FX Currency Arbitrage Detection via Bellman-Ford

Foreign exchange (FX) market makers continually scan inter-bank currency quotes for cross-rate pricing inefficiencies. You are given a list of $V$ currency names and a $V \times V$ matrix rates where rates[i][j] specifies the conversion rate from currency $i$ to currency $j$ (1 unit of currency $i$ yields rates[i][j] units of currency $j$).

An arbitrage opportunity exists if there is a cycle of conversions $c_0 \to c_1 \to \dots \to c_k \to c_0$ such that the compounding product of exchange rates strictly exceeds $1.0$: $\prod_{m=0}^{k-1} rates[c_m][c_{m+1}] \cdot rates[c_k][c_0] > 1.0$

Your Task

Implement has_arbitrage_opportunity(rates: list[list[float]]) -> bool that detects whether any currency arbitrage cycle exists.

Constraints

• $2 \le V \le 100$
• $rates[i][j] > 0.0$
• $rates[i][i] == 1.0$

Examples

rates = [
    [1.0, 0.85, 0.75],
    [1.18, 1.0, 0.88],
    [1.34, 1.14, 1.0]
]
# 0 -> 1 -> 2 -> 0: 0.85 * 0.88 * 1.34 = 1.00232 > 1.0 (arbitrage!)
# Output: True