Confession time: When I first tried applying Fourier transforms to price charts, two things happened.
My code crashed.
I briefly suspected my master’s in digital signal processing was a scam.
In this installment, we’re upgrading from “Fourier for Dummies” to “Fourier for People Who Pretend to Understand Wine Tasting Notes But Secretly Just Want the Alcohol.” We’ll cover:
Advanced applications (translation: "Cool stuff you can screenshot for LinkedIn flexing"),
4 pitfalls —think of them as landmines, but for your backtest results.
And how to avoid them—because I, too, enjoy not blowing up my results.
Full disclosure: After the second application, this becomes less “expert guide” and more “ape explaining rocket science”. I’m no quant overlord, so take my existential crisis over Fourier transforms with a grain of salt. Or a whisky😏. If you’re a pro and spot me hallucinating, please intervene. This isn’t a humblebrag — it’s a hostage situation.
But hey, let’s pretend competence! Onto the fun part…
5 Advanced Applications
Advanced Application #1: Phase Analysis & Prediction in Highly Correlated Asset Baskets
The Core Idea
To ease into the topic, let’s start by recalling that the purpose of Fourier analysis is to decompose a function into a sum of trigonometric functions. An important point is that this decomposition involves mathematically complex waves, where the real part represents the signal and the imaginary part corresponds to the phase. This phase captures the evolution of frequency/pulsation over time. Since the Fourier transform uses a complex exponential, we can retrieve the phase (as the imaginary part of the spectrum) after applying the inverse transform.
In a basket of tightly correlated assets (think oil majors, FAANG stocks, or Currencies), their price movements share DNA — but with time delays. Fourier’s phase component acts like a timestamp for these delays. If Asset A’s phase consistently leads the basket’s average, it’s either:
- A prophet (unlikely),
- A statistical fluke (probable),
- A candidate for mean reversion (cha-ching).🔔
How It Works
- Decompose each asset’s returns into Fourier components.
- Extract phase angles for dominant frequencies (the ones explaining >90% of variance for the nerd guy — Yes, You !👀).
- Calculate phase spreads relative to the basket’s aggregate phase (weighted average).
- Model phase deviations as Ornstein-Uhlenbeck processes — because everything mean-reverts until it doesn’t.
The Edge
- Sell the leader: When an asset’s phase drifts >1.5σ ahead of the basket, short it with the expectation its “lead” will collapse.
- Buy the lagger: If an asset trails by >2σ, go long — statistically, it’s playing catch-up.
- Correlation decay warning: If phase deviations persist despite high nominal correlation, your basket is fracturing. Exit before the divorce.
Pair this with Bollinger Bands-like thresholds for entries/exits, and you’ve got a tactical rebalancing tool.
Code Crash Course :
def phase_analysis(basket_returns):
phases = []
for asset in basket_returns:
fft = np.fft.fft(asset - np.mean(asset))
phase = np.angle(fft)[:len(fft)//2] # Ignore mirrored frequencies
phases.append(phase)
basket_phase = np.average(phases, weights=market_caps, axis=0)
deviations = [phase - basket_phase for phase in phases]
return deviations
(Disclaimer: This is the “IKEA manual” version. Actual code requires windowing, noise filtering and actual analysis)
Advanced Application #2: Spectral Correlation & Regime Detection
The Core Idea
Forget phase — this time, we’re weaponizing spectral similarity. Assets sharing nearly identical Fourier spectra ("energy" distribution across frequencies) likely move along to the same macroeconomic factors. By clustering assets based on spectral fingerprints, you can:
- Build baskets that move in unison (for pairs trading),
- Spot when “correlated” assets start diverging (early warning for regime shifts),
- Isolate frequency bands where correlations spike (e.g., panic selling = high-frequency syncing).
How It Works
- Chop returns into frequency buckets using FFT. Suppose you want to make a weekly bot, I would do it like this:
- Ultra-low (3+ month trends),
- Business-cycle (1-3 month swings),
- High-frequency noise (<1 week).
- Calculate “band-specific correlations” between assets.
- Cluster assets using spectral similarity (DTW or Wasserstein distance work better than Pearson here😌).
