### Concepts of Arbitrage, Replication and Risk Neutrality

The concepts of arbitrage, replication and risk neutrality are important to comprehend when pricing derivatives. We can use assumptions about arbitrage and investor risk preferences to determine derivative pricing.

## Arbitrage

Arbitrage is the practice of exploiting a price imbalance in the same asset that exists between two or more markets. For derivatives, this is taking advantage of the differences in prices of a unique asset, to make a risk-free profit. Arbitrage opportunities tend to be exploited very quickly which forces the convergence of prices. This is why identical assets should have just one price and gives rise to the law of one price. There are circumstances where the cost of exploiting the arbitrage opportunity may exceed the benefit in which case, the price discrepancy (which tends to be small) will persist.

Consider this simple example: Company ABC’s stock trades on the New York Stock Exchange for $10.00, and the equivalent of$11.00 on the London Stock Exchange. This sets up a perfect, risk-free arbitrage opportunity. The ‘arbitrageur’ can buy ABC’s stock on the New York Stock Exchange for $10.00 and simultaneously sell the stock on the London Stock Exchange for$11.00, making a \$1 per share ‘riskless’ profit. This action by market participants would force the two prices to converge back to one price.

## Replication

Replication is the creation of an asset or portfolio by using a combination of another asset, portfolio, and/or derivative. The following combinations produce the equivalent single asset:

$$\text{Long asset} + \text{Short derivative} = \text{Long risk free asset}$$

For example, let’s say an investor has just bought one troy ounce of gold at USD 1,500. Then, he short sells a futures contract on a troy ounce of gold at USD 1575 for one year so that he can sell his troy ounce of gold in twelve months at a higher price. This means that his return should be equivalent to the return he would earn on a risk-free asset:

$$USD 1500-USD 1575=USD 75$$

In other words, if the investor were to buy a risk-free bond for the same amount of USD 1500 and sell it one year later, he would earn an equivalent risk-free profit of USD 75 (or USD 75/USD 1500 = 5%).

We can rearrange the same formula as:

$$\text{Long asset}+\text{Short risk free asset}=\text{Long derivative}$$

$$\text{Short derivative}+\text{Short risk free asset}=\text{Short asset}$$

If assets are priced correctly to prohibit arbitrage, replication would seem to be a pointless exercise. However, if we relax the no-arbitrage assumption, we may identify opportunities where replication may be more profitable or have a lower transaction cost.

## Risk Neutrality

Most investors are risk averse and will not accept risk without the commensurate returns. However, as risk aversion is not relevant to the pricing of a derivative (unlike other assets), we can assume the investor is risk-neutral. This means the expected payoff of the derivative can be discounted at the risk-free rate rather than having to use the risk-free rate plus some premium.

Risk neutrality, otherwise known as risk-neutral derivative pricing, uses the fact that arbitrage opportunities guarantee that a risk-free portfolio consisting of the underlying and the derivative must earn the risk-free rate. The overall process of pricing derivatives by arbitrage and risk neutrality is called arbitrage-free pricing. We effectively determine the price of the derivative by assuming the market is free of arbitrage opportunities, sometimes referred to as the principle of no arbitrage.

## Question

When an arbitrage opportunity presents itself, what is most likely to happen?

B. Clearinghouses will restrict the transactions that can be arbitraged

C. An investor has the opportunity to earn a risk premium in the short run

Solution

The increased buying and selling by traders will eventually adjust the price to eliminate any arbitrage. As demand for the cheaper asset goes up, so does the price. When selling this asset in another market, the supply of that asset goes up, and the price of the asset goes down. Increased arbitrage activity will eventually neutralize the opportunity to take advantage of any price discrepancy.

Explain how the concepts of arbitrage, replication, and risk neutrality are used in pricing derivatives

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