Arbitrage and the law of one price

What is an arbitrage?

An arbitrage is a trade, or a set of trades, that gives a riskless profit with no net investment. You start with nothing, you finish with money, and no outcome of the market can leave you worse off. Two tests define it: zero net investment and zero risk.

A bet that is likely to pay is not an arbitrage. A stock with a high expected return is only a reward for taking risk. An arbitrage needs the cash flows to cancel in every state of the world.

Arbitrage is the pricing tool for derivatives because a derivative’s payoff can be copied with other assets. If the copy and the derivative sell for different prices, somebody can lock in the difference. Traders do exactly that, which moves the prices together. So derivative prices are set by what it costs to build the payoff, not by what anyone expects the market to do.

Two assumptions keep the argument clean: no transaction costs, and the ability to borrow and lend at the risk-free rate and to sell an asset you do not own. Selling an asset you do not own is short selling : you borrow the asset, sell it now, and buy it back later to return it. You pocket the sale proceeds today and owe the asset at the end.

Condition 1: identical cash flows must have the same price

The law of one price says that two assets that pay exactly the same cash flows at the same dates, with the same risk, must trade at the same price. If they do not, buy the cheaper one and sell the dearer one short.

Arbitrage opportunitya riskless profit with zero net investment

• Identical cash flows, different prices Memorize the law of one price fails
  • Buy the cheaper asset its price is pushed up
  • Sell short the dearer asset its price is pushed down
• Known future price off its present value Memorize compare S₀ with F₀(T)(1 + r)^−T
  • Spot too low borrow, buy the asset, sell it forward (cash-and-carry)
  • Spot too high sell the asset short, lend, buy it forward (reverse)

Condition 2: a known future price must equal spot compounded at the risk-free rate

Use it

Suppose you can lock in today the price at which you will sell an asset on a future date. That future price is now a known amount, so it must be worth its present value of a known future price today, discounted at the risk-free rate. The spot price cannot sit above or below it.

$$S_0 = F_0(T)\,(1 + r)^{-T}$$

The rate is the risk-free rate, because the payoff is certain. For most market participants the relevant risk-free rate is the repo rate , the rate on borrowing secured by highly liquid securities. With discrete compounding, T is the fraction of a year, as an exponent on $(1 + r)$.

$$P = F_0(T) - S_0(1 + r)^T$$

If P is positive, buy the asset with borrowed money and sell the forward. If it is negative, the same size of profit is available the other way round.

What if the spot price is too high?

When the spot price is above the present value of the known future price, spot is too dear. Reverse every leg: sell the asset short, lend the proceeds at the risk-free rate, and buy the asset forward at the agreed price.

Exam tip Always find the fair value first, then ask whether the quote is above or below it. Above fair: sell it. Below fair: buy it.

Go deeper Why do arbitrage gaps ever appear in practice?

Real trading has frictions. Transaction costs, limits on shorting, borrowing rates above the risk-free rate and the capital a trade needs all mean a small gap can persist if it is smaller than the cost of trading it. Large gaps are removed quickly, because many traders can see them.

Go deeper How arbitrage links to market efficiency

In an efficient market there are no riskless profits left after costs. Arbitrage is the force that gets prices there: each trade moves the mispriced price toward its fair value. That is why the exam can treat the no-arbitrage price as the price.