The Edge
- Regime-aware portfolios: Overweight assets dominant in low-frequency coherence during stable markets, switch to high-frequency decorrelated assets when volatility erupts.
- Stealth divergence detection: Traditional correlation matrices miss regime-dependent relationships. Spectral analysis catches when FAANG stocks start moving like crypto — before your risk model implodes.
Code Crash Course
def spectral_clustering(returns, n_freq_bands=3):
spectra = [np.abs(np.fft.fft(r - np.mean(r))) for r in returns]
bands = [np.array_split(spec, n_freq_bands) for spec in spectra] # Split spectra
distance_matrix = np.zeros((len(returns), len(returns)))
for i in range(len(returns)):
for j in range(len(returns)):
# Compare energy distribution across bands
distance_matrix[i,j] = wasserstein_distance(bands[i], bands[j])
return AgglomerativeClustering().fit(distance_matrix)
Actual implementation may require handling complex cross-spectral densities, not just magnitudes.
Advanced Application #3: Volatility Surface Analysis & Arbitrage
The Core Idea
And here we are, at the stage I find utterly perplexing (the infamous
volatility surface). I've never traded an option in my life—except for
one time, and that attempt was as mystifying as deciphering ancient
hieroglyphics with a broken decoder! —WTF are those greeks man!🤯
The volatility surface — that 3D monster plotting implied volatility against strike prices and maturities — is Wall Street’s Rorschach test. Fourier transforms let you see patterns in the inkblot. By applying a 2D Fourier transform to the surface:
- Low frequencies: Reveal broad market expectations.
- High frequencies: Expose micro-tremors.
These high-frequency “glitches” often mean:
- A market maker fat-fingered their vol curve
- A structural inefficiency exists (e.g., wallstreetbets piling into OTM meme calls)
How It (Sort Of) Works
- Preprocess the surface:
- Interpolate missing data (kernel methods > prayer).
- Zap illiquid options — if volume is lower than your dating app matches, ignore it.
- Apply 2D FFT: Transform the surface into frequency space.
- Hunt anomalies: Isolate high-frequency spikes that violate arbitrage bounds.
- Compare to models: If Heston says vol should be 42% but the surface screams 69%, someone’s wrong.
The Edge
- Volatility arbitrage: Exploit mismatches between surface distortions and SABR/Heston predictions.
- Example: Sell overpriced strangles where high-frequency noise inflates wings.
- Smirk trading: When the Fourier decomposition shows asymmetry in left vs. right tail frequencies, fade the panic.
Advanced Application #4: Spread Option Pricing via Fourier’s Dark Arts
The Core Idea
Spread options—those exotic bets on the difference between two assets (oil vs gas, Tesla vs BYD)—are the divas of derivatives. Their payoff: max(S₁ - S₂ - K, 0). Simple? Nope😑. Pricing them requires solving a double integral that would make even Riemann sweat.
Enter Hurd & Zhou’s Fourier method:
- Fourier-transform the payoff into frequency space, turning a calculus nightmare into a multiplication party in this space.
- Exploit "FFT acceleration" to compute prices faster.
- Extract Greeks without recalculating the entire model.
Why This is cool ?
- Speed: Prices a 100x100 grid of strikes/maturities faster than a PM can schedule a pointless meeting.
- Flexibility: Handles stoch vol (Heston), jumps (VG), and other models.
- Greeks on tap: the whole gang just fall out as byproducts — no finite difference required.
The Edge
- Model flexibility: Swap in stochastic vol, jumps, or your favorite model. FFT doesn’t care.
- Real-time risk: Recalculate fast the entire price/Greek matrice.
- Arbitrage sniffing: Spot mispricings across correlated assets before everyone arrives.
[put the article link here —Yes, it really is that simple!]
Advanced Application #5: Fourier Neural Networks for Transition Density Approximation
The Core Idea
Most quantitative finance problems boil down to this nightmare: “Find the transition density of this stochastic process, or perish trying.” Traditional methods like COS expansions or brute-force PDEs either explode or demand a PhD in witchcraft. FourNet sidesteps this by:
- Hijacking Fourier transforms: Directly approximates the characteristic function (the Fourier twin of transition densities).
- Using a single-layer neural network: with Gaussian activations.
