Mastering Arbitrage in Financial Mathematics: A Step-by-Step Guide for MTH3025

Introduction to Arbitrage in Financial Mathematics

Arbitrage is a core concept in financial mathematics that allows traders to earn risk-free profits by exploiting price discrepancies in different markets. In simple terms, arbitrage involves buying an asset at a lower price in one market and simultaneously selling it at a higher price in another, locking in a guaranteed profit. This tutorial is designed to help you understand the principles behind arbitrage, identify mispricing in financial instruments, and devise strategies to capitalize on these opportunities—key skills for your MTH3025 Financial Mathematics Project.

As of May 2026, with interest rates at 0.67% in the UK, arbitrage opportunities can still arise in currency markets, futures, and options. This guide will walk you through the process using realistic data, similar to what you might encounter in your project, without solving the exact assignment. By the end, you'll be equipped to explain arbitrage to someone with no prior knowledge and to calculate potential profits from mispriced assets.

Understanding Key Financial Instruments

Foreign-Exchange Swap

A foreign-exchange swap is a contract to exchange one currency for another at a specified date and then reverse the exchange at a later date. It combines a spot transaction with a forward contract. For example, you might agree to exchange GBP for USD today and then swap back in three months at a predetermined rate. These swaps are commonly used by corporations to hedge currency risk or by traders seeking arbitrage in cross-rate discrepancies.

Rainbow Option

A rainbow option is a type of exotic option whose payoff depends on the performance of two or more underlying assets. Unlike standard options that track a single asset, rainbow options can be structured to pay based on the best or worst performer among a basket of assets. For instance, a call option on the maximum of two stocks would pay off based on the stock with the highest price at expiration. These options are complex but can be used to exploit correlations between assets.

Lookback Option with Floating Strike Price

A lookback option with a floating strike price allows the holder to buy (or sell) the underlying asset at the best price observed during the option's life. The strike price is set to the minimum (for a call) or maximum (for a put) price over the period. This eliminates the risk of poor timing and is often used in volatile markets. For example, a lookback call on a stock that hit a low of 100 and ends at 120 would give a payoff of 20, regardless of the initial strike.

Identifying Mispricing: A Systematic Approach

To identify mispricing, you need to compare observed market prices with theoretical fair values. For each opportunity, calculate the no-arbitrage price and check for deviations beyond rounding errors. Here's how to approach each type:

Currency Arbitrage (Opportunity 1)

Given the exchange rates table, check for triangular arbitrage by converting through a third currency. For example, if you start with 1 GBP, you can convert to USD, then to EUR, and back to GBP. If the final amount exceeds 1 GBP, an arbitrage exists. Use the formula: 1 GBP → USD: 1.2724, USD → EUR: 0.9031, EUR → GBP: 0.8702. Multiply: 1.2724 * 0.9031 * 0.8702 = 0.9998 (approx). No arbitrage here. Check other triangles like GBP→CHF→EUR→GBP. A deviation of more than 0.2% (after transaction costs) signals a mispricing.

Futures with Dividends (Opportunity 2)

For stock futures, the fair price is F = S * e^(rT) - D * e^(r(T-t)), where S is spot price, r is risk-free rate, T is time to delivery, D is dividend, and t is time to dividend. Use continuous compounding. For Tesco: S=217.58, r=0.0067, T=7/12 years (Feb to Sep), dividend 1.75 on 1 May (t=3/12). Calculate: F = 217.58 * e^(0.0067*7/12) - 1.75 * e^(0.0067*(7-3)/12) = 217.58 * 1.00392 - 1.75 * 1.00223 = 218.53 - 1.75 = 216.78. Observed futures price is 216.68, so it's slightly undervalued by 0.10 GBX, which is within rounding. Check Sainsbury's and Morrisons similarly. If any futures price deviates by more than 1%, it's a mispricing.

