Consider the following Puzzle :
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:
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,ð‘]
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:
Kᵢⱼ(t) = exp(-((Vᵢ(t) - Vⱼ(t))²) / (2 ⋅ σ²))
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:
λₖ'(t) = λₖ(t) ⋅ (1 + ηₖ(t))
Where:
ðœ‚ð‘˜(ð‘¡) is Gaussian noise with variance proportional to the eigenvalue magnitude:
ðœ‚ð‘˜(ð‘¡)∼ð’©(0,ðœ†ð‘˜(ð‘¡)â‹…ðœˆ)​.
𜈠is a tunable noise coefficient.
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:
Ráµ¢(t) = ∑ₖ₌â‚áµ (⟨Káµ¢(t), vₖ⟩ â‹… vâ‚–)
Where
ð‘£ð‘˜ is the ð‘˜-th eigenvector.
Dynamic Threshold for Execution: Trades are executed when:
Signal(Cáµ¢) = Ráµ¢(t) / √(∑ⱼ₌â‚â¿ Râ±¼(t)²) > Θ(t)
The threshold Θ(ð‘¡) evolves dynamically:
Θ(t) = ð”¼[R(t)] + δ â‹… StdDev(R(t))
Where:
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.
#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);
}
