You are tasked with designing an algorithmic trading strategy for volatility arbitrage using a nonlinear volatility model. Your system must operate across ð‘=28 currency pairs, where the relationships between volatilities are nonlinear and highly entangled. The problem involves deciphering the following computational puzzle:
The Problem Volatility Web: Each currency pair ð¶ð‘– has an evolving volatility ð‘‰ð‘–(ð‘¡) over time ð‘¡, defined by the entangled volatility equation:
ð›¼ð‘–,ð›½ð‘–,ð›¾ð‘– are coefficients dependent on historical price movements. ð‘¡ represents discrete time steps (ticks). Kernel Transform: To uncover hidden arbitrage opportunities, a Gaussian kernel transformation is applied to the volatility spreads between pairs:
Where ð¾ð‘–ð‘—(ð‘¡)​ represents the similarity between volatilities ð‘‰ð‘– and ð‘‰ð‘—.
Dynamic Market Dynamics: At any tick ð‘¡, the kernel matrix ð¾(ð‘¡) evolves based on incoming price data. Your goal is to extract principal components from ð¾(ð‘¡) to identify arbitrage paths.
Principal Component Noise: The extracted components ðœ†1,ðœ†2,…,ðœ†ð‘› are influenced by noise-induced eigenvalue drift:
Profit Extraction: The trading signal for each pair ð¶ð‘– is generated from the reduced kernel matrix ð‘…ð‘–(ð‘¡) (t), calculated by projecting ð¾(ð‘¡) onto the top ð‘š m-principal components:
ð”¼[ð‘…(ð‘¡)] is the mean of ð‘…ð‘–(ð‘¡) across all pairs. 𛿠is a risk-adjustment coefficient.
Your Tasks Model the Volatility Web: Simulate ð‘‰ð‘–(ð‘¡) ​ (t) for all ð‘–∈[1,ð‘] over 1000 ticks, ensuring the coefficients ð›¼ð‘–,ð›½ð‘–,ð›¾ð‘– are randomly initialized but correlated to historical price changes.
Construct the Kernel Matrix: Compute ð¾ð‘–ð‘—(ð‘¡)​ at each tick using the Gaussian kernel formula.
Perform Kernel PCA: Decompose ð¾(ð‘¡) into eigenvalues ðœ†ð‘˜(ð‘¡) and eigenvectors ð‘£ð‘˜(ð‘¡)​. Extract the top ð‘š=3 components for trading signals.
Account for Noise: Simulate ðœ‚ð‘˜(ð‘¡) as Gaussian noise and apply it to the eigenvalues to observe how noise affects the trading signals.
Optimize the Threshold: Experiment with different values of 𛿠to maximize profitability while minimizing drawdowns.
Trading Logic: Implement a trading strategy that enters long or short positions based on the dynamic threshold Θ(ð‘¡). Evaluate performance using a back testing framework.
Additional Complexity
To further enhance the challenge, consider:
Dynamic Sigma for Kernel: Allow 𜎠in the Gaussian kernel to adapt based on volatility clustering:
σ(t) = σ₀ ⋅ (1 + κ ⋅ StdDev(V(t)))
Multi-Asset Dependencies: Introduce correlation across non-currency asset classes (e.g., equities, bonds) to impact ð‘‰ð‘–(ð‘¡) (t).
Optimization of Principal Components: Automatically optimize the number of components ð‘š to balance signal strength and noise robustness.
Puzzle Objective Your task is to:
Identify nonlinear arbitrage paths hidden in the noisy principal components. Design an efficient algorithm that adapts to the dynamic nature of the kernel matrix and volatility web. Maximize cumulative profitability while maintaining a Sharpe ratio above 2.0 over the simulated 1000-tick dataset.
Code
#define PAIRS 28 // Number of currency pairs
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"
};
vars VolatilityMatrix[PAIRS][PAIRS]; // Volatility relationship matrix
vars kernelMatrix[PAIRS][PAIRS]; // Kernel matrix for Kernel PCA
vars eigenvalues[PAIRS]; // Eigenvalues from Kernel PCA
vars eigenvectors[PAIRS][PAIRS]; // Eigenvectors from Kernel PCA
vars volatilities[PAIRS]; // Volatility for each pair
vars ReducedMatrix[PAIRS][PAIRS]; // Reduced matrix for all components
vars smoothedSignals[PAIRS]; // Smoothed signals for risk control
int lookback = 50; // Lookback period for volatility calculation
var sigma = 0.5; // Kernel width parameter
var dynamicThreshold; // Dynamic trading threshold
// Function to calculate volatilities for all pairs
function calculateVolatilities() {
for (int i = 0; i < PAIRS; i++) {
volatilities[i] = StdDev(series(price(CurrencyPairs[i]), lookback));
}
}
// Function to calculate the volatility matrix (volatility spreads)
function calculateVolatilityMatrix() {
for (int i = 0; i < PAIRS; i++) {
for (int j = 0; j < PAIRS; j++) {
if (i != j) {
VolatilityMatrix[i][j] = volatilities[i] - volatilities[j];
} else {
VolatilityMatrix[i][j] = 0; // Self-loops have no effect
}
}
}
}
// Function to calculate the kernel matrix using a Gaussian kernel
function calculateKernelMatrix(vars VolatilityMatrix[PAIRS][PAIRS], vars kernelMatrix[PAIRS][PAIRS], var sigma) {
for (int i = 0; i < PAIRS; i++) {
for (int j = 0; j < PAIRS; j++) {
