Arbitrage is the practice of capitalising on price discrepancies of identical or similar financial instruments across different markets serves as a fundamental mechanism in maintaining financial market efficiency. The No-Arbitrage Condition, which states that in efficient markets, such pricing discrepancies shouldn't exist since arbitrageurs would quickly rectify them, is at the heart of this idea. This article explains the no-arbitrage criterion, looks at real-world instances to show these ideas and investigates the mathematics underlying arbitrage.
Understanding Arbitrage and the No-Arbitrage Condition
Arbitrage is the practice of simultaneously buying and selling an item in two or more marketplaces in order to take advantage of price differences in the financial markets.According to the No-Arbitrage Condition, such pricing disparities shouldn't exist in an efficient market since arbitrageurs would swiftly remedy them. Accurate asset pricing and the creation of financial models depend on this idea.
Mathematical Foundation: The Fundamental Theorem of Asset Pricing
The no-arbitrage criteria has a formal mathematical foundation according to the Fundamental Theorem of Asset Pricing. It states that if and only if there is at least one risk-neutral probability measure under which the discounted price process of assets acts as a martingale, then a market is free of arbitrage opportunities. This indicates that, using this risk-neutral metric, the present value of an asset is equal to its anticipated future value, adjusted for the time value of money.
The Geometric Brownian Motion (GBM) technique is used to estimate the risk-neutral measure and show many simulated price paths for AAPL shares over a one-year period. Assuming that stock prices follow a continuous stochastic process in which the predicted growth rate coincides with the risk-free rate rather than the actual market return, each line depicts a possible trajectory. The model is crucial for pricing options and derivatives because of its risk-neutral assumption, which guarantees that price movements stay independent of investor risk preferences.
Triangular Arbitrage in Currency Markets
Triangular arbitrage takes advantage of differences in three different currencies. For example, an arbitrageur can convert USD to EUR, EUR to GBP, and then GBP back to USD, potentially making a risk-free profit, provided the exchange rates between USD/EUR, EUR/GBP, and GBP/USD are inconsistent.
E.G. an arbitrage opportunity exists in the charts below specifically a significant opportunity exists in May 2021
May 2021: The USD/EUR exchange rate was around 0.82–0.84.
May 2021: The EUR/GBP exchange rate was approximately 0.86–0.87.
May 2021: The GBP/USD exchange rate was around 1.41–1.42.
USD to EUR = 0.83
EUR to GBP = 0.87
GBP to USD = 1.42
Now, to calculate the implied rate:
0.83×0.87×1.42=1.025
Since this value is greater than 1, it suggests a potential arbitrage opportunity.
If this deviation persisted long enough, traders could profit by:
Converting USD to EUR at 0.83.
Converting EUR to GBP at 0.87.
Converting GBP back to USD at 1.42.
1 USD > 0.83 EUR
0.83×0.87=0.7221 GBP
0.7221×1.42=1.025 USD
This results in a 2.5% arbitrage profit(roughly).
Interest Rate Parity Arbitrage
Interest rate parity ensures that the difference in interest rates between two countries is equal to the differential between the forward exchange rate and the spot exchange rate.If this rule is broken, arbitrageurs can take advantage of the difference by using covered interest arbitrage, which involves borrowing money in a currency with a lower interest rate and investing in a currency with a higher interest rate, hedged by forward contracts.
The Covered Interest Rate Parity (CIP) equation states that the forward exchange rate should equal the spot exchange rate multiplied by the ratio of (1 + the foreign interest rate) to (1 + the domestic interest rate), ensuring no arbitrage opportunities exist.
This can also be measured using python. Below i have created a simulated Interest Rate parity scatter plot which could be used to measure arbitrage.
Plotting real market forward rates versus theoretical projections shows that there are no arbitrage opportunities when the market rates match the theoretical assumptions, as indicated by points that align with the red line. Significant departures from this line point to possible situations involving arbitrage. In particular, arbitrageurs can earn by entering forward contracts while leveraging different interest rates if the real forward rate is higher than the theoretical rate, which suggests the forward currency is overpriced. On the other hand, arbitrageurs can profit from shorting the currency and collecting gains through forward contracts if the real forward rate is lower than the theoretical rate, which suggests the forward currency is undervalued.
Traders can use Covered Interest Arbitrage (CIA) to take advantage of differences between theoretical forecasts and real market future rates. This tactic entails borrowing money in a currency with a lower interest rate for example, euros if rates in Europe are lower than those in the US and then converting it into a currency with a higher interest rate.
Commodity Futures Arbitrage: The Goldman Roll
The monthly rolling over of commodity futures contracts in the Goldman Sachs Commodity Index (S&P-GSCI) is known as the "Goldman Roll." With a Sharpe ratio as high as 4.4 between 2000 and 2010, this roll yield both generates and necessitates statistically significant arbitrage possibilities.
The fundamental equation for futures arbitrage, ensuring no arbitrage opportunity, is F = S(1 + r - y)t, where F is the futures price, S is the spot price, r is the risk-free interest rate, y is the dividend yield (or cost of carry), and t is the time to maturity.
Case Study: Arbitrage in Prediction Markets
When implied probabilities in several markets are inconsistent, there may be chances for arbitrage in prediction markets, which are platforms where users wager on the results of future events.For example, if one market predicts a 60% chance of an event occurring while another predicts 70%, an arbitrageur could bet accordingly to secure a risk-free profit. However, such opportunities highlight inefficiencies within these markets and can undermine their reliability.
Conclusion
The no-arbitrage condition, which encapsulates the mathematics of arbitrage, is essential to the efficiency and integrity of financial markets. Market players support price correction and market equilibrium by comprehending and spotting arbitrage opportunities. Real-world examples show how these concepts are applied in practice, but they also highlight how crucial complex mathematical models are to modern finance.
Reference List:
Delbaen, F., & Schachermayer, W. (2006). The Mathematics of Arbitrage. Springer Science & Business Media.
Hull, J. C. (2018). Options, Futures, and Other Derivatives (10th ed.). Pearson.
Investopedia. (n.d.). Arbitrage. Retrieved from https://www.investopedia.com/terms/a/arbitrage.asp
Wikipedia contributors. (2023, August 15). Fundamental theorem of asset pricing. In Wikipedia, The Free Encyclopedia. Retrieved from https://en.wikipedia.org/wiki/Fundamental_theorem_of_asset_pricing
Wikipedia contributors. (2023, July 10). Triangular arbitrage. In Wikipedia, The Free Encyclopedia. Retrieved from https://en.wikipedia.org/wiki/Triangular_arbitrage
Wikipedia contributors. (2023, July 5). Interest rate parity. In Wikipedia, The Free Encyclopedia. Retrieved from https://en.wikipedia.org/wiki/Interest_rate_parity
O'Connor, N. (2024, September 15). Arbitrage opportunity is bane of prediction markets. Financial Times. Retrieved from https://www.ft.com/content/3c9f2d7d-4b02-449f-b4d1-292d356dc21e
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