No-Arbitrage

Definition

No-arbitrage is the principle that in a frictionless market there exists no self-financing trading strategy that costs nothing (or negative) today yet delivers a payoff that is non-negative in every future state and strictly positive in at least one state — no "free lunch." Its weaker relative, the law of one price, requires only that any two portfolios with identical state-contingent payoffs carry identical prices; absence of arbitrage implies the law of one price but is strictly stronger (it also forbids dominance). The Fundamental Theorem of Asset Pricing (FTAP) makes the principle operational: absence of arbitrage is equivalent to the existence of a strictly positive Stochastic Discount Factor $M$ — equivalently, an equivalent martingale (risk-neutral) measure $Q$ under which every discounted asset price is a martingale. This equivalence, rather than any equilibrium or preference assumption, is the mathematical backbone of modern derivative and term-structure pricing.

Key Ideas

Law of one price ⊂ no-arbitrage. The law of one price is linearity of the pricing operator; no-arbitrage adds strict positivity (states with positive payoff have positive price). Positivity is what forces the discount factor / martingale density to be strictly positive.
Fundamental Theorem of Asset Pricing (triple equivalence): no-arbitrage ⇔ existence of a positive stochastic discount factor $M$ with $P_0 = E[M X]$ ⇔ existence of an equivalent martingale measure $Q$ under which discounted prices are martingales. The three statements are interchangeable descriptions of the same restriction.
Harrison–Kreps–Pliska. Harrison–Kreps (1979) and Harrison–Pliska (1981) established the martingale characterization: in a complete, frictionless market the absence of arbitrage is equivalent to the existence of a unique EMM, and asset pricing reduces to discounted $Q$ -expectation. This is "the unifying foundational theorem" of continuous-time finance (Sundaresan 2000).
Risk-neutral pricing. Any payoff $X$ at horizon $T$ is priced by the discounted risk-neutral expectation $P_0 = E^Q[e^{-rT} X]$ . Because preferences do not enter this expectation, prices are preference-free: one may compute them as if the representative agent were risk-neutral (Cox–Ross), even though real investors are risk-averse.
Completeness vs. incompleteness. When markets are complete (traded assets span all states) the EMM / SDF is unique and every claim has a single arbitrage-free price. When markets are incomplete — the generic case in discrete time — infinitely many positive SDFs are no-arbitrage compatible, and pricing an illiquid claim requires selecting one by convention (Gourieroux–Monfort 2007).
Cross-equation restrictions. In dynamic models no-arbitrage is not a single inequality but a web of restrictions linking the physical ( $P$ ) and risk-neutral ( $Q$ ) dynamics across all assets/maturities simultaneously — the source of the affine term-structure recursions and of testable over-identifying restrictions.

How It Works

From no-arbitrage to a martingale measure

Absence of arbitrage rules out any portfolio with non-positive cost and a non-negative, sometimes-positive payoff. By a separating-hyperplane argument this is equivalent to the existence of strictly positive state prices, i.e. a positive $M$ with $P_0 = E[M X]$ for every traded payoff $X$ . Normalizing $M$ by the riskless discount factor defines the Radon–Nikodym derivative $dQ/dP$ of an equivalent martingale measure $Q$ (equivalent because it assigns positive probability to exactly the same states as $P$ ), under which discounted prices are martingales. See Stochastic Discount Factor for the $P_0 = E[M X]$ representation and the $M = (dQ/dP)\,e^{-r}$ identity.

Replication and hedging in Black–Scholes

The cleanest no-arbitrage construction is dynamic replication (Black-Scholes Option Pricing). Holding one share long and $1/w_1$ calls short creates a portfolio whose instantaneous return has zero covariance with the market; if the hedge is rebalanced continuously the position is locally riskless, so no-arbitrage forces it to earn the riskless rate $r$ . Imposing this via Itô's lemma yields the Black–Scholes PDE $\tfrac{1}{2}v^2 x^2 w_{11} + r x w_1 + w_2 - rw = 0,$ whose solution is the closed-form option price. Risk-neutral pricing — $C = e^{-r\tau} E^Q[(S_T-K)^+]$ — emerges as a consequence of the ability to replicate, not as a separate assumption; because the hedge eliminates systematic risk, preference parameters cancel algebraically (Black–Scholes 1973; the same cancellation appears in their CAPM derivation, see Capital Asset Pricing Model). Rubinstein (1987) restates this through the Cox–Ross–Rubinstein binomial lattice: whenever a self-financing portfolio replicates the payoff each period, the claim is priced by discounted $Q$ -expectation, and put–call parity holds by pure static arbitrage regardless of the return distribution.