kernelMatrix[i][j] = exp(-pow(VolatilityMatrix[i][j], 2) / (2 * pow(sigma, 2))); // Gaussian kernel
}
}
}
// Perform Kernel PCA: Decompose the kernel matrix into eigenvalues and eigenvectors
function performKernelPCA(vars kernelMatrix[PAIRS][PAIRS], vars eigenvalues, vars eigenvectors) {
eigenDecomposition(kernelMatrix, eigenvalues, eigenvectors); // Decompose the kernel matrix
}
// Reduce data using the top principal components
function reduceKernelData(vars kernelMatrix[PAIRS][PAIRS], vars eigenvectors[PAIRS][PAIRS], vars ReducedMatrix[PAIRS][PAIRS], int numComponents) {
for (int i = 0; i < PAIRS; i++) {
for (int j = 0; j < numComponents; j++) {
ReducedMatrix[i][j] = dotProduct(kernelMatrix[i], eigenvectors[j]);
}
}
}
// Calculate a dynamic threshold based on standard deviation of reduced signals
function calculateDynamicThreshold(int topComponents) {
var sumSquares = 0;
int count = 0;
for (int i = 0; i < PAIRS; i++) {
for (int j = 0; j < topComponents; j++) {
sumSquares += ReducedMatrix[i][j] * ReducedMatrix[i][j];
count++;
}
}
return sqrt(sumSquares / count); // Standard deviation as threshold
}
// Smooth signals to reduce noise
function smoothSignals(int topComponents) {
for (int i = 0; i < PAIRS; i++) {
smoothedSignals[i] = SMA(series(ReducedMatrix[i][0]), lookback); // Smooth first component
}
}
// Trade logic based on Kernel PCA-reduced components
function tradeWithKernelPCA(int topComponents) {
for (int i = 0; i < PAIRS; i++) {
if (abs(smoothedSignals[i]) > dynamicThreshold) {
if (smoothedSignals[i] > 0) {
enterLong(CurrencyPairs[i]);
} else {
enterShort(CurrencyPairs[i]);
}
}
}
}
// Main trading function
function run() {
set(PLOTNOW);
calculateVolatilities(); // Step 1: Calculate volatilities
calculateVolatilityMatrix(); // Step 2: Compute volatility matrix
// Step 3: Compute kernel matrix
calculateKernelMatrix(VolatilityMatrix, kernelMatrix, sigma);
// Step 4: Perform Kernel PCA
performKernelPCA(kernelMatrix, eigenvalues, eigenvectors);
// Step 5: Optimize the number of components based on variance explained
int optimalComponents = optimizeComponents(eigenvalues, PAIRS, 0.90);
// Step 6: Reduce data using the top principal components
reduceKernelData(kernelMatrix, eigenvectors, ReducedMatrix, optimalComponents);
// Step 7: Calculate dynamic threshold
dynamicThreshold = calculateDynamicThreshold(optimalComponents);
// Step 8: Smooth signals for stability
smoothSignals(optimalComponents);
// Step 9: Execute trades based on Kernel PCA-reduced features
tradeWithKernelPCA(optimalComponents);
}
You are tasked with solving a nonlinear, multi-dimensional volatility arbitrage problem. Your goal is to design a strategy that detects complex relationships between currency pair volatilities using Kernel PCA and neural networks. The relationships between volatilities are highly entangled and nonlinear, requiring advanced mathematical techniques.
The Problem Volatility Evolution: Each currency pair ð¶ð‘– has a volatility ð‘‰ð‘–(ð‘¡), defined as:
ð‘—!=ð‘˜!=ð‘–, and ð‘—,ð‘˜âˆˆ[1,ð‘], with ð‘=28 currency pairs. ð›¼ð‘–,ð›½ð‘–,ð›¾ð‘– are coefficients based on historical data. ðœ–ð‘–(ð‘¡)∼ð‘(0,ðœŽ2)ϵ i ​ (t)∼N(0,σ2) is Gaussian noise.
Kernel Transformation: The volatility spreads are mapped into a high-dimensional kernel space:
ð¾ð‘–ð‘—(ð‘¡) is the similarity measure between volatilities ð‘‰ð‘– and ð‘‰ð‘—. 𜎠is the kernel width. Principal Component Analysis (PCA): Decompose the kernel matrix ð¾(ð‘¡) into eigenvalues and eigenvectors:
K(t) = sum_{k=1}^N (lambda_k * (v_k @ v_k'))
Extract the top ð‘š-principal components that explain at least 90% of the variance.
ðœ‚ð‘˜(ð‘¡)∼ð‘(0,ð‘™ð‘Žð‘šð‘ð‘‘ð‘Ž ð‘˜(ð‘¡)∗ð‘›ð‘¢)​ (t)∼N(0,lambda k(t)∗nu). ð‘›ð‘¢ is the noise scaling factor. Reduced Data: Project the kernel matrix onto the top ð‘š principal components:
R_i(t) = sum_{k=1}^m (dot(K_i(t), v_k) * v_k)
Dynamic Threshold: Neural networks predict thresholds for trade execution:
Theta_i(t) = NeuralNet(K_i(t), R_i(t))
Trade Signal: Normalize the reduced data and execute trades based on the signal:
Signal(C_i) = R_i(t) / sqrt(sum_{j=1}^N R_j(t)^2)
Trades are executed when:
abs(Signal(C_i)) > Theta_i(t)
Additional Complexity Dynamic Kernel Width: Allow 𜎠to adapt dynamically based on volatility clustering:
sigma(t) = sigma_0 * (1 + kappa * StdDev(V(t)))
Component Selection: Use a second neural network to optimize the number of components ð‘š:
Objective Simulate the market with ð‘=28 currency pairs and 1000 ticks. Train neural networks to predict thresholds (ð‘‡â„Žð‘’ð‘¡ð‘Ž(ð‘¡)) and optimize components (ð‘š). Execute the strategy and maximize the Sharpe ratio while maintaining a maximum drawdown below 5%.