- Training in Fourier space: Because why suffer in the time domain when you can flex in frequency?
Why Quants Care ?
FourNet isn’t just another “AI” buzzword, as I understand, it’s a precision tool for:
- Exotic option pricing.
- Stochastic volatility on steroids: Heston models with self-exciting jumps.
- Ultra-short expiries.
The Edge
- Beat traditional methods : Sorry COS !😏
- No "black-box" effect: FourNet’s simplicity means smart people can actually interpret the output.
Pro Tip:
Because, let's be honest, the authors explain it way better than I ever could—check out their expertise right here. Trust me, you’ll get it way faster!
4 pitfalls to avoid
1. Stationarity
The Fourier Transform works best under the assumption that the signal is stationary—that is, its statistical properties (such as mean and variance) remain constant over time (which, by the way, ensures integrability for those who remember 😉). However, in the real world—especially in finance—most signals (think stock prices) are non-stationary and need to be treated accordingly. For a signal to be stationary, it needs two things:
First things first—does your signal have a DC component? No, not the comic book universe—Direct Current (drop that term at your next cocktail party to sound fancy). Essentially, it’s like your signal is dragging along a constant bias (for example, an upward trend).
Then there’s heteroskedasticity (yes, it’s a real word, and yes, you’ll win Scrabble with it). This is a phenomenon where the signal’s variance changes over time, much like having unpredictable amplitude modulation. For instance, when you observe price returns, you'll notice periods of high volatility (positive or negative) interspersed with relatively flat stretches (say, overnight trading). This is just one more nail in the coffin for the idea that returns are Gaussian—because every now and then, the market throws a wild party with extreme events crashing the scene!
Why should you care?
Consider a stock price signal that, on average, trends upward over the long term (S&P, just as an example!). If you casually feed this into a Fourier Transform without any preprocessing, most of the spectral energy will concentrate in the low-frequency domain. In plain terms, it will tell you, "Hey, there’s a slow trend here," and—poof—it overlooks all the interesting finer oscillations. This can give you a distorted picture of the dominant frequencies. Even worse, if the variance is not constant, you end up introducing modulation lobes in the frequency domain. Why? Because temporal amplitude variations can mimic modulation effects, leading to intermodulation phenomena that spawn phantom frequencies—an analytical chaos, if you will. Without proper preparation, you might easily get misled by the spectrum and miss crucial details.
Fixes (because we’re problem-solvers, not problem-havers):
- Remove the DC Component: Center the data by subtracting the mean before performing any analysis, and apply detrending techniques (such as linear modeling or derivative filtering) to make the signal more stationary.
- Stabilize the Variance: If necessary, standardize the variance using, for example, GARCH models or by filtering out spectral modulations.
2. Windowing
The Fourier Transform is a beautiful thing, perfect for analyzing signals that stretch out into infinity. But in the real world, we're stuck dealing with signals of finite duration. When we chop a signal up (taking a chunk of it), it's like multiplying it by a time window. This windowing introduces artifacts in the frequency domain, known as spectral leakage. This is a real pain in the ass especially when you have to take a walk-forward approach...
Why should you care?
Solutions:
- The solution? Ditch the rectangular window.😏 Instead, use a more refined window function, like Hamming or Hann. These are designed to minimize those pesky side lobes. Think of them as noise-canceling headphones for your frequency spectrum.
- Choosing the right window is a delicate balancing act. You need to consider the trade-off between frequency resolution (how well you can distinguish closely spaced frequencies) and the level of side lobe suppression.
3. Sampling and Nyquist Theorem
The Nyquist theorem is simple: to avoid spectral folding (aliasing), your sampling frequency must be at least twice as high as the highest frequency in your signal. If you don't respect this, the high frequencies fold back and mess up your spectrum.
Why should you care?
Let's say you're working on a stock market signal with a key frequency of 1/1h, but you mess up and sample it at 1/45 min instead of the minimum 1/30 min required by Daddy Nyquist —Bad Idea. This frequency of 1/1h will fold back and appear in your spectrum under a fake frequency of 1/15 min. And just like that—boom! Your analysis crashes and burns in a spectacular mess of errors. The frequencies too high and badly sampled create artifacts that distort everything, making your analysis completely useless as soon as you exceed half the sampling frequency.
