The no-arbitrage principle requires prices to exclude a feasible portfolio with no net investment, no possible loss, and a positive payoff in at least one state.
The no-arbitrage principle says that market prices should not permit a feasible portfolio with no net investment, no possible future loss, and a positive payoff in at least one possible state. It is a consistency condition used to value derivatives and related cash flows, not a claim that markets are perfectly efficient or that prices equal intrinsic value.
In textbook models, an arbitrage can be locked in through simultaneous trades. In real markets, an apparent opportunity may disappear after bid-ask spreads, financing, short-sale constraints, margin, taxes, timing, settlement, and counterparty risk are included.
In a simple one-period model, a portfolio is an arbitrage if its initial value (V_0) and terminal payoff (V_T) satisfy:
with a strictly positive payoff in at least one state that can occur:
The formulation allows either zero initial cost or an initial cash inflow. Equivalent definitions may state the condition using discounted gains, self-financing strategies, or stricter positivity requirements. The exact definition must match the mathematical setting.
This is stronger than merely expecting a positive average return. A portfolio that can lose money is not a pure arbitrage even if its expected return is attractive.
If two tradable portfolios produce exactly the same cash flows in every relevant state, they should have the same current price. Otherwise, a trader could buy the cheaper portfolio and sell the more expensive one.
The equivalence must be genuine. Two instruments can differ in:
If any difference affects the payoff, the observed price gap may compensate for a real basis rather than represent arbitrage.
Assume a default-free one-year zero-coupon bond pays $100 at maturity. The one-year risk-free gross return is 1.04, and borrowing and lending are available at that rate with no costs.
The no-arbitrage price is:
Borrow $94 and buy the bond. At maturity:
| Cash flow | Amount |
|---|---|
| Bond payment | +$100.00 |
| Loan repayment, (94\times1.04) | -$97.76 |
| Locked-in terminal gain | +$2.24 |
The strategy has zero initial net investment and a positive terminal payoff under the assumptions.
Short the bond for $99 and invest the proceeds at 4%. At maturity:
| Cash flow | Amount |
|---|---|
| Investment value, (99\times1.04) | +$102.96 |
| Bond payment owed | -$100.00 |
| Locked-in terminal gain | +$2.96 |
This second trade requires the ability to short the bond and invest the proceeds. If shorting is prohibited, costly, or subject to recall, the textbook arbitrage may not be feasible.
Replication can establish bounds even when an exact price is unavailable. For a non-dividend-paying stock and a European call with strike (K), one-period gross risk-free return (R), and current stock price (S_0):
The upper bound reflects that a call cannot be worth more than the underlying stock in this simplified setting. The lower bound follows by comparing the call with a portfolio involving the stock and the present value of the strike.
Bounds change when dividends, early exercise, negative rates, funding costs, settlement, or contract terms differ. A formula should not be applied outside its assumptions.
Suppose a derivative payoff can be reproduced exactly by trading the underlying asset and a risk-free asset. The derivative and replicating portfolio must have the same price; otherwise, buying the cheaper and selling the more expensive creates an arbitrage within the model.
This logic leads to Risk-Neutral Probabilities. Under the pricing measure, discounted traded-asset prices behave consistently with no arbitrage. The pricing measure changes probability weights; it does not say investors are indifferent to risk.
Under suitable technical conditions:
In an incomplete market, several pricing measures may satisfy no-arbitrage restrictions, so no arbitrage can produce a range rather than one unique value.
| Concept | Core claim | What it does not establish |
|---|---|---|
| No arbitrage | No feasible zero-investment strategy has nonnegative payoffs in all states and a gain in at least one state | Prices are correct forecasts or equal fundamental value |
| Law of one price | Identical tradable cash flows should have identical prices | Similar-looking instruments are economically identical |
| Market efficiency | Prices reflect a specified information set under a stated hypothesis | Every price is permanent, stable, or free from risk |
| Relative-value trade | Related prices may converge based on a model or catalyst | Convergence is guaranteed or risk-free |
A market can contain forecasting errors, risk premia, illiquidity, and volatility without offering a pure arbitrage.
Many strategies called “arbitrage,” including merger arbitrage and statistical arbitrage, retain material event, model, and funding risk. Their names do not make them textbook arbitrages.
This article provides general financial education. It does not identify an executable trade or provide personalized trading, investment, legal, tax, or financing advice.