Code
#define PAIRS 28 // Number of currency pairs
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"
};
vars kernelMatrix[PAIRS][PAIRS]; // Kernel matrix for Kernel PCA
vars ReducedMatrix[PAIRS][PAIRS]; // Reduced matrix for principal components
vars Thresholds[PAIRS]; // Neural network predicted thresholds
vars smoothedSignals[PAIRS]; // Smoothed signals for risk control
vars Predictions[PAIRS]; // Neural network predictions
vars volatilities[PAIRS]; // Volatility for each pair
vars eigenvalues[PAIRS]; // Eigenvalues from PCA
vars eigenvectors[PAIRS][PAIRS]; // Eigenvectors from PCA
int lookback = 50; // Lookback period for volatility calculation
int NeuralModelThreshold; // Handle for neural network predicting thresholds
int NeuralModelSigma; // Handle for neural network predicting kernel width
var sigma = 0.5; // Default kernel width parameter
var dynamicThreshold; // Dynamic trading threshold
// Function to calculate volatilities for all pairs
function calculateVolatilities() {
for (int i = 0; i < PAIRS; i++) {
volatilities[i] = StdDev(series(price(CurrencyPairs[i]), lookback));
}
}
// Function to calculate the kernel matrix using a Gaussian kernel
function calculateKernelMatrix(vars volatilities[PAIRS], vars kernelMatrix[PAIRS][PAIRS], var sigma) {
for (int i = 0; i < PAIRS; i++) {
for (int j = 0; j < PAIRS; j++) {
kernelMatrix[i][j] = exp(-pow(volatilities[i] - volatilities[j], 2) / (2 * pow(sigma, 2))); // Gaussian kernel
}
}
}
// Perform Kernel PCA: Decompose the kernel matrix into eigenvalues and eigenvectors
function performKernelPCA(vars kernelMatrix[PAIRS][PAIRS], vars eigenvalues, vars eigenvectors) {
eigenDecomposition(kernelMatrix, eigenvalues, eigenvectors); // Decompose the kernel matrix
}
// Reduce data using the top principal components
function reduceKernelData(vars kernelMatrix[PAIRS][PAIRS], vars eigenvectors[PAIRS][PAIRS], vars ReducedMatrix[PAIRS][PAIRS], int numComponents) {
for (int i = 0; i < PAIRS; i++) {
for (int j = 0; j < numComponents; j++) {
ReducedMatrix[i][j] = dotProduct(kernelMatrix[i], eigenvectors[j]);
}
}
}
// Neural network training function for dynamic threshold prediction
function trainNeuralNetworks() {
// Train for thresholds
LayerPerceptron(10); // 10 nodes in hidden layer
LayerNormalize();
LayerPerceptron(1); // Output layer for threshold prediction
NeuralModelThreshold = Train(
"threshold_model", // Model name
Features, // Input features (kernel matrix and volatilities)
Targets, // Target outputs (thresholds from historical data)
NumSamples // Number of samples
);
// Train for sigma adjustment
LayerPerceptron(10); // 10 nodes in hidden layer
LayerNormalize();
LayerPerceptron(1); // Output layer for kernel width (sigma) prediction
NeuralModelSigma = Train(
"sigma_model", // Model name
Features, // Input features (historical volatilities and returns)
Targets, // Target outputs (optimal sigma values)
NumSamples // Number of samples
);
}
// Predict dynamic thresholds using the neural network
function predictThresholds(vars kernelMatrix[PAIRS][PAIRS], vars Thresholds[PAIRS]) {
for (int i = 0; i < PAIRS; i++) {
Thresholds[i] = Predict(NeuralModelThreshold, kernelMatrix[i]); // Predict for each row in kernel matrix
}
}
// Predict dynamic sigma using the neural network
function predictSigma(vars volatilities[PAIRS]) {
return Predict(NeuralModelSigma, volatilities); // Predict sigma dynamically
}
// Calculate a dynamic threshold based on reduced signals
function calculateDynamicThreshold(int topComponents) {
var sumSquares = 0;
int count = 0;
for (int i = 0; i < PAIRS; i++) {
for (int j = 0; j < topComponents; j++) {
sumSquares += ReducedMatrix[i][j] * ReducedMatrix[i][j];
count++;
}
}
return sqrt(sumSquares / count); // Standard deviation as threshold
}
// Smooth signals to reduce noise
function smoothSignals(vars Predictions[PAIRS], vars smoothedSignals[PAIRS]) {
for (int i = 0; i < PAIRS; i++) {
smoothedSignals[i] = SMA(series(Predictions[i]), lookback);
}
}
// Trade logic with neural network-predicted thresholds
function tradeWithNeuralNetwork(vars smoothedSignals[PAIRS], vars Thresholds[PAIRS]) {
for (int i = 0; i < PAIRS; i++) {
if (smoothedSignals[i] > Thresholds[i]) {
enterLong(CurrencyPairs[i]);
} else if (smoothedSignals[i] < -Thresholds[i]) {
enterShort(CurrencyPairs[i]);
}
}
}
// Main trading function
function run() {
set(PLOTNOW);
// Step 1: Calculate volatilities
calculateVolatilities();
// Step 2: Predict sigma using the neural network
sigma = predictSigma(volatilities);
// Step 3: Compute kernel matrix
calculateKernelMatrix(volatilities, kernelMatrix, sigma);
// Step 4: Perform Kernel PCA
performKernelPCA(kernelMatrix, eigenvalues, eigenvectors);
// Step 5: Optimize the number of components based on variance explained
int optimalComponents = optimizeComponents(eigenvalues, PAIRS, 0.90);
// Step 6: Reduce data using the top principal components
reduceKernelData(kernelMatrix, eigenvectors, ReducedMatrix, optimalComponents);
// Step 7: Predict thresholds using the neural network
predictThresholds(kernelMatrix, Thresholds);
// Step 8: Generate predictions (e.g., signal strength)
for (int i = 0; i < PAIRS; i++) {
Predictions[i] = dotProduct(kernelMatrix[i], Thresholds);
}
// Step 9: Smooth predictions for stable signals
smoothSignals(Predictions, smoothedSignals);
// Step 10: Execute trades based on predictions and thresholds
tradeWithNeuralNetwork(smoothedSignals, Thresholds);
}
Problem Context You are managing a portfolio of ð‘=28 currency pairs, denoted as ð¶1,ð¶2,…,ð¶ð‘. Your goal is to identify volatility arbitrage opportunities by analyzing volatility spreads and return correlations, projecting them into a nonlinear feature space, and recursively refining the results to adapt to market dynamics.
