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In quantitative finance, combining statistical filtering techniques with machine learning can provide robust insights into market dynamics. In this article, we explore two powerful tools — Neural Networks and the Kalman Filter — and show how they can be used together to predict the direction of asset price movements. We then outline a trading strategy that uses these predictions, backtests its performance, and compares it to a simple buy-and-hold approach.
1. Theoretical Background
1.1 Neural Networks
Neural networks are a class of machine learning models inspired by biological neural structures. They consist of layers of interconnected nodes (neurons) that transform inputs into outputs through a series of linear and nonlinear operations.
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Feed-Forward Neural Network Model
A basic multilayer perceptron (MLP) can be mathematically described as follows:
Input Layer:
The network receives an input vector:
2. Hidden Layers:
Each hidden layer performs a linear transformation followed by a nonlinear activation function (e.g., ReLU or sigmoid). For layer l:
where:
W(l) is the weight matrix.
b(l) is the bias vector.
σ is an activation function.
For l=1, h(0)=x.
3. Output Layer:
The final layer computes the output:
The softmax function is often used for classification tasks to convert raw scores into probabilities.
4. Training via Backpropagation:
The network parameters {W(l),b(l)} are optimized by minimizing a loss function L(y^,y) (e.g., cross-entropy for classification) using gradient descent:
where θ represents all the parameters and η is the learning rate.
In our code, we use Python’s MLPClassifier from scikit-learn, which implements a multilayer perceptron with hidden layers (in our case, with sizes 32 and 16 neurons) to predict the direction of asset price movements.
1.2 Kalman Filter
The Kalman filter is a recursive algorithm used for estimating the state of a dynamic system from noisy observations. It is especially useful in financial applications where price signals are noisy.
Kalman Filter Equations
The filter works in two main steps: prediction and update.
Prediction Step:
State Prediction:
Error Covariance Prediction:
Here, F is the state transition model, and Q is the process noise covariance.
2. Update Step:
Kalman Gain:
State Update:
Error Covariance Update:
In these equations, H is the observation model, R is the measurement noise covariance, and zₖ is the measurement at time k.
In our code, we use a custom KalmanFilter class to smooth the price series. The filter produces two outputs: a smoothed price and an estimated rate of change, which serve as features for the neural network.
2. The Trading Strategy
The core idea is to predict the price direction for the next week (7 days) using the neural network. The target is defined as:
Trading Signal Generation
Long Position (+1):
If the model predicts a 1, the strategy goes long by buying at the current close and selling 7 days later.Short Position (-1):
If the model predicts -1, the strategy goes short by selling (or taking a short position) at the current close and buying back 7 days later.No Trade (0):
If the model’s confidence is below a threshold (e.g., 80%), the signal is set to 0, meaning no position is taken.
Backtesting the Strategy
For backtesting:
7-Day Returns:
The asset’s 7‑day return is computed as:
Strategy Return:
The trading return is given by:
The backtest aggregates these returns, computes cumulative performance (equity curve), and then compares the strategy to a buy-and-hold approach.
3. Code Walkthrough
Below we break down key sections of the code, explaining how each component contributes to the overall strategy.
The full end-to-end workflow is available in a Google Colab notebook, exclusively for paid subscribers of my newsletter. Paid subscribers also gain access to the complete article, including the full code snippet in the Google Colab notebook, which is accessible below the paywall at the end of the article. Subscribe now to unlock these benefits!
3.1 Data Acquisition and Preprocessing
from binance.client import Client
import pandas as pd
import numpy as np
import ta# Download price data from Binance
client = Client()
pair = 'ETHUSDC'
data = pd.DataFrame(client.get_historical_klines(pair, '1d', '1 year ago'))
data.columns = ['timestamp', 'open', 'high', 'low', 'close', 'volume',
'close_time', 'quote_asset_volume', 'trades',
'taker_buy_base', 'taker_buy_quote', 'ignore']
data['timestamp'] = pd.to_datetime(data['timestamp'], unit='ms')
ohlcv_columns = ['open', 'high', 'low', 'close', 'volume']
data[ohlcv_columns] = data[ohlcv_columns].astype(float)
data.set_index('timestamp', inplace=True)# Shift data to avoid lookahead bias in indicator calculations
data = data.shift()Explanation:
Data is fetched using the Binance API and converted into a DataFrame with proper datetime indexing.
The
.shift()function is used to avoid using current day data for calculations that would normally be computed using past data.
3.2 Smoothing with the Kalman Filter…
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