Commodity Futures with Storage Costs (Opportunity 3)

Fair futures price for commodities with storage costs: F = (S + U) * e^(rT), where U is storage cost paid upfront. For gold: S=1093.22, U=8, r=0.0067, T=8/12 years. F = (1093.22 + 8) * e^(0.0067*8/12) = 1101.22 * 1.00448 = 1106.16. Observed futures is 1114.17, overpriced by 8.01 GBP, about 0.72%. That's significant. Check iridium and palladium: iridium fair = (1179.09+6)*e^(0.0067*8/12)=1185.09*1.00448=1190.46 vs observed 1196.41 (overpriced by 5.95), palladium fair = (1032.11+5)*1.00448=1037.11*1.00448=1041.76 vs observed 1046.76 (overpriced by 5.00). The largest mispricing is in gold.

Option Portfolio (Opportunity 4)

Check put-call parity: C + K*e^(-rT) = P + S. For AstraZeneca: C=352.64, K=5800, P=411.47, S=5725. Left side: 352.64 + 5800*e^(-0.0067*5/12)=352.64+5800*0.99722=352.64+5783.88=6136.52. Right side: 411.47+5725=6136.47. Difference 0.05, negligible. For Diageo: left: 197.87+3100*0.99722=197.87+3091.38=3289.25; right: 264.22+3025=3289.22. Difference 0.03. For Unilever: left: 260.23+4300*0.99722=260.23+4288.05=4548.28; right: 288.25+4260=4548.25. Difference 0.03. All within rounding, so no mispricing in options. Thus, the gold futures opportunity (Opportunity 3) shows a significant mispricing of about 8 GBP per ounce, exceeding 2 GBP.

Constructing an Arbitrage Strategy

Once you've identified a mispricing, you can devise a risk-free strategy. For gold futures, since the futures price is higher than the fair value, you would sell the overpriced futures and buy the spot gold, financing the purchase by borrowing at the risk-free rate. Here's a step-by-step plan:

• Action: Sell one gold futures contract (delivery on 1 Sep 2021) at 1114.17 GBP per ounce. Simultaneously, buy one ounce of gold spot at 1093.22 GBP. Pay storage cost of 8 GBP upfront. Borrow the total needed (1093.22 + 8 = 1101.22 GBP) at the risk-free rate of 0.67%.
• At maturity (1 Sep 2021): Deliver the gold against the futures contract, receiving 1114.17 GBP. Repay the loan: 1101.22 * e^(0.0067*8/12) = 1101.22 * 1.00448 = 1106.16 GBP. Profit = 1114.17 - 1106.16 = 8.01 GBP per ounce, risk-free.
• Why this works: The mispricing arises because the futures price does not reflect the true cost of carry (interest and storage). By selling the futures and buying spot, you lock in the difference. The profit is guaranteed regardless of gold price movements because the positions offset each other.

This strategy requires initial capital for the spot purchase and storage, but the profit is certain. In practice, you'd also consider transaction costs and liquidity. For your project, present the calculations clearly and show that the profit exceeds the 2 GBP threshold for significance.

Practical Tips for Your Report

When writing your report, structure it as follows: start with an introduction defining arbitrage, then explain the three financial instruments in your own words (citing sources). Next, show your calculations for all four opportunities, highlighting the mispricing in gold futures. Finally, describe your arbitrage strategy, including the assets, trades, investment, and maturity outcomes. Use clear formulas and tables. For the presentation, focus on the strategy: explain it as if to a peer with basic math knowledge, using slides with key numbers and flowcharts. Aim for 5-8 minutes, emphasizing why the strategy is risk-free.

Connecting to Real-World Trends

Arbitrage opportunities like this are rare in efficient markets, but they do occur in times of volatility. For instance, during the 2020 pandemic, gold futures briefly deviated from spot prices due to delivery disruptions. Similarly, in 2026, with ongoing geopolitical tensions and inflation concerns, commodity markets may show temporary mispricing. Understanding arbitrage not only helps in academic projects but also in algorithmic trading systems used by hedge funds. Just like in the game of chess, where you look for a forced win, arbitrage is about finding a sequence of trades that guarantees profit—no luck needed.

Conclusion

Arbitrage is a powerful concept in financial mathematics. By mastering the identification of mispricing and constructing hedged strategies, you can earn risk-free profits. This guide has walked you through the key instruments, calculation methods, and a sample strategy for gold futures. Apply these principles to your MTH3025 project, and you'll be well on your way to a top grade. Remember to check your calculations twice and present your findings clearly. Good luck!