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bearish
Posts: 5180
Joined: February 3rd, 2011, 2:19 pm

Re: Complex numbers in finance

July 16th, 2021, 11:29 am

I am very much a beginner in finance, so not the answer you are looking for - however i am considering whether there might be a (future) role for Hodge theory , whereby algebraic cycles / subvarieties would be somehow identified with reference instruments with the context of a complex manifold representing the 'market' itself, with interesting (?) results (possibly) arising from cohomology computations, and mixed Hodge theory/intersection cohomology possibly useful (?) in the case of 'incomplete markets' due to its ability to deal with singularities (eg jumps).
I have nothing against solutions looking for problems, but since I don’t understand some of the key words you are using, I can’t contribute all that much. But a coauthor of mine, who knows much about such things, used to occasionally speak fondly about Kähler manifolds, so maybe you are on to something. Maybe you could speculate a little further about what you mean with reference instruments and market? We have been modeling jumps in finance since (at least) the late 70’s though, so that by itself would not be a novelty.