Arbitrage & the Law of One Price

An arbitrage is a free lunch: a self-financing trading strategy that costs nothing today, can never lose money, and makes money with positive probability. Formally, a portfolio with value V_t is an arbitrage on [0, T] if

V_0 = 0, \qquad \mathbb{P}(V_T \ge 0) = 1, \qquad \mathbb{P}(V_T > 0) > 0.

You put in nothing, you are guaranteed to walk away no worse off, and sometimes you walk away richer — a riskless profit conjured from nowhere. The no-arbitrage principle is the assumption that, in an efficient market, no such opportunity persists: if one appeared, traders would pile in until prices adjusted and the gap closed. It is the single axiom from which almost all of derivative pricing flows.

The law of one price, derived by contradiction

The flagship consequence of no-arbitrage is the law of one price: any two portfolios that produce identical payoffs at maturity must have identical prices today. If they didn't, we could manufacture an arbitrage. Suppose two portfolios A and B have the same terminal value in every state of the world,

V_T^A = V_T^B \quad \text{with certainty,}

but trade at different prices today. Without loss of generality, say A is the cheaper one: P_A < P_B. We build a free lunch in three lines.

Step 1 — buy the cheap, short the dear. Today, go long A (pay P_A) and short B (receive P_B). The net cash you collect at t = 0 is

P_B - P_A > 0.

Step 2 — bank the surplus risklessly. Set that strictly positive amount aside (in the risk-free account, where it can only grow). Your net position now costs nothing — every dollar to open it came from the short — yet you are holding a guaranteed cushion.

Step 3 — unwind at maturity. At time T you owe the short position V_T^B and you collect V_T^A from the long. Because the payoffs were assumed identical, these cancel exactly:

V_T^A - V_T^B = 0.

Step 4 — read off the contradiction. You opened the trade for zero net cost, the maturity cashflows wash out to zero, and yet you pocketed P_B - P_A > 0 at the start with certainty. That is precisely an arbitrage — V_0 = 0, V_T \ge 0 always, and a sure positive profit. No-arbitrage forbids it, so the premise P_A \ne P_B is impossible. Hence

P_A = P_B.

The cashflow table below lays the same argument out column by column — today versus maturity — so you can see the riskless profit fall out with nothing owed at the end. Step through it.

In a market with no arbitrage, if two portfolios A and B have the same payoff at maturity in every state, V_T^A = V_T^B \quad\text{(with probability 1)}, then they must have the same price today: P_A = P_B. Equivalently — any two ways of manufacturing the same future cashflow must cost the same now, because a price gap is a money pump.

No-arbitrage does more than equate identical payoffs — it brackets prices that merely dominate one another. For a European call with strike K and maturity T, under continuous compounding at the risk-free rate r,

\big(S_0 - K e^{-rT}\big)^+ \;\le\; C \;\le\; S_0.

The upper bound is intuitive: a call can never be worth more than the stock itself, since owning the call gives you at most one share — if C > S_0, sell the call, buy the stock, and the difference is free money. The lower bound comes from comparing the call against "buy the stock, borrow K e^{-rT}": that portfolio costs S_0 - K e^{-rT} and pays S_T - K \le (S_T-K)^+, so the call, paying at least as much, can't be cheaper (and a price is never negative, giving the (\cdot)^+). Each bound is just the law of one price loosened to an inequality.

Behind the elementary arguments sits a deep theorem. The Fundamental Theorem of Asset Pricing says that the absence of arbitrage is equivalent to the existence of a special probability measure under which every discounted asset price is a fair game — a martingale. Pricing then becomes an expectation under that measure, and completeness (every payoff replicable) makes it unique. We unpack this as no arbitrage ⇔ a risk-neutral measure; for now, hold the slogan: no free lunch is the same fact as "prices are discounted expectations".

The free lunch, column by column

Lay out the cashflows of "buy A, short B" in two columns — today and maturity. The two maturity legs cancel because the payoffs are equal; the two opening legs leave a strictly positive P_B - P_A in your pocket. Step through to watch the riskless profit emerge with nothing owed at T.