Detect Arbitrage in Currency Exchange with Bellman-Ford Algorithm

How to Identify Arbitrage Opportunities in Currency Markets with Efficient Graph-Based Solutions

Problem

Suppose you are given a table of currency exchange rates, represented as a 2D array. Determine whether there is a possible arbitrage: that is, whether there is some sequence of trades you can make, starting with some amount A of any currency, so that you can end up with some amount greater than A of that currency. There are no transaction costs and you can trade fractional quantities.

Problem Explanation

You are given a 2D array representing currency exchange rates between different currencies. Each element of the array, rates[i][j], represents the exchange rate from currency i to currency j. The goal is to determine if there is an arbitrage opportunity, meaning if you can start with an amount of currency, perform a series of trades, and end up with more of that currency than you started with.

Approach

This problem can be solved using graph theory, specifically Bellman-Ford algorithm. Here's how it works:

Graph Representation:

• Treat each currency as a node in a graph.

• Each exchange rate between two currencies is treated as a directed edge between two nodes.

Edge Weights Transformation:

• The exchange rates are multiplicative. To use Bellman-Ford (which operates on sums), we need to transform the weights.
• Use logarithms to convert the problem to an additive form:

weight(i,j) = −log⁡(rate[i][j])

By using logarithms and negating, we turn the problem into detecting a negative weight cycle in a graph. A negative weight cycle corresponds to an arbitrage opportunity.

Cycle Detection:

If we find a negative weight cycle in this transformed graph, it means there is an arbitrage opportunity. The Bellman-Ford algorithm can detect negative weight cycles in a graph.

Solution Code:

function detectArbitrage(rates: number[][]): boolean {
    const n = rates.length;

    // Convert rates to log space: log(-rate)
    const logRates: number[][] = Array.from({ length: n }, () => Array(n).fill(0));

    for (let i = 0; i < n; i++) {
        for (let j = 0; j < n; j++) {
            logRates[i][j] = -Math.log(rates[i][j]);
        }
    }

    // Run Bellman-Ford to detect negative cycle
    const distances = Array(n).fill(Infinity);
    
    // We can start from any node, here we use node 0
    distances[0] = 0;

    for (let k = 0; k < n - 1; k++) {
        for (let i = 0; i < n; i++) {
            for (let j = 0; j < n; j++) {
                if (distances[i] + logRates[i][j] < distances[j]) {
                    distances[j] = distances[i] + logRates[i][j];
                }
            }
        }
    }

    // Check for negative weight cycles
    for (let i = 0; i < n; i++) {
        for (let j = 0; j < n; j++) {
            if (distances[i] + logRates[i][j] < distances[j]) {
                return true; // Arbitrage detected
            }
        }
    }

    return false; // No arbitrage found
}

Explanation of Code

Convert Rates to Logarithmic Space:

We first convert the rates to a logarithmic space using -Math.log(rates[i][j]), which allows us to transform the problem from a multiplicative relationship into an additive one.

Bellman-Ford Algorithm:

We initialize the distances from the start node (chosen arbitrarily as node 0) to all other nodes as infinity (Infinity), except for the start node itself which is 0. The Bellman-Ford algorithm then runs for n-1 iterations, where n is the number of currencies. In each iteration, we relax all edges and update the distances.

Negative Cycle Detection:

After n-1 iterations, we perform one more iteration to check if we can still relax any edge. If so, it indicates the presence of a negative weight cycle, which corresponds to an arbitrage opportunity.

Time Complexity

• Time Complexity: O(n^3) where n is the number of currencies. This is because we perform n-1 iterations over all pairs of nodes, and for each pair, we update the distances.
• Space Complexity: O(n^2) because we store the logRates matrix and the distance array.

Conclusion

The above solution detects whether an arbitrage opportunity exists using the Bellman-Ford algorithm. If a negative weight cycle is found, it indicates the presence of arbitrage. The transformation to logarithmic space allows us to handle the multiplicative nature of exchange rates.

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