You are tasked with designing an advanced algorithmic trading strategy that integrates Kernel PCA and Graph Neural Networks (GNNs) to uncover hidden trading opportunities in a highly interconnected financial system of ð‘=28 currency pairs. The relationships between these pairs are nonlinear and dynamic, creating a computational challenge to extract actionable trading signals.
The Problem
1. Volatility Web Each currency pair ð¶ð‘– has a volatility ð‘‰ð‘–(ð‘¡) at time ð‘¡, evolving as:
Vᵢ(t) = αᵢ ⋅ sin((π / 2) ⋅ Vⱼ(t-1)) + βᵢ ⋅ cos((π / 3) ⋅ Vₖ(t-2)) + γᵢ
Where:
ð‘—≠ð‘˜â‰ ð‘– and ð‘—,ð‘˜âˆˆ[1,28] j,k∈[1,28],ð›¼ð‘–,ð›½ð‘–,ð›¾ð‘– : Randomly initialized coefficients influenced by historical price movements,
ð‘¡: Discrete time steps.
2. Kernel Transformation
The relationships between currency pairs are represented using a Gaussian kernel matrix
ð¾(ð‘¡), defined as:
Kᵢⱼ(t) = exp(-((Vᵢ(t) - Vⱼ(t))²) / (2 ⋅ σ²))
Where:
ð¾ð‘–ð‘—(ð‘¡): Similarity between volatilities
ð‘‰ð‘– and ð‘‰ð‘—, ðœŽ: Kernel bandwidth, dynamically adjusted based on volatility clustering:
σ(t) = σ₀ ⋅ (1 + κ ⋅ StdDev(V(t)))
3. Enhanced PCA Extract principal components from ð¾(ð‘¡) using Kernel PCA:
Eigenvalue Decomposition:
K(t) = V(t) ⋅ Λ(t) ⋅ V(t)^T
Where:
Λ(ð‘¡): Diagonal matrix of eigenvalues
ðœ†ð‘˜(ð‘¡),ð‘‰(ð‘¡): Matrix of eigenvectors ð‘£ð‘˜(ð‘¡).
Projection: Compute the PCA-reduced features ð‘ð‘–(ð‘¡) for each currency pair:
Záµ¢(t) = [⟨Káµ¢(t), vâ‚⟩, ⟨Káµ¢(t), v₂⟩, ⟨Káµ¢(t), v₃⟩]
Where
ð‘ð‘–(ð‘¡)∈R3 are the top-3 principal components.
4. Graph Construction
Construct 28 graphs, one for each GNN, using PCA-reduced features
ð‘ð‘–(ð‘¡) :
Adjacency Matrices:
Aᵢⱼâ¿(t) = exp(-||Záµ¢(t) - Zâ±¼(t)||² / (2 â‹… σ²))
Where
ð´ð‘–ð‘—ð‘›(ð‘¡) is the weight of the edge between
ð¶ð‘– and ð¶ð‘— in the ð‘›ð‘¡â„Ž graph.
Normalization: Normalize ð´ð‘–ð‘—ð‘›(ð‘¡) row-wise:
Aᵢⱼâ¿(t) = Aᵢⱼâ¿(t) / ∑ⱼ Aᵢⱼâ¿(t)
5. GNN Refinement Refine
ð‘ð‘–(ð‘¡) using a 2-layer Graph Neural Network:
Initialization:
Háµ¢â¿(0)(t) = Záµ¢(t)
Propagation: Update features at each layer ð‘™:
Háµ¢â¿(l+1)(t) = ReLU(∑ⱼ Aᵢⱼâ¿(t) â‹… Hâ±¼â¿(l)(t) â‹… Wâ¿(l))
Where:
ð‘Šð‘›(ð‘™): Trainable weight matrix for the ð‘›ð‘¡â„Ž GNN.
