The Weighted Dynamic Arbitrage Graph
You are given a directed weighted graph ðº=(ð‘‰,ð¸) where: 𑉠represents currency pairs (e.g., EURUSD, GBPUSD, etc.).
ð¸ represents the relationships between the currency pairs (e.g., price ratios, arbitrage opportunities).
Each edge ð‘’(ð‘–,ð‘—) between nodes ð‘£ð‘– and ð‘£ð‘— has a weight ð‘¤(ð‘–,ð‘—) representing the logarithm of the price ratio between
ð‘£ð‘– and ð‘£ð‘—.
Problem:
The graph ðº is time-dependent. At each time step ð‘¡, the edge weights ð‘¤(ð‘–,ð‘—,ð‘¡) change according to a function:
ð‘¤(ð‘–,ð‘—,ð‘¡) = log(ð‘ƒð‘–(ð‘¡)ð‘ƒð‘—(ð‘¡))−Δ(ð‘–,ð‘—,ð‘¡) where: ð‘ƒð‘–(ð‘¡) and ð‘ƒð‘—(ð‘¡) are the prices of ð‘£ð‘– and ð‘£ð‘— at time ð‘¡. Δ(ð‘–,ð‘—,ð‘¡) is a dynamic threshold, influenced by volatility.
A negative weight cycle in ðº represents an arbitrage opportunity, where the product of the weights along the cycle is greater than
1: âˆ(ð‘–,ð‘—)∈ð¶ð‘’ð‘¤(ð‘–,ð‘—,ð‘¡)>1.
Your goal is to: Identify all negative weight cycles in ðº for ð‘¡1 ≤ 𑡠≤ð‘¡2 .
For each negative cycle ð¶, calculate the maximum profit factor:
Profit(ð¶)=âˆ(ð‘–,ð‘—)∈ð¶ð‘’ð‘¤(ð‘–,ð‘—,ð‘¡).
Find the most profitable cycle across all time steps ð‘¡.
Additional Constraints:
The graph ðº has ∣ð‘‰âˆ£=28 nodes (one for each currency pair) and ∣ð¸âˆ£=∣ð‘‰âˆ£Ã—(∣ð‘‰âˆ£âˆ’1)
∣E∣=∣V∣×(∣V∣−1) edges (fully connected directed graph).
Edge weights ð‘¤(ð‘–,ð‘—,ð‘¡) vary dynamically with ð‘¡ according to: Δ(ð‘–,ð‘—,ð‘¡)=ðœŽð‘–(ð‘¡)â‹…ðœŽð‘—(ð‘¡), where
ðœŽð‘–(ð‘¡) is the standard deviation of prices for ð‘£ð‘– over a rolling window of 20 time steps.
You must use an efficient algorithm (e.g., Bellman-Ford for detecting negative cycles) to handle the graph's dynamic nature.
#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 CorrelationMatrix[PAIRS][PAIRS]; // Correlation adjacency matrix
vars ArbitrageMatrix[PAIRS][PAIRS]; // Arbitrage adjacency matrix
vars VolatilityMatrix[PAIRS]; // Volatility levels for each pair
vars DynamicThresholdMatrix[PAIRS][PAIRS]; // Dynamic arbitrage thresholds
vars RiskAdjustedWeights[PAIRS]; // Risk-adjusted portfolio weights
int StateMatrix[PAIRS][3][3]; // Transition matrix for each pair
vars CurrentDrawdown; // Current drawdown level
int LookBack = 200; // Lookback period for calculations
// Function to calculate correlation between two series
function calculateCorrelation(vars series1, vars series2, int len) {
var mean1 = SMA(series1, len);
var mean2 = SMA(series2, len);
var numerator = 0, denom1 = 0, denom2 = 0;
for (int i = 0; i < len; i++) {
numerator += (series1[i] - mean1) * (series2[i] - mean2);
denom1 += pow(series1[i] - mean1, 2);
denom2 += pow(series2[i] - mean2, 2);
}
return numerator / sqrt(denom1 * denom2);
}
// Initialize the correlation network
function initializeCorrelationNetwork() {
for (int i = 0; i < PAIRS; i++) {
for (int j = 0; j < PAIRS; j++) {
if (i != j) {
vars series1 = series(price(CurrencyPairs[i]));
vars series2 = series(price(CurrencyPairs[j]));
CorrelationMatrix[i][j] = calculateCorrelation(series1, series2, LookBack);
} else {
CorrelationMatrix[i][j] = 1; // Self-correlation
}
}
}
}
// Calculate arbitrage opportunities between pairs with dynamic thresholds
function calculateDynamicArbitrage() {
for (int i = 0; i < PAIRS; i++) {
for (int j = 0; j < PAIRS; j++) {
if (i != j) {
var priceRatio = log(price(CurrencyPairs[i]) / price(CurrencyPairs[j]));
var threshold = max(0.01, VolatilityMatrix[i] * 0.5); // Dynamic threshold
DynamicThresholdMatrix[i][j] = threshold;
ArbitrageMatrix[i][j] = (abs(priceRatio) > threshold) ? priceRatio : 0;
} else {
ArbitrageMatrix[i][j] = 0; // No arbitrage within the same pair
}
}
}
}
// Calculate volatility levels for each pair
function calculateVolatilityMatrix() {
for (int i = 0; i < PAIRS; i++) {
VolatilityMatrix[i] = StdDev(series(price(CurrencyPairs[i])), LookBack);
}
}
// Risk adjustment based on drawdown and portfolio composition
function adjustRiskWeights() {
var TotalWeight = 0;
for (int i = 0; i < PAIRS; i++) {
var riskFactor = max(0.1, 1 - CurrentDrawdown / 0.2); // Reduce risk if drawdown exceeds 20%
RiskAdjustedWeights[i] = (1 / VolatilityMatrix[i]) * riskFactor;
TotalWeight += RiskAdjustedWeights[i];
}
for (int i = 0; i < PAIRS; i++) {
RiskAdjustedWeights[i] /= TotalWeight; // Normalize weights
}
}
// Execute trades based on dynamic arbitrage and risk-adjusted weights
function executeDynamicTrades() {
for (int i = 0; i < PAIRS; i++) {
for (int j = 0; j < PAIRS; j++) {
if (ArbitrageMatrix[i][j] != 0) {
var WeightI = RiskAdjustedWeights[i];
var WeightJ = RiskAdjustedWeights[j];
if (ArbitrageMatrix[i][j] > 0) { // Long-Short arbitrage
enterLong(CurrencyPairs[i], WeightI);
enterShort(CurrencyPairs[j], WeightJ);
} else if (ArbitrageMatrix[i][j] < 0) { // Short-Long arbitrage
enterShort(CurrencyPairs[i], WeightI);
enterLong(CurrencyPairs[j], WeightJ);
}
}
}
}
}
// Track drawdown levels
function monitorDrawdown() {
CurrentDrawdown = max(0, 1 - (Equity / MaxEquity));
if (CurrentDrawdown > 0.2) { // Emergency shutdown at 20% drawdown
exitLong();
exitShort();
printf("Emergency risk controls triggered. All positions closed.");
}
}
// Main trading function
function run() {
set(PLOTNOW);
// Update and calculate all matrices
initializeCorrelationNetwork();
calculateVolatilityMatrix();
calculateDynamicArbitrage();
adjustRiskWeights();
monitorDrawdown();
// Execute trades based on advanced analysis
executeDynamicTrades();
}
