Recursive Fourier Transform Applied to Financial Signals By Steve Rogers, member, EE-Pub Published: February 13, 2006
Abstract In the financial industry spectral estimation is useful for estimation of cycles. Cycling refers to repetitive sinusoidal signals which dominate financial instruments in lieu of other news worthy events of the day. Cycling may be estimated in numerous non model-based methods. These are generally based on the Fourier Transform. For high frequency trading applications, the standard computation of a discrete Fourier Transform (DFT) or a fast Fourier Transform (FFT) may be too computationally intensive. Hence there has been some interest in recursive approaches to speed up computation. In this paper we will explain a recursive approach to the DFT and apply it to a stock index.
Article Information Field of Study—Signal Processing Keywords—recursive-Fourier-Transform cycle-estimation,
I. INTRODUCTION The Fourier Transform is the basis for much of the spectral estimation conducted. As the computer has gained in capability more emphases has been placed on complex algorithms to gain additional insight into the spectral characteristics of financial signals. The discrete Fourier transform is the most commonly used method for frequency domain studies of signals. It relies on manipulating an available data record length and is computationally intensive due to the redundancy and long length of data required. The objective of cycle estimation is to determine the periodic nature of the signal and to determine the dominant cycle. This is basically the same objective in pitch estimation in speech analyses, in which the dominant cycle is presumed to be the pitch [1]. In pitch determination an all pole model is used for frequency magnitude estimation. The maximum magnitude is determined at each update. The maximum magnitude is then associated with its frequency, which becomes the latest pitch frequency estimate. This process may be applied to other signals of interest, including financial instruments. In this case we may be interested in more than one cycle, as multiple cycles commonly exist in financial instruments. Note that pitch cycles are usually estimated using all pole models by linear prediction [1]. Morelli [2] has introduced a recursive discrete Fourier Transform (RDFT) that is very effective in cycle estimation and frequency based parameter estimation. In this paper we will focus on the recursive discrete Fourier Transform (RDFT).
II. RECURSIVE DISCRETE FOURIER TRANSFORM The general approach is shown below in Figure1 based on Morelli’s approach.
Figure 1 RDFT visualization A more detailed explanation is presented in Figure 2. Morelli’s approach is clarified and shown to be a simple low pass filter technique, which is very easy to implement.
Figure 2 RDFT Development
III. EXAMPLES In the case of financial cycle estimation we are generally looking for a range of cycles that would be useful for trading guidance. If a certain dominant cycle is present we may judge when the stock trend may change up or down. We can also observe any changes in the dominant cycles over time. With an RDFT method the cycle estimations will be current each sample. End of day data for several stocks traded on the NYSE is used for the proof of concept. The matlab code that implements the RDFT described in Figure 2 above is shown in Figure 3 below. Raytheon end-of-day opening stock price data is shown in Figure 4. The Raytheon data cycle estimate is shown in Figure 5.
Figure 3 RDFT matlab code
Figure 4 Raytheon daily stock prices
Figure 5 Raytheon Cycle Estimate in Days The equivalent Boeing and Exxon results are shown in Figure 6 through Figure 9.
Figure 6 Boeing daily stock prices
Figure 7 Boeing Cycle Estimate in Days
Figure 8 Exxon daily stock prices
Figure 9 Exxon Cycle Estimate in Days A recursive Welch spectral estimation algorithm was developed for testing with good correlation of results. These results are not shown here, however, the relevant matlab code is shown in Figure 10.
Figure 10 Matlab code for Recursive Welch Spectral Estimator
IV. CONCLUSION
In the financial industry spectral estimation is useful for estimation of cycles. A recursive discrete Fourier transform (RDFT) is presented and clarified. Results were presented using end of day stock data for three stocks. Results were compared to a Welch spectral estimation approach with reasonable correlations. The RDFT approach is especially useful when the frequencies of interest are known. Future research involves experimentation with various test frequencies and testing with portfolios.
V. REFERENCES [1] Deller, J., etal, Discrete-Time Processing of Speech Signals, 1993, MacMillan, ISBN 0-02-328301-7. [2] Morelli, E. A., High Accuracy Evaluation of the Finite Fourier Transform Using Sampled Data, June, 1997, NASA Technical Memorandum 110340. [3] Franses, P., and Dijk, D., Non-Linear Time Series Models in Empirical Finance, 2000, Cambridge University Press, ISBN 0-521-77965-0.