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Lesson 1 of 6 · 11 min

Arbitrage and the law of one price

Prices must leave no riskless profit on the table: identical cash flows must cost the same, and an asset with a known future price must trade at that price discounted at the risk-free rate.

Video lesson · 5 min. Watch it first, then use the text below to go deeper.

In short

Arbitrage is a riskless profit with no net investment. It exists when the law of one price fails: the same thing trades at two prices at the same time.
For derivatives there are two arbitrage conditions: assets with identical future cash flows must have the same price today, and an asset with a known future price must trade at the present value of that price.
With no other costs or benefits of ownership, the right discount rate is the risk-free rate.
Discrete compounding, $(1+r)^T$ , is used for single assets; continuous compounding, $e^{rT}$ , for portfolios such as indexes and for foreign exchange.
Arbitrageurs buy the cheap side and sell the dear side, which pushes the prices together until the profit disappears.

What arbitrage is

An arbitrage opportunity is a set of trades that costs nothing (or pays you) today and can never lose money later. In well-functioning markets with no transaction costs, such opportunities should not last.

The simplest version is a breach of the law of one price: one asset, or two assets that are economically identical, trading at different prices at the same moment. You buy where it is cheap, sell where it is dear, and pocket the gap.

Derivative pricing relies on this idea. A derivative's value comes from the future cash flows of its underlying, so if we can find another position with exactly the same future cash flows, the two must cost the same today.

Arbitrage is not 'buy low, sell high later'. Both legs are locked in at the same time, so the profit does not depend on where prices go next.

Case 1: identical cash flows, different prices

Take two zero-coupon bonds from the same issuer, with the same maturity, par value and default risk. Bond A trades at EUR 97.40 and Bond B at EUR 97.65 per 100 of par. Both will pay exactly 100 at maturity.

Short-sell B for 97.65 and buy A for 97.40. You receive EUR 0.25 today. At maturity A pays 100, which is exactly what you need to buy back and return B. The future cash flows cancel, so the 0.25 is riskless profit.

Others will spot the same gap. Selling pushes B's price down, buying pushes A's price up, and the trade stops working once the two prices are equal.

Arbitrage corrects itself: the very trades that capture the gap close it. That is why we can assume no-arbitrage prices when valuing derivatives.

Case 2: a known future price

Now suppose an asset will certainly be worth a known amount at time T, for example because you can lock in its sale price today. Then its spot price must equal that future price discounted at the risk-free rate: $S_0 = S_T(1+r)^{-T}$ .

If the spot price is below that present value, borrow at r, buy the asset, and lock in the sale. At T the sale proceeds exceed the loan repayment. If the spot price is above it, do the reverse: short the asset, lend the proceeds, and lock in a purchase at the known price.

The risk-free rate is the right discount rate here because, ignoring any other costs or benefits of holding the asset, the only thing you give up by owning it is the interest you could have earned on the cash.

Spot SGD 1,990 is below the PV of the known SGD 2,040 price (2,010.07 at 3%). Borrowing to buy costs nothing today and leaves SGD 20.37 at T whatever happens.

Discrete and continuous compounding

Moving cash through time can be done two ways. With discrete compounding, $FV = PV(1+r)^N$ , interest is added once per period. With continuous compounding, $FV = PV e^{rT}$ , the periods shrink towards zero, which gives slightly more growth for the same stated rate.

Both are valid for pricing derivatives. The convention in this topic: use discrete compounding for a single underlying asset (one share, one bond, one commodity), and continuous compounding for underlyings that are portfolios (equity, bond, commodity or credit indexes) and for foreign exchange, where two currencies' rates are involved.

EUR 1,000 for two years at 4%
Method	Formula	Future value
Discrete (annual)	$1{,}000 \times 1.04^{2}$	1,081.60
Continuous	$1{,}000 \times e^{0.04 \times 2}$	1,083.29

Formulas

Identical cash flows: $S_0^A = S_0^B$
Two assets with the same future cash flows and risk must have the same price today.
Known future price: $S_0 = S_T(1+r)^{-T}$
S_T = the known price at T, r = annual risk-free rate, T = years. Assumes no other costs or benefits of owning the asset.
Future value: discrete and continuous: $FV_N = PV(1+r)^N \qquad FV_T = PV e^{rT}$
Discrete for single assets; continuous for index (portfolio) underlyings and FX.

Worked examples

Example 1 Two identical bonds

Bond A and Bond B are zero-coupon bonds from the same issuer, maturing on the same date at 100. A trades at EUR 97.40 and B at EUR 97.65. What trades capture the arbitrage, and how large is the profit?

Example 2 Spot below the PV of a known future price

An asset can be sold for SGD 2,040 in six months under a contract agreed today. It trades at SGD 1,990 now, the risk-free rate is 3%, and holding it has no other costs or benefits. Is there an arbitrage, and how large is the profit at T?

On the exam

Expect either a definition item on the law of one price or a short calculation of the riskless profit when spot differs from the discounted future price. Wrong answers usually ignore the interest on the loan.

Taking the raw gap (future price − spot) as the profit. The financing cost of buying the asset must be subtracted.
Discounting over a full year when the horizon is shorter. T is in years, so six months is 0.5.
Thinking an arbitrage needs a view on future prices. Both legs are set today, so the outcome is known.

Check yourself

Q1The law of one price is violated when:
Q2Two zero-coupon bonds with identical issuer, maturity and par of 100 trade at EUR 96.10 and EUR 96.35. The arbitrage profit per bond is:
Q3An asset's price in one year is known to be 515. It trades at 495 today, the risk-free rate is 3% and there are no other costs or benefits. The arbitrage profit at T is closest to:
Q4Under the convention used in this topic, a forward on an equity index is most likely priced using:
Q5An asset's spot price is below the risk-free present value of its known future price. As arbitrageurs trade, the spot price will most likely:

0/5Check yourself

Answer the 5 questions: +10 XP for each one you get right first time.

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