Market Dynamics 1. Volatility Spread Calculation The volatility spread between any two currency pairs ð¶ð‘– and ð¶ð‘— is given by:
V_ij(t) = sigma_i(t) - sigma_j(t),
where:
ð‘ ð‘–ð‘”ð‘šð‘Žð‘–(ð‘¡): Rolling standard deviation of currency pair ð¶ð‘– over the past ð¿ time steps. ð‘‰ð‘–ð‘—(ð‘¡): The relative volatility between ð¶ð‘– and ð¶ð‘—. 2. Return Correlation The return correlation between ð¶ð‘– and ð¶ð‘— evolves according to:
rho_ij(t+1) = rho_ij(t) + eta_ij(t),
where:
ð‘Ÿâ„Žð‘œð‘–ð‘—(ð‘¡): Pearson correlation coefficient between the returns of ð¶ð‘– and ð¶ð‘—. ð‘’ð‘¡ð‘Žð‘–ð‘—(ð‘¡)∼ð‘(0,ð‘›ð‘¢2): Gaussian noise representing market shocks.
Kernel Transformation To capture nonlinear relationships, both the volatility spread matrix (ð‘‰) and the correlation matrix (ð‘Ÿâ„Žð‘œ) are projected into a high-dimensional feature space using Gaussian kernels.
ð‘…(ð‘‰)ð‘–ð‘—​: Reduced features from the volatility kernel matrix ð¾(ð‘‰).
Kernel PCA Each kernel matrix (ð¾(ð‘‰), ð¾(ð‘Ÿâ„Žð‘œ) is decomposed using Kernel PCA:
K(t) = sum_{k=1}^m lambda_k * (v_k @ v_k'),
where:
ð‘™ð‘Žð‘šð‘ð‘‘ð‘Žð‘˜ : Eigenvalues of the kernel matrix. ð‘£ð‘˜ : Eigenvectors of the kernel matrix. ð‘š: Number of principal components that explain >=90 of the variance. The reduced feature matrices are:
Overview You are tasked with designing a multi-layer volatility arbitrage strategy for a portfolio of 28 currency pairs. The strategy leverages Kernel PCA to extract latent structures in pairwise volatility relationships and their interplay with profitability signals. The challenge is to optimize the combination of kernel components to maximize risk-adjusted profitability while managing noise and feedback loops between dependent volatility and profitability spaces.
Problem Statement Volatility Pair Dynamics: Each pair ð¶ð‘–,ð¶ð‘— exhibits a volatility relationship:
V_{i,j}(t) = |σ_i(t) - σ_j(t)|,
where ðœŽð‘–(ð‘¡) (t) is the volatility of currency pair ð¶ð‘–, computed as:
σ_i(t) = StdDev(P_i(t), \text{lookback}), and ð‘ƒð‘–(ð‘¡) is the price series of ð¶ð‘–.
Profitability Metric: For each pair combination (ð‘–,ð‘—), define a profitability measure Î ð‘–,ð‘—(ð‘¡):
Î _{i,j}(t) = |R_i(t) - R_j(t)|, where ð‘…ð‘–(ð‘¡) is the return:
R_i(t) = P_i(t) / P_i(t-1) - 1.
Kernel Transformation: Use a Gaussian kernel to map volatility and profitability measures into high-dimensional spaces:
Principal Component Extraction: Decompose the kernel matrices ð¾ð‘‰ and ð¾Î into their eigenvalues ðœ†ð‘˜â€‹ and eigenvectors ð‘£ð‘˜â€‹ using:
K = Σ_{k=1}^N λ_k * (v_k ⊗ v_k), where ð‘=378 is the number of pair combinations.
Feedback Loop: The reduced components ð‘…𑉠and ð‘…Π​ are recursively coupled:
Implement the kernel matrices ð¾ð‘‰ and ð¾Î . Extract ð‘š-principal components for both matrices. Train neural networks ð‘ð‘1, ð‘ð‘2, and MetaNN.
Recursive Feedback: Establish recursive coupling between ð‘…𑉠V and ð‘…Î .
Ensure convergence within the specified tolerance. Profitability Optimization:
Optimize the selection of pair combinations to maximize the cumulative profitability.
Backtesting:
Simulate the strategy over 10,000 ticks and report: Sharpe ratio. Maximum drawdown. Trade frequency.
Code
#define PAIRS 28 // Number of currency pairs
#define COMBINATIONS 378 // Total number of unique pair combinations
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"
};
vars volatilities[PAIRS]; // Volatility for each pair
vars volatilityRatios[COMBINATIONS]; // Volatility ratios for pair combinations
vars kernelMatrix1[COMBINATIONS][COMBINATIONS]; // Kernel matrix for PCA-1
vars kernelMatrix2[COMBINATIONS][COMBINATIONS]; // Kernel matrix for PCA-2
vars ReducedMatrix1[COMBINATIONS][COMBINATIONS]; // Reduced matrix for PCA-1
vars ReducedMatrix2[COMBINATIONS][COMBINATIONS]; // Reduced matrix for PCA-2
vars profitabilityMatrix[COMBINATIONS]; // Profitability matrix for pair combinations
vars Threshold1[COMBINATIONS]; // Thresholds from NN-1
vars Threshold2[COMBINATIONS]; // Thresholds from NN-2
vars MetaFeatures[COMBINATIONS]; // Meta-features for MetaNN
vars FinalSignals[COMBINATIONS]; // Final trade signals
int combinationIndex[COMBINATIONS][2]; // Index mapping for combinations
int NN1, NN2, MetaNN; // Neural network handles
int lookback = 50; // Lookback period
// Step 1: Calculate volatilities for all currency pairs
function calculateVolatilities() {
for (int i = 0; i < PAIRS; i++) {
volatilities[i] = StdDev(series(price(CurrencyPairs[i])), lookback);
}
}
// Step 2: Generate all pair combinations
function generateCombinations() {
int index = 0;
for (int i = 0; i < PAIRS - 1; i++) {
for (int j = i + 1; j < PAIRS; j++) {
combinationIndex[index][0] = i; // First pair in the combination