Output: After ð¿=2 layers, obtain the refined features:
Háµ¢â¿(L)(t)
6. Signal Generation
Generate trading signals using the first refined feature
ð» ð‘–ð‘› (ð¿)(ð‘¡)[0]: Dynamic Threshold: Calculate a dynamic threshold Θ(ð‘¡):
Θ(t) = ð”¼[Háµ¢â¿(L)(t)[0]] + δ â‹… StdDev(Háµ¢â¿(L)(t)[0])
Signals: Assign signals ð‘†ð‘–ð‘›(ð‘¡) for each GNN ð‘›:
Sáµ¢â¿(t) ={ +1 if Háµ¢â¿(L)(t)[0] > Θ(t), -1 if Háµ¢â¿(L)(t)[0] < -Θ(t), 0 otherwise }
Final Signal Aggregation: Combine signals across all 28 GNNs:
Sáµ¢(t) = Median(Sáµ¢â¿(t) ∀ n ∈ [1,28])
Optimization Tasks
Parameter Optimization: Tune ðœŽ0,ðœ…,ð›¿,ð‘Šð‘›(ð‘™)σ 0​ ,κ,δ,W n (l) to maximize profitability.
Noise Robustness: Account for noise ðœ‚ð‘˜(ð‘¡) in eigenvalues:
λₖ'(t) = λₖ(t) ⋅ (1 + ηₖ(t))
Where ðœ‚ð‘˜(ð‘¡)∼ð’©(0,ðœ†ð‘˜(ð‘¡)â‹…ðœˆ)η k​ (t)∼N(0,λ k​ (t)⋅ν), and 𜈠is a noise coefficient.
Sharpe Ratio: Maximize the Sharpe ratio:
Sharpe = Mean(Returns) / StdDev(Returns)
Puzzle Objective
Simulate ð‘‰ð‘–(ð‘¡) over 1000 ticks for all 28 currency pairs.
Construct 28 GNNs using PCA-reduced features and adjacency matrices.
Refine features through GNN propagation.
Generate trading signals based on dynamic thresholds.
Maximize cumulative returns with a Sharpe ratio above 2.0.
#define PAIRS 28
#define COMPONENTS 3 // Number of principal components from PCA
#define GNN_LAYERS 2 // Number of GNN layers per GNN
string CurrencyPairs[PAIRS] = {
"EURUSD", "GBPUSD", "USDJPY", "GBPJPY", "USDCAD", "EURAUD", "EURJPY",
"AUDCAD", "AUDJPY", "AUDNZD", "AUDUSD", "CADJPY", "EURCAD", "EURCHF",
"EURGBP", "EURNZD", "GBPCAD", "GBPCHF", "NZDCAD", "NZDJPY", "NZDUSD",
"USDCHF", "CHFJPY", "AUDCHF", "GBPNZD", "NZDCHF", "CADCHF", "GBPAUD"
};
// Variables
vars volatilities[PAIRS]; // Volatility for each pair
vars adjacencyMatrices[PAIRS][PAIRS][28]; // Adjacency matrices for 28 GNNs
vars pcaReducedFeatures[PAIRS][COMPONENTS]; // PCA-reduced features
vars gnnWeights[28][GNN_LAYERS][COMPONENTS][COMPONENTS]; // GNN weights for each GNN
vars gnnRefinedFeatures[PAIRS][COMPONENTS]; // Refined features from GNNs
vars kernelMatrix[PAIRS][PAIRS]; // Kernel matrix for PCA
vars eigenvalues[PAIRS]; // Eigenvalues from PCA
vars eigenvectors[PAIRS][PAIRS]; // Eigenvectors from PCA
vars reducedMatrix[PAIRS][COMPONENTS]; // Final reduced components
vars signals[PAIRS]; // Final trading signals
int lookback = 50;
int propagationDepth = 3; // GNN propagation depth
// Step 1: Calculate volatilities for all pairs
function calculateVolatilities() {
for (int i = 0; i < PAIRS; i++) {
volatilities[i] = StdDev(series(price(CurrencyPairs[i])), lookback);
}
}
// Step 2: Construct adjacency matrices for each GNN
function constructAdjacencyMatrices(vars features[PAIRS][COMPONENTS]) {
for (int gnn = 0; gnn < 28; gnn++) { // Each GNN has its own adjacency matrix
for (int i = 0; i < PAIRS; i++) {
for (int j = 0; j < PAIRS; j++) {
