Cross Rates, Currency Arbitrage, and Settlement Systems

1. Cross Rates: Meaning, Necessity, and Mechanics

A Cross Rate is an exchange rate between two non-standard currencies calculated through their respective exchange rates against a common third vehicle currency (almost always the US Dollar).

Why Are Cross Rates Necessary?

There are over 180 sovereign currencies in the world, creating over 16,000 potential direct currency pairs. It is impossible and illiquid for commercial banks to maintain active, continuous trading desks for every single bilateral pair (e.g., Thai Baht against Indian Rupee, or Danish Krone against Chilean Peso).

Instead, banks trade almost all national currencies against the highly liquid US Dollar (USD). Any non-dollar currency pair is derived mathematically through the Chain Rule.

2. Mathematical Calculation of Cross Rates

Case A: Single Mid-Rate Calculation

If we are given:

• USD / INR = 84.00 (meaning 1 USD = 84.00 INR)
• EUR / USD = 1.0800 (meaning 1 EUR = 1.0800 USD)

To calculate the cross rate for EUR / INR:

EUR / INR = (EUR / USD) × (USD / INR) = 1.0800 × 84.00 = 90.72 INR per EUR

Case B: Two-Way Bid-Ask Cross Rates (The Chain Rule)

When calculating cross rates with two-way Bid and Ask spreads, apply the golden banking rule:
The derived Bid must reflect the cheapest purchase price, and the derived Ask must reflect the most expensive selling price.

Derived Bid: (EUR / INR) Bid = (EUR / USD) Bid × (USD / INR) Bid

Derived Ask: (EUR / INR) Ask = (EUR / USD) Ask × (USD / INR) Ask

Examination Numerical Example:

Given the following interbank quotes:

• EUR / USD = 1.0820 / 1.0850
• USD / INR = 84.10 / 84.30

Calculate the cross rate quote for EUR / INR:

• Step 1: Calculate EUR/INR Bid:

• (EUR / INR) Bid = 1.0820 × 84.10 = 90.9962 INR

• Step 2: Calculate EUR/INR Ask:

• (EUR / INR) Ask = 1.0850 × 84.30 = 91.4655 INR

• Final Quotation:

  Derived Cross Quote (EUR / INR): 90.9962 / 91.4655

3. Currency Arbitrage: Foundations & Types

Arbitrage is the simultaneous purchase and sale of an identical financial asset in different markets to exploit pricing discrepancies and earn a guaranteed, riskless profit without capital commitment.

In the foreign exchange market, high-speed electronic algorithmic trading ensures that arbitrage opportunities are fleeting, lasting only fractions of a second before the trading volume pushes exchange rates back to equilibrium.

4. Two-Point (Geographic) Arbitrage with Numerical Example

Suppose the US Dollar is quoted differently in London and New York at the exact same instant:

• In London: 1 GBP = USD 1.3000
• In New York: 1 GBP = USD 1.3200

Arbitrage Strategy:

1. Identify the cheaper market: In London, 1 GBP costs only USD 1.3000. In New York, 1 GBP sells for USD 1.3200. The Pound is cheaper in London and dearer in New York.
2. Execute Simultaneous Trades:
   • Take USD 1,300,000 in London and buy British Pounds:
     • Pounds Purchased = USD 1,300,000 / 1.3000 = GBP 1,000,000
   • Simultaneously sell GBP 1,000,000 in New York at USD 1.3200:
     • USD Received = GBP 1,000,000 × 1.3200 = USD 1,320,000
3. Net Risk-Free Profit:

   Net Arbitrage Profit = USD 1,320,000 - USD 1,300,000 = USD 20,000 (Risk-Free Gain)

4. Market Correction: Buying Pounds in London pushes its price UP, while selling Pounds in New York pushes its price DOWN until both quotes converge identically.

5. Three-Point (Triangular) Arbitrage with Numerical Example

Triangular arbitrage occurs when the direct quote between two currencies does not align with their indirect cross rate calculated through a third currency.

Consider the following simultaneously available quotes in three financial centers:

1. In New York: EUR / USD = 1.1000 (1 EUR = 1.1000 USD)
2. In Frankfurt: EUR / GBP = 0.8800 (1 EUR = 0.8800 GBP)
3. In London: GBP / USD = 1.2000 (1 GBP = 1.2000 USD)

Step 1: Calculate the Theoretical Cross Rate

From the New York and London quotes, what should 1 EUR cost in terms of GBP?

Theoretical EUR / GBP = (EUR / USD) / (GBP / USD) = 1.1000 / 1.2000 = 0.9167 GBP

Step 2: Compare with the Actual Market Quote

• Actual Market Quote in Frankfurt: 1 EUR = 0.8800 GBP
• Theoretical Equilibrium Cross Rate: 1 EUR = 0.9167 GBP
• Because the actual market price (0.8800) is lower than the theoretical rate (0.9167), the Euro is undervalued against the Pound in Frankfurt, creating an arbitrage opportunity!

Step 3: Execute the Triangular Trade with USD 1,000,000

• Wait! In Leg 3, USD 960,000 < USD 1,000,000 (A Loss of USD 40,000).
• Rule: If circling one direction produces a loss, reverse the direction of the cycle immediately to earn the profit!

Corrected Reverse Triangular Cycle:

1. Leg 1 (Start in London: USD → GBP):
   Convert USD 1,000,000 to GBP at 1.2000:
   • GBP Received = USD 1,000,000 / 1.2000 = GBP 833,333.33
2. Leg 2 (Frankfurt: GBP → EUR):
   Convert GBP 833,333.33 to EUR at 0.8800:
   • EUR Received = GBP 833,333.33 / 0.8800 = EUR 946,969.70
3. Leg 3 (New York: EUR → USD):
   Convert EUR 946,969.70 to USD at 1.1000:
   • USD Received = EUR 946,969.70 × 1.1000 = USD 1,041,666.67
4. Net Risk-Free Profit:

   Net Triangular Profit = USD 1,041,666.67 - USD 1,000,000 = USD 41,666.67