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Complex numbers in finance

July 8th, 2021, 9:31 pm

To what degree are complex numbers useful in finance? I have not seen much discussed on it, a little here:

"The use of complex numbers enables to move from the positive to the negative rates environment without creating prices discrepancies that are surely obtained when using the normal pricing structure." 

Black’s model in a negative interest rate environment, with application to OTC derivatives

"Figure 1 shows an illustrative example of the cumulative standard normal distribution in the field of complex numbers."
 
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Re: Complex numbers in finance

July 8th, 2021, 9:38 pm

this is very complex!
 
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Re: Complex numbers in finance

July 8th, 2021, 10:18 pm

Not long ago, I saw a lecture by Peter Carr and he referenced complex numbers.  

A quick Google search yielded this paper - it's a bit older (2003), but does make use of complex number z, alongside Laplace transforms and Bessel processes.

It may be interesting to you and a deeper search on Carr, some of his coauthors, and also other Courant folks might be fruitful.

Bessel processes, the integral of geometric Brownian motion, and Asian options - ArXiv Peter Carr and Michael Schroeder
 
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Re: Complex numbers in finance

July 9th, 2021, 6:18 am

First comprehensive text to discuss the fundamentals of complex-valued modeling in economics and finance 2012

"Very little has been published on this topic and its applications within the fields of economics and finance, and this volume appeals to graduate-level students studying economics, academic researchers in economics and finance, and economists."
 
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Re: Complex numbers in finance

July 9th, 2021, 11:23 am

For those who have been paying attention the last while on this forum (the non-OT section which is BTW page 3 :-)) will know that the normal CDF can be analytically continued to the complex plane (the Monodramy theorem). Then [$]N(z)[$] is an entire function and can be defined for complex [$]z[$] using the Faddeeva (Kramp) function.
Next, we can use the new form to give a representation of BS formula for _any_ complex value of its parameters. This, in conjunction with the Complex Step Method (CSM) to compute 1st-order greeks, obviating the need to use traditional calculus as in Haug 2007 and Appendix of Bramante et al.

// CSM >> divided differences and it is a competitor of Automatic Differentiation. It can be generalised to gradients and Jacobians.

BTW section 3 of Bramante et al is not clear, especially eqs. (15)-(18). (NO page numbers...)

I have a chapter on this stuff in my forthcoming PDE/FDM book. As well as a crash course on complex analysis etc.

also a discussion here
https://www.datasim.nl/application/file ... hesis_.pdf
 
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Re: Complex numbers in finance

July 9th, 2021, 11:41 am

 
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Re: Complex numbers in finance

July 9th, 2021, 2:07 pm

To what degree are complex numbers useful in finance? I have not seen much discussed on it, a little here:
They are routine in option valuation theory: stochastic vol models, SVJ models, affine models, etc. See: this thread
Of course, also routine in discussions of (analytic) characteristic functions, and whenever integral transforms are useful. 

Basically, most stock price evolution models are invariant under [$]S_t \rightarrow k S_t[$], which means translational invariance in [$]x_t = \log S_t[$], which leads to  Fourier transforms for dimensional reduction.  
 
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Re: Complex numbers in finance

July 9th, 2021, 2:38 pm

The Geometry of the complex plane first invented by my neighbor, Caspar Wessel  
Screen Shot 2021-07-09 at 4.37.37 PM.png
Screen Shot 2021-07-09 at 4.37.37 PM.png (9.4 KiB) Viewed 2984 times
 
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Re: Complex numbers in finance

July 9th, 2021, 3:10 pm

Den glemte norske helten av matematikk: Caspar Wessel

"Han innførte altså et koordinatsystem med reelle tall på én akse og imaginære tall på den andre aksen. Dermed benyttet han det komplekse plan for første gang i historien, og han løste et problem som matematikere hadde slitt med siden 1500-tallet, forteller Johansen."

Able worked as private math teacher just a few miles away, and Munch close bye also, not at the same time.

Caspar Wessel was educated lawyer, (trustable? or did he use a lot of imaginary tricks?))
 
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Re: Complex numbers in finance

July 9th, 2021, 6:46 pm

The Geometry of the complex plane first invented by my neighbor, Caspar Wessel  

Screen Shot 2021-07-09 at 4.37.37 PM.png
He worked for the cadastre? which is two-dimensional. In UK, estate system is 4-dimensional (x,y,z,t).
 
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Re: Complex numbers in finance

July 10th, 2021, 6:23 am

 
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Re: Complex numbers in finance

July 10th, 2021, 6:36 pm

This thread didn't live up to its title.

[$]\mathbb{C}[$] is isomorphic to [$]\mathbb{R}^2[$].

www.youtube.com/watch?v=8fO_-4TecSg

Matrix representation of complex numbers

https://www.nagwa.com/en/explainers/152196980513/
 
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Re: Complex numbers in finance

July 11th, 2021, 4:17 pm

"This thread didn't live up to its title." there is more to it than the real part!
 
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Re: Complex numbers in finance

July 12th, 2021, 9:01 pm

"This thread didn't live up to its title." there is more to it than the real part!
Wilmott threads are not Cauchy sequences in general (not complete space).

But each thread sequence has a convergent subsequence.(Bolzano–Weierstrass theorem)
 
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Re: Complex numbers in finance

July 16th, 2021, 3:52 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).