combinationIndex[index][1] = j; // Second pair in the combination
index++;
}
}
}
// Step 3: Calculate volatility ratios for all combinations
function calculateVolatilityRatios() {
for (int k = 0; k < COMBINATIONS; k++) {
int i = combinationIndex[k][0];
int j = combinationIndex[k][1];
volatilityRatios[k] = abs(volatilities[i] - volatilities[j]); // Volatility difference
}
}
// Step 4: Compute kernel matrices
function calculateKernelMatrices(var sigma1, var sigma2) {
for (int i = 0; i < COMBINATIONS; i++) {
for (int j = 0; j < COMBINATIONS; j++) {
kernelMatrix1[i][j] = exp(-pow(volatilityRatios[i] - volatilityRatios[j], 2) / (2 * pow(sigma1, 2)));
kernelMatrix2[i][j] = exp(-pow(profitabilityMatrix[i] - profitabilityMatrix[j], 2) / (2 * pow(sigma2, 2)));
}
}
}
// Step 5: Perform Kernel PCA
function performKernelPCA(vars kernelMatrix[COMBINATIONS][COMBINATIONS], vars eigenvalues, vars eigenvectors) {
eigenDecomposition(kernelMatrix, eigenvalues, eigenvectors);
}
// Step 6: Reduce kernel data
function reduceKernelData(vars kernelMatrix[COMBINATIONS][COMBINATIONS], vars eigenvectors[COMBINATIONS][COMBINATIONS], vars ReducedMatrix[COMBINATIONS][COMBINATIONS], int components) {
for (int i = 0; i < COMBINATIONS; i++) {
for (int j = 0; j < components; j++) {
ReducedMatrix[i][j] = dotProduct(kernelMatrix[i], eigenvectors[j]);
}
}
}
// Step 7: Train neural networks
function trainNeuralNetworks() {
// Train NN-1
LayerPerceptron(10);
LayerNormalize();
LayerPerceptron(1);
NN1 = Train("NN1", ReducedMatrix1, Threshold1, COMBINATIONS);
// Train NN-2
LayerPerceptron(10);
LayerNormalize();
LayerPerceptron(1);
NN2 = Train("NN2", ReducedMatrix2, Threshold2, COMBINATIONS);
// Train Meta Neural Network
LayerPerceptron(5);
LayerNormalize();
LayerPerceptron(1);
MetaNN = Train("MetaNN", [Threshold1, Threshold2], FinalSignals, COMBINATIONS);
}
// Step 8: Predict thresholds and signals
function predictThresholdsAndSignals() {
for (int i = 0; i < COMBINATIONS; i++) {
Threshold1[i] = Predict(NN1, ReducedMatrix1[i]);
Threshold2[i] = Predict(NN2, ReducedMatrix2[i]);
// Meta Neural Network combines predictions
MetaFeatures[i] = [Threshold1[i], Threshold2[i]];
FinalSignals[i] = Predict(MetaNN, MetaFeatures[i]);
}
}
// Step 9: Execute trades
function executeTrades() {
for (int i = 0; i < COMBINATIONS; i++) {
int pair1 = combinationIndex[i][0];
int pair2 = combinationIndex[i][1];
// Example trade logic based on meta signal
if (FinalSignals[i] > 0.5) {
enterLong(CurrencyPairs[pair1]);
enterShort(CurrencyPairs[pair2]);
} else if (FinalSignals[i] < -0.5) {
enterShort(CurrencyPairs[pair1]);
enterLong(CurrencyPairs[pair2]);
}
}
}
// Main function
function run() {
set(PLOTNOW);
// Step 1: Calculate volatilities
calculateVolatilities();
// Step 2: Generate pair combinations
generateCombinations();
// Step 3: Calculate volatility ratios
calculateVolatilityRatios();
// Step 4: Compute kernel matrices
calculateKernelMatrices(0.5, 0.5); // Example sigma values
// Step 5: Perform Kernel PCA
vars eigenvalues1, eigenvectors1;
vars eigenvalues2, eigenvectors2;
performKernelPCA(kernelMatrix1, eigenvalues1, eigenvectors1);
performKernelPCA(kernelMatrix2, eigenvalues2, eigenvectors2);
// Step 6: Reduce kernel data
reduceKernelData(kernelMatrix1, eigenvectors1, ReducedMatrix1, 3); // Top 3 components
reduceKernelData(kernelMatrix2, eigenvectors2, ReducedMatrix2, 3);
// Step 7: Train neural networks
trainNeuralNetworks();
// Step 8: Predict thresholds and signals
predictThresholdsAndSignals();
// Step 9: Execute trades
executeTrades();
}
Problem: Recursive Kernel PCA with GNN for Profitability Optimization
You are tasked to design a trading strategy that integrates two Kernel PCAs (for volatility and profitability relationships) with a Graph Neural Network (GNN). The strategy dynamically updates node embeddings and kernel matrices to extract dominant patterns and optimize trading signals for a portfolio of ð‘›=28 currency pairs.
The goal is to maximize profitability Î , subject to market dynamics and constraints.
Step 1: Graph Construction Graph Definition: Let the currency pairs be represented as a graph ðº=(ð‘‰,ð¸), where:
ð‘‰: Nodes, ð‘£ð‘– , represent currency pairs. ð¸: Edges represent relationships between pairs. Node Features: Each node ð‘£ð‘– is assigned a feature vector ð‘¥ð‘–, defined as:
x_i = [σ_i(t), R_i(t), M_i(t), Ï_{i,j}(t)]
where:
ðœŽð‘–(ð‘¡): Volatility of ð¶ð‘–, ð‘…ð‘–(ð‘¡): Return of ð¶ð‘–,ð‘€ð‘–(ð‘¡): Momentum of ð¶ð‘–, ðœŒð‘–,ð‘—(ð‘¡): Correlation between ð¶ð‘– and ð¶ð‘—.
Edge Weights: The weighted adjacency matrix ð´ is given by:
A[i][j] = exp(-||x_i - x_j||^2 / (2 * σ^2))
where 𜎠is a kernel width parameter.
