regards to the last point. Yeah, it’s off. I really should replace “terminal object” with the more pedestrian notion of an attracting state or a sink. I’ll do it later today. Thanks Andy.

      1. I know we’re two years removed from this post, but it’s utter gold. It’s only a shame I didn’t stumble on to this earlier. Is this original, or a summary of a more expansive treatment of financial math using category theory. I’d love to know more.

    1. Thanks Jesse+Tyler! Sadly, I can’t offer you a bibliography. The content is basically original and written on a whim. I was preparing for job interviews at the time and I became frustrated with my textbook. Perhaps I could reach out to somebody more entrenched in the applied category theory movement (e.g. David Spivak) to see if they recognize any of this.

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