#### Subject 1. The Principle of Arbitrage

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.

Risk Aversion, Risk Neutrality, and Arbitrage-Free Pricing

Risk-seeking investors give away a risk premium because they enjoy taking risk. Risk-averse investors expect a risk premium to compensate for the risk. Risk-neutral investors neither give nor receive a risk premium because they have no feelings about risk.

Risk-neutral pricing: Suppose you want to price a derivative. The payoff of this derivate can be replicated using the underlying asset and risk-free rate. The market price of this derivative and the replicating strategy must be exactly the same under the principle of no arbitrage, regardless of risk preferences.

To obtain the derivative price we should assume the investor is risk-neutral, because an investor's risk aversion is not a factor in determining the derivative price. Risk can be eliminated by dynamic hedging in a situation where there is no arbitrage possible. Once risk is eliminated in this way the expected return becomes equal to the risk-free rate for all investors. Assets can be assumed to grow at the risk-free rate and also discounted at the risk-free rate.