double distance = 0;
for (int k = 0; k < COMPONENTS; k++) {
distance += pow(features[i][k] - features[j][k], 2);
}
adjacencyMatrices[i][j][gnn] = exp(-distance / (2 * pow(0.5, 2))); // Gaussian kernel
}
}
}
}
// Step 3: Initialize GNN weights
function initializeGNNWeights() {
for (int gnn = 0; gnn < 28; gnn++) {
for (int l = 0; l < GNN_LAYERS; l++) {
for (int i = 0; i < COMPONENTS; i++) {
for (int j = 0; j < COMPONENTS; j++) {
gnnWeights[gnn][l][i][j] = random() * 0.1; // Small random initialization
}
}
}
}
}
// Step 4: Propagate features through each GNN
function propagateGNN(vars features[PAIRS][COMPONENTS]) {
vars tempFeatures[PAIRS][COMPONENTS];
for (int gnn = 0; gnn < 28; gnn++) { // Process each GNN
for (int t = 0; t < propagationDepth; t++) { // Propagation steps
for (int i = 0; i < PAIRS; i++) {
for (int k = 0; k < COMPONENTS; k++) {
tempFeatures[i][k] = 0;
for (int j = 0; j < PAIRS; j++) {
for (int m = 0; m < COMPONENTS; m++) {
tempFeatures[i][k] += adjacencyMatrices[i][j][gnn] * features[j][m] * gnnWeights[gnn][t][m][k];
}
}
tempFeatures[i][k] = max(0, tempFeatures[i][k]); // ReLU activation
}
}
// Update features for the next layer
for (int i = 0; i < PAIRS; i++) {
for (int k = 0; k < COMPONENTS; k++) {
features[i][k] = tempFeatures[i][k];
}
}
}
}
// Copy the refined features
for (int i = 0; i < PAIRS; i++) {
for (int k = 0; k < COMPONENTS; k++) {
gnnRefinedFeatures[i][k] = features[i][k];
}
}
}
// Step 5: Perform Kernel PCA
function performKernelPCA() {
eigenDecomposition(kernelMatrix, eigenvalues, eigenvectors);
for (int i = 0; i < PAIRS; i++) {
for (int j = 0; j < COMPONENTS; j++) { // Use top COMPONENTS
reducedMatrix[i][j] = dotProduct(kernelMatrix[i], eigenvectors[j]);
}
}
}
// Step 6: Generate trading signals
function generateSignals() {
double mean = 0, stddev = 0;
for (int i = 0; i < PAIRS; i++) {
mean += reducedMatrix[i][0];
}
mean /= PAIRS;
for (int i = 0; i < PAIRS; i++) {
stddev += pow(reducedMatrix[i][0] - mean, 2);
}
stddev = sqrt(stddev / PAIRS);
double threshold = mean + 1.0 * stddev;
for (int i = 0; i < PAIRS; i++) {
if (reducedMatrix[i][0] > threshold) signals[i] = 1; // Buy
else if (reducedMatrix[i][0] < -threshold) signals[i] = -1; // Sell
else signals[i] = 0; // Hold
}
}
// Main function
function run() {
set(PLOTNOW);
// Step 1: Calculate volatilities
calculateVolatilities();
// Step 2: Perform Kernel PCA to reduce features
performKernelPCA();
// Step 3: Construct adjacency matrices for GNNs
constructAdjacencyMatrices(reducedMatrix);
// Step 4: Initialize GNN weights
initializeGNNWeights();
// Step 5: Apply GNN propagation
propagateGNN(reducedMatrix);
// Step 6: Generate trading signals
generateSignals();
// Step 7: Execute trades
for (int i = 0; i < PAIRS; i++) {
if (signals[i] > 0) enterLong(CurrencyPairs[i]);
else if (signals[i] < 0) enterShort(CurrencyPairs[i]);
}
}