Step 2: Kernel Matrices
Volatility Kernel ð¾ð‘‰ : Construct ð¾ð‘‰K using a Gaussian kernel:
You are tasked with designing a multi-layer trading strategy based on recursive Kernel PCA and interdependent ð‘-Layer GNNs to analyze ð‘›=28 currency pairs. Each GNN layer specializes in capturing different graph dynamics (e.g., volatility, profitability, momentum), while layers interact nonlinearly and recursively. The outputs from each GNN layer influence:
The adjacency matrices for the other layers. The kernel matrices for volatility (ð¾ð‘‰) and profitability (ð¾Î ). The node embeddings, which are combined and propagated across layers. The challenge is to optimize the system by dynamically balancing these interactions while maximizing risk-adjusted profitability.
Extracting dominant volatility and profitability components with Kernel PCA. Propagating and aggregating embeddings across ð‘-interdependent GNN layers. Dynamically refining kernels and thresholds using recursive feedback.
Code
#define PAIRS 28
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"
};
vars volatilities[PAIRS]; // Volatility for each pair
vars adjacencyMatrices[PAIRS][PAIRS][28]; // Adjacency matrices for 28 GNNs
vars nodeEmbeddings[PAIRS][5][28]; // Node embeddings for 28 GNNs
vars updatedEmbeddings[PAIRS][5]; // Final node embeddings after aggregation
vars kernelMatrix[PAIRS][PAIRS]; // Kernel matrix for PCA
vars eigenvalues[PAIRS]; // Eigenvalues from PCA
vars eigenvectors[PAIRS][PAIRS]; // Eigenvectors from PCA
vars reducedMatrix[PAIRS][3]; // Reduced components from PCA
vars thresholds[PAIRS]; // Thresholds for trade execution
vars signals[PAIRS]; // Final trading signals
int lookback = 50;
int NN; // Neural network handle for threshold prediction
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 28 GNNs
function constructAdjacencyMatrices() {
for (int l = 0; l < 28; l++) { // For each GNN layer
for (int i = 0; i < PAIRS; i++) {
for (int j = 0; j < PAIRS; j++) {
if (i != j) {
adjacencyMatrices[i][j][l] = exp(-pow(volatilities[i] - volatilities[j], 2) / (2 * 0.5^2)); // Gaussian kernel
} else {
adjacencyMatrices[i][j][l] = 0; // No self-loops
}
}
}
}
}
// Step 3: Propagate embeddings for each GNN
function propagateEmbeddings() {
for (int l = 0; l < 28; l++) { // For each GNN
for (int t = 0; t < propagationDepth; t++) { // Propagation steps
vars newEmbeddings[PAIRS][5];
for (int i = 0; i < PAIRS; i++) {
for (int k = 0; k < 5; k++) {
newEmbeddings[i][k] = 0;
for (int j = 0; j < PAIRS; j++) {
newEmbeddings[i][k] += adjacencyMatrices[i][j][l] * nodeEmbeddings[j][k][l];
}
newEmbeddings[i][k] = ReLU(newEmbeddings[i][k]);
}
}
// Update embeddings
for (int i = 0; i < PAIRS; i++) {
for (int k = 0; k < 5; k++) {
nodeEmbeddings[i][k][l] = newEmbeddings[i][k];
}
}
}
}
}
// Step 4: Aggregate embeddings across all GNNs
function aggregateEmbeddings() {
for (int i = 0; i < PAIRS; i++) {
for (int k = 0; k < 5; k++) {
updatedEmbeddings[i][k] = 0;
for (int l = 0; l < 28; l++) {
updatedEmbeddings[i][k] += nodeEmbeddings[i][k][l];
}
updatedEmbeddings[i][k] /= 28; // Average across GNNs
}
}
}
// Step 5: Update kernel matrix using aggregated embeddings
function updateKernelMatrix() {
for (int i = 0; i < PAIRS; i++) {
for (int j = 0; j < PAIRS; j++) {
kernelMatrix[i][j] = exp(-dotProduct(updatedEmbeddings[i], updatedEmbeddings[j]) / (2 * 0.5^2)); // Gaussian kernel
}
}
}
// Step 6: Perform Kernel PCA
function performKernelPCA() {
eigenDecomposition(kernelMatrix, eigenvalues, eigenvectors);
for (int i = 0; i < PAIRS; i++) {
for (int j = 0; j < 3; j++) { // Top 3 components
reducedMatrix[i][j] = dotProduct(kernelMatrix[i], eigenvectors[j]);
}
}
}
// Step 7: Train neural network for threshold prediction
function trainNeuralNetwork() {
LayerPerceptron(10); // Hidden layer
LayerNormalize();
LayerPerceptron(1); // Output layer
NN = Train("ThresholdNN", reducedMatrix, thresholds, PAIRS);
}
// Step 8: Generate trade signals
function generateSignals() {
for (int i = 0; i < PAIRS; i++) {
signals[i] = Predict(NN, reducedMatrix[i]);
}
}
// Step 9: Execute trades
function executeTrades() {
for (int i = 0; i < PAIRS; i++) {
if (signals[i] > 0.5) {
enterLong(CurrencyPairs[i]);
} else if (signals[i] < -0.5) {
enterShort(CurrencyPairs[i]);
}
}
}
// Main function
function run() {
set(PLOTNOW);
// Step 1: Calculate volatilities
calculateVolatilities();
// Step 2: Construct adjacency matrices
constructAdjacencyMatrices();
// Step 3: Propagate embeddings for each GNN
propagateEmbeddings();
// Step 4: Aggregate embeddings
aggregateEmbeddings();
// Step 5: Update kernel matrix
updateKernelMatrix();
// Step 6: Perform Kernel PCA
performKernelPCA();
// Step 7: Train neural network for thresholds
trainNeuralNetwork();
// Step 8: Generate trade signals
generateSignals();
// Step 9: Execute trades
executeTrades();
}
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:
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:
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.
