Neural Network Cost Functions for Price Prediction

Understanding cost functions in financial prediction

Cost functions in financial neural networks serve as the mathematical compass that guides model optimization. They quantify prediction errors and provide the gradient signals needed for backpropagation. For price prediction tasks, the choice of cost function significantly impacts how the model learns from market data and what types of errors it prioritizes minimizing.

The most common cost functions include:

Mean Squared Error (MSE)

The MSE cost function calculates the average squared difference between predicted and actual prices:

$MSE = \frac{1}{n}\sum_{i=1}^{n}(y_i - \hat{y}_i)^2$

Where:

• $y_i$ is the actual price
• $\hat{y}_i$ is the predicted price
• $n$ is the number of samples

MSE heavily penalizes large prediction errors due to the squared term, making it particularly useful for applications where large price deviations are especially costly.

Asymmetric loss functions for trading

In financial markets, the cost of prediction errors is often asymmetric - underestimating a price move might be more costly than overestimating it, or vice versa. This leads to specialized asymmetric cost functions:

Asymmetric MSE

$AMSE = \frac{1}{n}\sum_{i=1}^{n}[α(y_i - \hat{y}_i)^2 \cdot 1_{y_i > \hat{y}_i} + β(y_i - \hat{y}_i)^2 \cdot 1_{y_i ≤ \hat{y}_i}]$

Where:

• α and β are asymmetry parameters
• $1_{condition}$ is the indicator function

This function allows different penalties for over-prediction versus under-prediction, aligning with specific trading strategy requirements.

Directional accuracy and trend prediction

For trend-following algorithms, directional accuracy may be more important than precise price levels. Specialized cost functions can emphasize directional correctness:

Directional Cost Function

$DCF = -\frac{1}{n}\sum_{i=1}^{n}sign(y_{i+1} - y_i) \cdot sign(\hat{y}_{i+1} - \hat{y}_i)$

This function focuses on predicting price movement directions correctly, regardless of the magnitude of the moves.

Incorporating market microstructure

Neural networks for price prediction often need to account for market microstructure effects. Cost functions can be modified to include:

• Bid-ask spread considerations
• Transaction costs
• Market impact estimates

Spread-Aware Cost Function

$SACF = \frac{1}{n}\sum_{i=1}^{n}(y_i - \hat{y}_i)^2 + λ \cdot spread_i$

Where:

• $spread_i$ is the bid-ask spread
• λ is a weighting parameter

Time-series specific considerations

Financial price prediction often involves time series analysis, requiring cost functions that account for temporal dependencies:

Temporal Cost Function

$TCF = \frac{1}{n}\sum_{i=1}^{n}(y_i - \hat{y}_i)^2 + γ\sum_{i=1}^{n-1}||\hat{y}_{i+1} - \hat{y}_i||$

The second term penalizes excessive volatility in predictions, promoting smoother forecast sequences.

Risk-adjusted cost functions

For applications in portfolio optimization, cost functions can incorporate risk measures:

Risk-Weighted MSE

$RWMSE = \frac{1}{n}\sum_{i=1}^{n}(y_i - \hat{y}_i)^2 \cdot (1 + δ \cdot σ_i)$

Where:

• $σ_i$ is a measure of market volatility
• δ is a risk sensitivity parameter

Implementation considerations

When implementing neural network cost functions for price prediction:

1. Normalization: Ensure input prices are properly normalized to prevent scaling issues
2. Gradient stability: Avoid cost functions that may lead to vanishing or exploding gradients
3. Batch effects: Consider the impact of batch size on cost function behavior
4. Regularization: Include appropriate regularization terms to prevent overfitting

Market adaptation and dynamic cost functions

Markets evolve over time, requiring adaptive cost functions that can respond to changing conditions:

Adaptive Cost Function

$ACF_t = w_t \cdot MSE + (1-w_t) \cdot DCF$

Where:

• $w_t$ is a time-varying weight
• The weight adapts based on market regime indicators

This approach allows the model to adjust its learning objectives based on market conditions, improving robustness across different market environments.

Best practices for cost function selection

1. Strategy alignment: Choose cost functions that match trading strategy objectives
2. Risk management: Incorporate risk preferences into the cost function design
3. Market conditions: Consider how cost functions perform in different market regimes
4. Computational efficiency: Balance complexity with computational requirements
5. Robustness: Test cost function behavior across various market scenarios