Arbitrage

Arbitrage is the practice of taking advantage of a difference in prices in two or more marketsstriking a combination of matching deals to capitalize on the difference, the profit being the difference between the market prices at which the unit is traded. Arbitrage has the effect of causing prices of the same or very similar assets in different markets to converge.
When used by academics in economics, an arbitrage is a transaction that involves no negative cash flow at any probabilistic or temporal state and a positive cash flow in at least one state; in simple terms, it is the possibility of a risk-free profit after transaction costs. For example, an arbitrage opportunity is present when there is the possibility to instantaneously buy something for a low price and sell it for a higher price.
In principle and in academic use, an arbitrage is risk-free; in common use, as in statistical arbitrage, it may refer to expected profit, though losses may occur, and in practice, there are always [|risks] in arbitrage, some minor, some major. In academic use, an arbitrage involves taking advantage of differences in price of a single asset or identical cash-flows; in common use, it is also used to refer to differences between similar assets, as in merger arbitrage.
The term is mainly applied in the financial field. People who engage in arbitrage are called arbitrageurs.

Etymology

"Arbitrage" is a French word and denotes a decision by an arbitrator or arbitration tribunal. It was first defined as a financial term in 1704 by French mathematician Mathieu de la Porte in his treatise "La science des négociants et teneurs de livres" as a consideration of different exchange rates to recognise the most profitable places of issuance and settlement for a bill of exchange

Arbitrage equilibrium

If the market prices do not allow for profitable arbitrage, the prices are said to constitute an arbitrage equilibrium, or an arbitrage-free market. An arbitrage equilibrium is a precondition for a general economic equilibrium. The 'no-arbitrage assumption' is used in quantitative finance to calculate a unique risk neutral price for derivatives.

Arbitrage-free pricing approach for bonds

Arbitrage-free pricing for bonds is the method of valuing a coupon-bearing financial instrument by discounting its future cash flows by multiple discount rates. By doing so, a more accurate price can be obtained than if the price is calculated with a present-value pricing approach. Arbitrage-free pricing is used for bond valuation and to detect arbitrage opportunities for investors.
For the purpose of valuing the price of a bond, its cash flows can each be thought of as packets of incremental cash flows with a large packet upon maturity, being the principal. Since the cash flows are dispersed throughout future periods, they must be discounted back to the present. In the present-value approach, the cash flows are discounted with one discount rate to find the price of the bond. In arbitrage-free pricing, multiple discount rates are used.
The present-value approach assumes that the bond yield will stay the same until maturity. This is a simplified model because interest rates may fluctuate in the future, which in turn affects the yield on the bond. For this reason, the discount rate may differ for each cash flow. Each cash flow can be considered a zero-coupon instrument that pays one payment upon maturity. The discount rates used should be the rates of multiple zero-coupon bonds with maturity dates the same as each cash flow and similar risk as the instrument being valued. By using multiple discount rates, the arbitrage-free price is the sum of the discounted cash flows. Arbitrage-free price refers to the price at which no price arbitrage is possible.
The idea of using multiple discount rates obtained from zero-coupon bonds and discounting a similar bond's cash flow to find its price is derived from the yield curve, which is a curve of the yields of the same bond with different maturities. This curve can be used to view trends in market expectations of how interest rates will move in the future. In arbitrage-free pricing of a bond, a yield curve of similar zero-coupon bonds with different maturities is created. If the curve were to be created with Treasury securities of different maturities, they would be stripped of their coupon payments through bootstrapping. This is to transform the bonds into zero-coupon bonds. The yield of these zero-coupon bonds would then be plotted on a diagram with time on the x-axis and yield on the y-axis.
Since the yield curve displays market expectations on how yields and interest rates may move, the arbitrage-free pricing approach is more realistic than using only one discount rate. Investors can use this approach to value bonds and find price mismatches, resulting in an arbitrage opportunity. If a bond valued with the arbitrage-free pricing approach turns out to be priced higher in the market, an investor could have such an opportunity:

Investor shorts the bond at price at time t₁.
Investor longs the zero-coupon bonds making up the related yield curve and strips and sells any coupon payments at t₁.
As t>t₁, the price spread between the prices will decrease.
At maturity, the prices will converge and be equal. Investor exits both the long and short positions, realising a profit.