Code
#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]);
}
}
In the realm of market energies, 28 currency pairs, each represented as a celestial entity ð¶1,ð¶2,…,ð¶28, dance to the rhythms of volatility and interconnectedness. These entities are bound by the threads of a Volatility Web, which symbolizes the fluctuating relationships between their inner energies (volatilities). The goal is to achieve a state of market equilibrium, a harmonious alignment where profitable opportunities can be uncovered while respecting the natural dynamics of the system. ( Attached problem description file.)
Code
#define PAIRS 28
#define COMPONENTS 3 // Number of PCA components
#define GNN_LAYERS 2 // Number of GNN layers
#define ACTIONS 3 // Buy, Sell, Hold
// Variables
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"
};
// Enhanced PCA, GNN, and Signal Variables
vars kernelMatrix[PAIRS][PAIRS]; // Kernel matrix for PCA
vars pcaReducedFeatures[PAIRS][COMPONENTS]; // PCA-reduced features for each pair
vars adjacencyMatrices[PAIRS][PAIRS]; // Adjacency matrices for GNNs
vars gnnWeights[GNN_LAYERS][COMPONENTS][COMPONENTS]; // GNN weights
vars gnnOutputs[PAIRS][ACTIONS]; // GNN probabilities for Buy/Sell/Hold
vars similarityMatrix[PAIRS][PAIRS]; // Cross-Entropy similarity matrix
vars refinedOutputs[PAIRS][ACTIONS]; // Refined GNN probabilities
vars signals[PAIRS]; // Final trading signals
// Step 1: 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
pcaReducedFeatures[i][j] = dotProduct(kernelMatrix[i], eigenvectors[j]);
}
}
}
// Step 2: Initialize GNN Weights
function initializeGNNWeights() {
for (int l = 0; l < GNN_LAYERS; l++) {
for (int i = 0; i < COMPONENTS; i++) {
for (int j = 0; j < COMPONENTS; j++) {
gnnWeights[l][i][j] = random() * 0.1; // Small random initialization
}
}
}
}
// Step 3: GNN Propagation
function propagateGNN() {
vars tempFeatures[PAIRS][COMPONENTS];
for (int l = 0; l < GNN_LAYERS; l++) { // GNN propagation layers
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] * pcaReducedFeatures[j][m] * gnnWeights[l][m][k];
}
}
tempFeatures[i][k] = max(0, tempFeatures[i][k]); // ReLU activation
}
}
// Update PCA features for the next layer
for (int i = 0; i < PAIRS; i++) {
for (int k = 0; k < COMPONENTS; k++) {
pcaReducedFeatures[i][k] = tempFeatures[i][k];
}
}
}
// Generate probabilities (Buy/Sell/Hold) based on final GNN outputs
for (int i = 0; i < PAIRS; i++) {
for (int k = 0; k < ACTIONS; k++) {
gnnOutputs[i][k] = random() * 0.1; // Placeholder for GNN probability outputs
}
}
}
// Step 4: Compute Cross-Entropy Similarity
function computeCrossEntropySimilarity() {
for (int i = 0; i < PAIRS; i++) {
for (int j = 0; j < PAIRS; j++) {
similarityMatrix[i][j] = 0;
for (int k = 0; k < ACTIONS; k++) {
similarityMatrix[i][j] -= gnnOutputs[i][k] * log(gnnOutputs[j][k] + 1e-8);
}
}
}
}
// Step 5: Refine GNN Outputs Using Similarity
function refineGNNOutputs() {
for (int i = 0; i < PAIRS; i++) {
for (int k = 0; k < ACTIONS; k++) {
refinedOutputs[i][k] = 0;
double weightSum = 0;
for (int j = 0; j < PAIRS; j++) {
refinedOutputs[i][k] += similarityMatrix[i][j] * gnnOutputs[j][k];
weightSum += similarityMatrix[i][j];
}
refinedOutputs[i][k] /= (weightSum + 1e-8); // Normalize
}
}
}
// Step 6: Generate Trading Signals
function generateSignals() {
for (int i = 0; i < PAIRS; i++) {
signals[i] = refinedOutputs[i][0] - refinedOutputs[i][1]; // Buy-Sell difference
}
}
// Step 7: Execute Trades
function executeTrades() {
for (int i = 0; i < PAIRS; i++) {
if (signals[i] > 0) enterLong(CurrencyPairs[i]);
else if (signals[i] < 0) enterShort(CurrencyPairs[i]);
}
}
// Main Function
function run() {
set(PLOTNOW);
// Step 1: Perform Kernel PCA
performKernelPCA();
// Step 2: Initialize GNN weights
initializeGNNWeights();
// Step 3: Propagate GNN
propagateGNN();
// Step 4: Compute Cross-Entropy Similarity
computeCrossEntropySimilarity();
// Step 5: Refine GNN outputs
refineGNNOutputs();
// Step 6: Generate trading signals
generateSignals();
// Step 7: Execute trades
executeTrades();
}
A Game-Theoretic and Abstract Perspective: The Dance of Volatility
In the ever-unfolding theater of financial markets, imagine a dynamic stage where 28 players—currency pairs—engage in an intricate and ceaseless dance. Each player moves to the rhythm of the global economy, its steps choreographed by the forces of supply, demand, sentiment, and speculation. Yet, the rules of this dance are not static; they shift in response to unseen pressures, from political tremors to macroeconomic tides.
The problem before us is akin to deciphering the secret language of this dance. It is not merely about observing the movements but about understanding the hidden relationships—the subtle cues and responses that bind the dancers together. Game theory offers a lens through which we might glimpse the rules of this game. It teaches us that no dancer moves in isolation; every step, every pause, every flourish is shaped by the anticipation of others' moves.
In this game, volatility becomes a signal, a whisper of intent, echoing through the financial theater. Each player’s volatility reflects its internal tensions and external interactions, and collectively, these signals form a web of interdependencies. The challenge lies in untangling this web, not to still the dance, but to predict its next evolution.
The problem is also an abstraction of mathematical elegance. Imagine the players as entities in a multidimensional space, their positions defined by intricate patterns of volatility. This space is not flat but warped by nonlinearities, where proximity signifies similarity, and distance hints at independence. Yet, even the closest dancers may diverge when the music shifts.
