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Question 1164876: A dealer has PKR 100 million and receives the following rates of foreign exchange from the market. Find out how dealer can earn maximum arbitrage profit in FX market transaction. Solve and explain step by step.
USD/PKR 138/139
GBP/PKR 199/201
GBP/USD 1.2729/35
EUR/GBP 1.1520/33
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website! To determine if an arbitrage opportunity exists, we need to compare the **direct rates** (provided by the market) with the **cross rates** (calculated using other currency pairs). Arbitrage is possible if the cross rate is significantly different from the market rate, allowing us to "buy low" in one market and "sell high" in another.
Since we are starting with **PKR 100 million**, we are looking for a **Triangular Arbitrage** path.
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### Step 1: Analyze the Market Rates (Bid/Ask)
The rates are given as **Bid / Ask**.
* **Bid:** The rate at which the bank buys the base currency (left side).
* **Ask:** The rate at which the bank sells the base currency (left side).
| Pair | Bid (Bank Buys) | Ask (Bank Sells) |
| --- | --- | --- |
| **USD/PKR** | 138.00 | 139.00 |
| **GBP/PKR** | 199.00 | 201.00 |
| **GBP/USD** | 1.2729 | 1.2735 |
---
### Step 2: Calculate the Implied Cross Rate
Let's see what the GBP/PKR rate *should* be based on the USD/PKR and GBP/USD rates.
**Comparison:**
* The market **GBP/PKR Ask** is **201.00**.
* Our calculated **Implied Ask** is **177.0165**.
* The market **GBP/PKR Bid** is **199.00**.
Because the market's Bid for GBP (199.00) is much higher than the cost to "create" a GBP through USD (177.0165), an arbitrage opportunity exists. We should buy GBP using the USD route and sell it directly for PKR.
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### Step 3: Execute the Arbitrage Path
The goal is to go: **PKR USD GBP PKR**.
**1. Convert PKR to USD:**
We must buy USD from the bank at the **Ask** rate (139.00).
**2. Convert USD to GBP:**
We need to buy GBP. The rate GBP/USD is 1.2729/35. The bank sells GBP at the **Ask** (1.2735).
**3. Convert GBP back to PKR:**
We sell our GBP to the market at the **GBP/PKR Bid** rate (199.00).
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### Step 4: Calculate the Profit
* **Ending PKR:** 112,418,898.91
* **Starting PKR:** 100,000,000.00
* **Total Profit:** **PKR 12,418,898.91**
The dealer earns a profit of approximately **12.42%** in one cycle.
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### Summary Table of Transactions
| Action | Currency Pair used | Rate | Result |
| --- | --- | --- | --- |
| **Start** | PKR | — | 100,000,000.00 |
| **Sell PKR for USD** | USD/PKR (Ask) | 139.00 | 719,424.46 |
| **Sell USD for GBP** | GBP/USD (Ask) | 1.2735 | 564,919.09 |
| **Sell GBP for PKR** | GBP/PKR (Bid) | 199.00 | 112,418,898.91 |
**Note on EUR/GBP:** The rate EUR/GBP 1.1520/33 was provided but not needed for this specific triangular loop. It would only be used if we were attempting a four-way arbitrage involving the Euro.
Would you like me to check if adding the Euro into the cycle (PKR USD GBP EUR PKR) would yield a higher profit?
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