# Applications of Fourier theory in trading

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.

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References with a connection to trading, especially in high frequency are most interesting. Thank you! –  Richard Feb 26 '13 at 15:22
Causal wavelets which are similar to STFT can be used as an online bandpass filter to denoise a signal. –  user2763361 Oct 6 '13 at 14:06

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):

http://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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Thanks Freddy, I edited the question. What I am looking for are applications in a time series analysis spirit. Sorry for any confusion. –  Richard Feb 26 '13 at 14:48
@Richard, I updated my answer, still digging for some more academic treatise as I have not applied FFT to time series forecasting yet. (I am not a huge believer in applying any sort of bias on market patterns, hence my slight distaste for FFT as applied to forecasting prices). –  Matt Wolf Feb 26 '13 at 15:02
I am afraid, I have to say that you found some nice papers! I have googled and found at least 20 papers and went through them, but the ones that you post look good. I will read them as number 21 - 23. thanks! –  Richard Feb 26 '13 at 15:16
Any references for trading, especially HFT? Thanks! –  Richard Feb 26 '13 at 15:22
@Richard, the 2 papers by Pollock are for obvious reasons quite similar, I think the second one is more general while the first focuses more on circulant matrices. I included some stuff re hft but those are just findings I never read nor touched them. –  Matt Wolf Feb 26 '13 at 15:25

The main application I know of is in option pricing. Peter Carr has done some research here. For an introductory article see this one:

Option valuation using the fast Fourier transform by Peter Carr and Dilip B. Madan:

In this paper the authors show how the fast Fourier transform may be used to value options when the characteristic function of the return is known analytically.

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you beat me to it, my reference relies on your cited paper. –  Matt Wolf Feb 26 '13 at 14:41
Thank you for this link. I know the Fourier methods in option pricing (and e.g. calculation of risk measures) but I mean application in time series analysis. I should note this in the question. –  Richard Feb 26 '13 at 14:42

The direct filter approach (DFA) is a time series filter which is calculated in Fourier space. DFA minimizes the mean square error of a time series $y_t$ compared to a filter estimate $\hat{y_t}$

$E[(y_t - \hat{y_t})^2] = \frac{1}{2 \pi} \int_{-\pi}^{\pi} |\Gamma(\omega)- \hat{\Gamma}(\omega)|^2 h(\omega) d\omega$

The minimization is done in the frequency domain, where $h(\omega)$ is the periodogram and the $\Gamma, \hat{\Gamma}$ are the Fourier coefficients. A detailed discussion of DFA and related frequency-based approaches can be found in the blog link below.

There are applications of DFA to trading and high-frequency trading. I think that the method was originally proposed to forecast economic time series. See the SEF-blog maintained by Marc Wildi for more information.

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In Hamilton's book there is a chapter on Spectral Analysis. It is equivalent to Fourier Analysis of deterministic functions, but now in a stochastic setting.

Intuitively, it is similar to the 'construction' of a Brownian motion as the limit of a Fourier series with random (but carefully selected) coefficients. Extracting and studying these coefficients can shed light on the underlying dynamics.

It can be used as an alternative to Least Squares or Maximum Likelihood for the estimation of model parameters. I imagine that there might be cases where this is more robust, but I do not have first hand experience (have seen it being applied to fractional integration). For example, I can see these techniques yielding alternative co-integration estimators for HFT (i.e. co-integration in time-domain will translate into specific testable/tradeable conditions in frequency-domain).

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I have created some Fourier Analysis of stocks here: http://www.gregthatcher.com/Stocks/Default.aspx

I turn the raw data into a series of sines and cosines, show the Fourier approximation as a graph, and then allow you to "turn off" the various sines and cosines, so that you can see how the various "frequencies" contribute to the graph of the stocks values.

I'm currently working on creating some predictions from the Fourier Series.

I'm not a trader (just a programmer interested in data analysis), so I'd be very interested in any feedback any of you real traders have on what more you'd like to see with Fourier Analysis.

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In general all the applications of filtering and time series analysis can be included; from the standard periodogram analysis to determine the order of a model to the more complex hidden markov model estimations.

An example of the later can be found in Alizera Javaheri's book. In there, he poses the problem of estimating a stochastic volatility problem as a filtering one, and hence all the filtering tools come into hand. How you use the results to generate profit (if that is what your looking for) depends more on you. I have seen people using these estimations combined with credit grades kind of model to try to calibrate the credit and equity markets jointly, and find mis-pricings among them.

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