If the outcome from the valuation were the reverse case, the opposite positions would be taken in the bonds. This arbitrage opportunity comes from the assumption that the prices of bonds with the same properties will converge upon maturity. This can be explained through market efficiency, which states that arbitrage opportunities will eventually be discovered and corrected. The prices of the bonds in t₁ move closer together to finally become the same at t_T.

Conditions for arbitrage

Arbitrage may take place when:

the same asset does not trade at the same price on all markets.
two assets with identical cash flows do not trade at the same price.
an asset with a known price in the future does not today trade at its future price discounted at the risk-free interest rate.

Arbitrage is not simply the act of buying a product in one market and selling it in another for a higher price at some later time. The transactions must occur simultaneously to avoid exposure to market risk, or the risk that prices may change in one market before both transactions are complete. In practical terms, this is generally possible only with securities and financial products that can be traded electronically, and even then, when each leg of the trade is executed, the prices in the market may have moved. Missing one of the legs of the trade is an 'execution risk' referred to as 'leg risk'.
In the simplest example, any good sold in one market should sell for the same price in another. Traders may, for example, find that the price of wheat is lower in agricultural regions than in cities, purchase the good, and transport it to another region to sell at a higher price. This type of price arbitrage is the most common, but this simple example ignores the cost of transport, storage, risk, and other factors. "True" arbitrage requires that there is no market risk involved. Where securities are traded on more than one exchange, arbitrage occurs by simultaneously buying in one and selling on the other.
See rational pricing, particularly § arbitrage mechanics, for further discussion.
Mathematically it is defined as follows:
where, denotes the portfolio value at time t and T is the time the portfolio ceases to be available on the market. This means that the value of the portfolio is never negative, and guaranteed to be positive at least once over its lifetime.
Negative, or anti-, arbitrage is similarly defined as
and occurs naturally in arbitrage relations as the seller view as opposed to the buyer view.

Price convergence

Arbitrage has the effect of causing prices, and thus purchasing power, in different markets to converge. As a result of arbitrage, the currency exchange rates and the prices of securities and other financial assets in different markets tend to converge. The speed at which they do so is a measure of market efficiency. Arbitrage tends to reduce price discrimination by encouraging people to buy an item where the price is low and resell it where the price is high.
Arbitrage moves different currencies toward purchasing power parity. Assume that a car purchased in the United States is cheaper than the same car in Canada. Canadians would buy their cars across the border to exploit the arbitrage condition. At the same time, Americans would buy US cars, transport them across the border, then sell them in Canada. Canadians would have to buy American dollars to buy the cars and Americans would have to sell the Canadian dollars they received in exchange. Both actions would increase demand for US dollars and supply of Canadian dollars. As a result, there would be an appreciation of the US currency. This would make US cars more expensive and Canadian cars less so until their prices were similar. On a larger scale, international arbitrage opportunities in commodities, goods, securities, and currencies tend to change exchange rates until the purchasing power is equal.
In reality, most assets exhibit some difference between countries. These, transaction costs, taxes, and other costs provide an impediment to this kind of arbitrage. Similarly, arbitrage affects the difference in interest rates paid on government bonds issued by the various countries, given the expected depreciation in the currencies relative to each other.

Risks

Arbitrage transactions in modern securities markets involve fairly low day-to-day risks, but can face extremely high risk in rare situations, particularly financial crises, and can lead to bankruptcy. Formally, arbitrage transactions have negative skew – prices can get a small amount closer, while they can get very far apart. The day-to-day risks are generally small because the transactions involve small differences in price, so an execution failure will generally cause a small loss. The rare case risks are extremely high because these small price differences are converted to large profits via leverage, and in the rare event of a large price move, this may yield a large loss.
The principal risk, which is typically encountered on a routine basis, is classified as execution risk. This transpires when an aspect of the financial transaction does not materialize as anticipated. Infrequent, albeit critical, risks encompass counterparty and liquidity risks. The former, counterparty risk, is characterized by the failure of the other participant in a substantial transaction, or a series of transactions, to fulfill their financial obligations. Liquidity risk, conversely, emerges when an entity is necessitated to allocate additional monetary resources as margin, but encounters a deficit in the required capital.
In the academic literature, the idea that seemingly very low-risk arbitrage trades might not be fully exploited because of these risk factors and other considerations is often referred to as limits to arbitrage.