What are fashionable applications of Fourier analysis in trading? I have heard vague ideas of applications in High Frequency Trading but can somebody provide an example, maybe a reference?
Just for clarification: The approach to split up a stock price in its cosines and to apply this for forecasts or anything similar seems theoretically not justified as we can not assume the stock price to be periodic (outside of the period of observation). So I don't really mean such applications.
Put differently: are there useful, theoretically valid applications of Fourier theory in trading? I am curious for any comments, thank you!
EDIT: I am aware of (theoretically $100\%$ valid) applications in option pricing and calculation of risk measures in the context of Lévy processes (see e.g. here p.11 and following and references therein). This is well established, I guess. What I mean are applications in time series analysis. Sorry for any confusions.
Answer
I can think of an application in options pricing. I came across the following paper a long time ago but think it explains FT very eloquently as applied to pricing options under BS:
http://maxmatsuda.com/Papers/2004/Matsuda%20Intro%20FT%20Pricing.pdf
The fun starts on page 112 but it relies on the 1998 paper by Madan and Carr.
What I like about the paper is that it gives a thorough introduction to FT and only when the groundwork is set it applies it to option pricing. Not a bad approach vs many other papers which make a lot of assumption and assume the reader can jump right into it.
Edit to reflect OP's clarification
There was a question on SO (curious why there but nonetheless its there):
https://stackoverflow.com/questions/4479463/using-fourier-analysis-for-time-series-prediction
In the following couple papers I came across, some more applicable to engineering (signal processing) but I think you have to make very similar assumptions when applying those to financial time series analysis:
Applied to hft:
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