Arbitrage is a process through which an investor can buy an asset or combination of assets at one price and concurrently sell at a higher price, thereby earning a profit without investing any money or being exposed to any risk.

In a well-functioning market, arbitrage opportunities should not exist. If they do exist, arbitrage activities would quickly eliminate the price differential. The no-arbitrage principle states that any rational price for a financial instrument must exclude arbitrage opportunities. This is the minimal requirement for a feasible or rational price for any financial instrument. There is no free money.

The role of arbitrage:

• It facilitates the determination of prices. The combined actions of many investors engaging in arbitrage result in rapid price adjustments that eliminate any arbitrage opportunities, thereby bringing prices back.

• It promotes market efficiency. Efficient markets are those in which it is impossible to earn abnormal returns, which are returns that are in excess of the return required for the risk assumed. Arbitrage activities will quickly eliminate arbitrage opportunities available in the market, thereby promoting market efficiency.

Arbitrage means taking advantage of price differences in different markets. In well-functioning markets, arbitrage opportunities are quickly exploited, and the resulting increased buying of underpriced assets and increased selling of overpriced assets return prices to equivalence.

Arbitrage and Derivatives

Assume the risk-free rate is 5%. The current price of gold is $300 per ounce and the forward price of gold is $330 in one year's time. Is there an arbitrage opportunity?

Here is what you can do:

• Borrow $300 at 5% today.
• Buy one ounce of gold (price $300).
• Enter into a short forward to sell one ounce of gold for $330 in one year's time.
• After one year you sell the gold for $330, and repay the bank $300 plus $15 interest.

Hence, a profit of $15 can be made without any risk!

In fact, any delivery price above $315 will result in a risk-free profit using this strategy.

What if the delivery price is $310?

• Sell one ounce of gold for $300.
• Deposit the $300 in the bank at 5% interest.
• Enter into a forward to buy one ounce of gold in one year's time for the delivery price ($310).
• After one year, buy one ounce of gold for $310 and keep the $5 profit.

Again, a profit of $5 can be made without any risk.

Investors in the gold market will take advantage of any forward price that is not equal to $315, eventually bring the price to $315, which is known as the arbitrage-free price.

The arbitrage principle is the essence of derivative pricing models.

Arbitrage and Replication

A portfolio composed of the underlying asset and the riskless asset could be constructed to have exactly the same cash flows as a derivative. This portfolio is called the replicating portfolio. Since they have the same cash flows, they would have to sell at the same price (the law of one price).

Assume the forward price of gold is $315 in one year's time, and the spot price is $300. You have $300.

• You can deposit $300 in the bank at 5% interest. One year later you will get $315.
• You can also buy one ounce of gold, and a forward contract to sell it in one year for $315. One year later you will also get $315.

Why replicate?

• To explore pricing differentials
• Lower transaction costs

Replication is the essence of arbitrage.

Replication generally has more costs than simply taking a position in the asset. So, if all the assets are priced correctly, the replication seems useless. However, the ability to replicate something with something else can be valuable to investors as it helps them in making arbitrage profits. Sometimes, replication can even have lower transaction costs. For example, replicating return of an index fund has lower transaction costs with derivative and risk-free rate rather than purchasing every security in the index fund.