The task is to map this space, not merely by observing the surface but by delving into its deeper dimensions. Enhanced techniques like principal component analysis allow us to distill the essence of the dance, reducing its complexity while preserving its soul. Graph structures emerge to connect the players, each edge a reflection of shared rhythms and mutual influence. These connections are not static; they evolve, adapting as the dancers respond to new melodies.
But here lies the heart of the puzzle: how do we assign meaning to these connections? How do we ensure that what we perceive as a pattern is not an illusion conjured by noise? The game is adversarial in nature, with uncertainty itself playing the role of a cunning opponent. It introduces randomness to confound our predictions, demanding that we refine our understanding, filter the signals, and discard the noise.
To navigate this space, we employ strategies that echo the principles of cooperation and competition. Graph neural networks, as players in their own right, enter the stage, modeling relationships and predicting movements. Yet, they, too, are fallible, requiring constant refinement through feedback and comparison. Cross-entropy becomes the arbiter, a measure of alignment between their outputs, guiding the networks to harmonize with the dance.
At its core, this problem challenges us to think deeply about strategy and foresight. It is not merely about predicting individual moves but about understanding the grand choreography. What happens when the music changes? When the connections between players weaken or intensify? How do we balance our desire for precision with the inevitability of uncertainty?
The solution lies not in imposing control but in embracing the fluidity of the game. It requires humility to recognize the limits of our understanding and the creativity to imagine new ways to learn. It demands an appreciation for the interplay of chaos and order, a respect for the dancers' autonomy, and the wisdom to act when the signals align, and the next move becomes clear. In this, the problem transcends finance, touching upon the universal dance of interconnection and adaptation.
Code
#define PAIRS 28
#define COMPONENTS 3 // Number of PCA components
#define GNN_LAYERS 2 // Number of GNN layers
#define ACTIONS 3 // Buy, Sell, Hold
// Variables
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"
};
// Kernel PCA, GNN, and Signal Variables
vars kernelMatrix[PAIRS][PAIRS]; // Kernel matrix for PCA
vars pcaReducedFeatures[PAIRS][COMPONENTS]; // PCA-reduced features for each pair
vars adjacencyMatrices[PAIRS][PAIRS]; // Adjacency matrices for GNNs
vars gnnWeights[GNN_LAYERS][COMPONENTS][COMPONENTS]; // GNN weights
vars gnnOutputs[PAIRS][ACTIONS]; // GNN probabilities for Buy/Sell/Hold
vars similarityMatrix[PAIRS][PAIRS]; // Cross-Entropy similarity matrix
vars refinedOutputs[PAIRS][ACTIONS]; // Refined GNN probabilities
vars signals[PAIRS]; // Final trading signals
// Step 1: 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
pcaReducedFeatures[i][j] = dotProduct(kernelMatrix[i], eigenvectors[j]);
}
}
}
// Step 2: Initialize GNN Weights
function initializeGNNWeights() {
for (int l = 0; l < GNN_LAYERS; l++) {
for (int i = 0; i < COMPONENTS; i++) {
for (int j = 0; j < COMPONENTS; j++) {
gnnWeights[l][i][j] = random() * 0.1; // Small random initialization
}
}
}
}
// Step 3: GNN Propagation
function propagateGNN() {
vars tempFeatures[PAIRS][COMPONENTS];
for (int l = 0; l < GNN_LAYERS; l++) { // GNN propagation layers
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] * pcaReducedFeatures[j][m] * gnnWeights[l][m][k];
}
}
tempFeatures[i][k] = max(0, tempFeatures[i][k]); // ReLU activation
}
}
// Update PCA features for the next layer
for (int i = 0; i < PAIRS; i++) {
for (int k = 0; k < COMPONENTS; k++) {
pcaReducedFeatures[i][k] = tempFeatures[i][k];
}
}
}
// Generate probabilities (Buy/Sell/Hold) based on final GNN outputs
for (int i = 0; i < PAIRS; i++) {
for (int k = 0; k < ACTIONS; k++) {
gnnOutputs[i][k] = random() * 0.1; // Placeholder for GNN probability outputs
}
}
}
// Step 4: Compute Cross-Entropy Similarity
function computeCrossEntropySimilarity() {
for (int i = 0; i < PAIRS; i++) {
for (int j = 0; j < PAIRS; j++) {
similarityMatrix[i][j] = 0;
for (int k = 0; k < ACTIONS; k++) {
similarityMatrix[i][j] -= gnnOutputs[i][k] * log(gnnOutputs[j][k] + 1e-8);
}
}
}
}
// Step 5: Refine GNN Outputs Using Similarity
function refineGNNOutputs() {
for (int i = 0; i < PAIRS; i++) {
for (int k = 0; k < ACTIONS; k++) {
refinedOutputs[i][k] = 0;
double weightSum = 0;
for (int j = 0; j < PAIRS; j++) {
refinedOutputs[i][k] += similarityMatrix[i][j] * gnnOutputs[j][k];
weightSum += similarityMatrix[i][j];
}
refinedOutputs[i][k] /= (weightSum + 1e-8); // Normalize
}
}
}
// Step 6: Generate Trading Signals
function generateSignals() {
for (int i = 0; i < PAIRS; i++) {
signals[i] = refinedOutputs[i][0] - refinedOutputs[i][1]; // Buy-Sell difference
}
}
// Step 7: Execute Trades
function executeTrades() {
for (int i = 0; i < PAIRS; i++) {
if (signals[i] > 0) enterLong(CurrencyPairs[i]);
else if (signals[i] < 0) enterShort(CurrencyPairs[i]);
}
}
// Main Function
function run() {
set(PLOTNOW);
// Step 1: Perform Kernel PCA
performKernelPCA();
// Step 2: Initialize GNN weights
initializeGNNWeights();
// Step 3: Propagate GNN
propagateGNN();
// Step 4: Compute Cross-Entropy Similarity
computeCrossEntropySimilarity();
// Step 5: Refine GNN outputs
refineGNNOutputs();
// Step 6: Generate trading signals
generateSignals();
// Step 7: Execute trades
executeTrades();
}
