A Comparison of Four Noise Reduction Algorithms as Applied to the BSE Sensex index.

Chart 1: Noise Reduction Algorithms Compared. Data as of 12 June 2025

As compared with established and mature stock exchanges such as the New York, Japanese and European stock exchanges, frontier and emerging stock exchanges have a higher level of ‘noise’. In financial markets, "noise" refers to the erratic and often unpredictable fluctuations in price and volume that can make it difficult to discern genuine market trends. This noise can stem from various sources, including news reports, market sentiment, and even algorithmic trading. It essentially obscures the underlying fundamental value of an asset, making it difficult for investors to make sound decision.

In this post, I am applying four types of de-noising algorithms on the Bombay Stock Exchange Sensex Index (“BSE” or “Sensex”). My hypothesis is that the Sensex  has  a high level of noise. This is because a large proportion of the investors on the approximately 5000 stocks listed on the BSE are retail investors steeped in a speculative trading culture based on rumours, recommendations of friends, news of corporate ties to political bigwigs etc. While exact figures can vary, estimates suggest retail investors contribute around 35-40% of the total trading volume.

Using Julius (https://julius.ai) the Statistics-trained GPT I instructed it to calculate and plot the following noise reduction algorithms (a) Wavelet (b) Fourier (c) Ledoit-Wolf (d) Exponential Moving Average on the Sensex and produce a chart from which I can clearly see the different degrees of noise reduction of each of the four algorithms. (2) Using the output tables of each algorithm, separately generate trading signals for each algorithm. Trading signals: Buy when algorithm crosses Sensex from below and Sell when algorithm crosses Sensex from above. Back test and display the results for each algorithm in terms of profitability using $1.00 per 1 point of the Sensex an initial investment of $1000. (3) I leave to Julius the decision on what values to use for the algorithm parameters/coefficients as well how to partition the data for back testing or forward testing on unseen data.

In Chart 1above, we observe that:

• EMA (mint green) tracks price closely—minimal smoothing.

• Wavelet (lime) dampens high-frequency wiggles a bit more.

• Fourier (red) keeps only the first 10 frequency components—very smooth.

• Ledoit–Wolf (purple) hardly departs from price because shrinkage stays near zero in this small, one-asset context

Table 1 Initial results on performance of each algorithm

Quick take-aways from Table 1 above

• Fourier low-pass filtering generated the highest net profit (~$33 k) under the simple crossover rule, because the extreme smoothing created a few, large trades that captured big swings.

• Wavelet also did well but a bit less aggressive (~$24.8 k).

• EMA produced many small trades for a modest gain (~$4.4 k).

• Ledoit–Wolf added no value here; its curve mirrors price, so crossovers are rare.

In the next stage, we eliminate Exponential Moving Average and Ledoit-Wolf due to their poor performance and focus on Wavelet and Fourier. I instructed Julius to re-run the back test splitting the data to training set=70% and testing set=30%. In addition to absolute profit, I wanted to see which one (Fourier or Wavelet) has larger drawdown. Also calculate not the Sharpe Ratio but the Treynor for each algorithm.

Chart 2 below shows the Equity Curve for the test set of 30% of the data.

Equity curves on the test portion confirm what the metrics imply

• Fourier (red) climbs more gently but almost never gives back gains.
• Wavelet (lime) spikes higher when it’s right, but sinks a bit when momentum signals whipsaw.

Table 2: Profitability and Drawdown on stage 2 Test Set

Key take-aways of stage 2

• Absolute profit: Fourier wins on the test set; Wavelet still wins on the train set.

• Risk: Fourier’s peak drawdown is smaller in both splits—safer capital curve.

• Most importantly, the drawdowns of 3.2 % and 7.7% were incredibly small and only at the beginning part of the Equity Curves. 

• Treynor: both negative on the test sample (inverse market exposure); Wavelet positive on the training sample.

• Overall, if you favour smoother ride and lower drawdown, Fourier filtering is preferable. If you accept deeper dips for the chance of larger rewards, Wavelet remains attractive.

Epilogue

Stage 3: Optimizing the thresholds for trading signals by imposing a band of from 1 to 200 points did not produce better results.  Simple crossover signals work best.