Fixes:
- Choose a suitable sampling frequency by respecting the Nyquist theorem. Basically, if you're trying to learn all the juicy details about a frequency , you need to sample it twice as fast. 👌
- If the signal contains unwanted high-frequency components, apply a low-pass filter before sampling to avoid aliasing.
4. Overfitting or the Pitfalls of Perfect Function Description
In the realm of spectral analysis, overfitting occurs when you push things too far by attempting to capture every single frequency. Instead of isolating the true trends of your signal, you start modeling the random noise. In effect, you're crafting a model so exquisitely detailed that it ends up being completely off target. This misstep also happens when you use windows that are too short or improperly chosen intervals—essentially amplifying trivial fluctuations until they masquerade as significant patterns.
Why is this a problem?
Imagine a stock market signal laden with random noise. If you segment this signal into slices that are too brief in your quest for cycles, you'll inevitably assign precise frequencies to the noise itself. The result? Spurious cycles that don't actually exist and conclusions that miss the mark. If you treat every spectral component as equally important without any filtering, you end up reconstructing your original noisy signal—noise included😏. This means that your spectral analysis will yield "fake" frequencies that bear no relation to the true dynamics of the signal. Ultimately, models built on such analyses are utterly useless for forecasting or generalization. It’s a classic pitfall, especially when backtesting with Fourier methods on a dataset that’s as thin as overcooked pasta.
Solutions:
- Pre-filter or Smooth the Signal: Before diving into the spectral analysis, clean up your signal to remove noise.
- Statistical Validation: Rigorously validate your spectral results using robust statistical methods to distinguish genuine components from noise artifacts. You can go all fancy with a quantitative score like r² — or just roll with the classic "looks good, what's next?" 😎 It’s up to you!
- Occam's Razor: Stick to only the essential variables—throw away the extra fluff. If it doesn’t add predictive power, why include it?
Conclusion
Fourier analysis is like switching to a secret decoder ring for reality—it lets you reframe problems in ways that make even chaos look like it’s following a well-rehearsed dance routine. By decomposing complex phenomena into neat, localized frequency components, you can spot periodic structures and hidden orders lurking in systems that, on the surface, seem to be throwing a wild rave without a plan. In the trading world, breaking down financial signals into their frequency ingredients paves the way for some really cool applications: noise reduction, cycle and trend detection, risk prediction, and even the analysis of volatility and inter-asset correlations. Basically, it’s the Swiss Army knife of signal processing for those of us who’d rather not wing it with our investments. 😁
But—and here’s the kicker—there are some hurdles that can trip you up faster than a caffeinated squirrel. You’ve got to worry about the stationarity of your signals, apply the right window functions to dodge spectral leakage (because no one likes unexpected cameos in their frequency spectrum), honor Daddy Nyquist to avoid sampling disasters, and steer clear of overfitting that could turn your backtest results into a tragic comedy. Fortunately, there are fixes for these issues: ditch the DC component, slap on the appropriate windows, and employ anti-aliasing filters for that pristine sampling you’ve always dreamed of.
But wait—there’s more! Enter wavelet analysis, the cooler, more flexible—and infinitely more stylish—cousin of the Fourier transform. Wavelets tackle non-stationary signals with a variable time-frequency resolution that opens up even more exciting opportunities for algorithmic trading. It’s like upgrading from a flip phone to a smartphone: more features, more fun, and way less fumbling.
Congratulations to those brave souls who slogged through this article—writing it wasn’t exactly a walk in the park, and I can only imagine that reading it felt like deciphering an alien jazz solo. Now, take a well-deserved break, and then dive into the follow-up article on denoising and forecasting. That’s where we put all these fancy math tricks into practical use—and trust me, your trading strategy will thank you for it.😊
Don’t hesitate to comment, share, and most importantly, code!
I wish you an excellent day and lots of success in your trading projects!
La Bise et à très vite! ✌️
References:
- "A Fourier Transform Method for Spread Option Pricing". (2010)
- Rong Du and Duy-Minh Dang. "Fourier Neural Network Approximation of Transition Densities in Finance". (2024)
Comments