No-arbitrage restrictions in affine term-structure models

In an Affine Term Structure Model no-arbitrage is imposed through a log-normal pricing kernel $m_{t+1} = \exp(-r_t - \tfrac12\lambda_t'\lambda_t - \lambda_t'\epsilon_{t+1})$ with affine market prices of risk $\lambda_t = \lambda_0 + \lambda_1 X_t$ . Bond prices $p_t^n = E_t[m_{t+1}p_{t+1}^{n-1}]$ are exponential-affine, $p_t^n = \exp(\bar A_n + \bar B_n' X_t)$ , and the coefficients are not free: they satisfy recursive no-arbitrage equations $\bar A_{n+1} = \bar A_n + \bar B_n'(\mu - \Sigma'\lambda_0) + \tfrac12\bar B_n'\Sigma\Sigma'\bar B_n - \delta_0,\qquad \bar B_{n+1}' = \bar B_n'(\Phi - \Sigma\lambda_1) - \delta_1'.$ These recursions are exactly the cross-equation restrictions linking the physical dynamics $(\mu,\Phi)$ to the risk-neutral dynamics $(\mu-\Sigma'\lambda_0,\ \Phi-\Sigma\lambda_1)$ : the same state process and risk prices must price every maturity at once. Empirically these restrictions act as regularization — Ang–Piazzesi (2001) find imposing them improves out-of-sample yield forecasts by roughly 25% RMSE over an unrestricted VAR, while pricing the entire curve, not just the maturities in the VAR. See Term Structure of Interest Rates.

Selecting a measure under incompleteness

When traded returns do not span all states, no-arbitrage pins down a set of admissible SDFs rather than one. Gourieroux–Monfort (2007) resolve the indeterminacy by convention, restricting the SDF to the exponential-affine (Esscher) class $M_{t,t+1} = \exp(a_t' r_{t+1} + b_t)$ , which guarantees positivity and — when the only state variables are traded-asset returns — uniquely pins down the risk-neutral distribution. The choice is an identifying restriction, analogous to a normalization in a structural VAR, not something the data can reveal.

Why It Matters

Unifies asset pricing. CAPM, ICAPM, consumption-CAPM, Black–Scholes, and affine term-structure models are all special cases of $P_0 = E[M X]$ / discounted $Q$ -expectation — one framework prices options, bonds, and equities consistently.
Preference-free valuation. Because risk-neutral pricing removes preference parameters, derivatives can be priced from observables (prices and volatilities) alone, even where expected returns are unobservable. This is why Black–Scholes is operationally usable and empirically durable.
Discipline and testable restrictions. No-arbitrage cross-equation restrictions reduce parameter proliferation and generate over-identifying restrictions; imposing them improves forecasting (Ang–Piazzesi 2001) and yields internally consistent term-premium decompositions.
A benchmark for mispricing. Deviations from arbitrage identities (put–call parity, index-futures fair value, model prices) flag either genuine mispricing or limits to arbitrage — Rubinstein (1987) reads persistent S&P index-futures deviations as evidence of insufficient arbitrage capital.

Open Questions

Incomplete-market selection. Which SDF/EMM to use when markets are incomplete is a convention, not identified by prices; different admissible measures give different prices for illiquid claims, and no consensus criterion exists (Gourieroux–Monfort 2007).
Limits to arbitrage. Real markets exhibit persistent, apparently arbitrageable deviations (index-futures mispricing, post-1987 volatility skew) that pure no-arbitrage theory cannot explain without frictions, funding constraints, or bounded arbitrage capital.
Robustness of replication. The continuous, frictionless, complete-market hedge that makes pricing preference-free fails under transaction costs, jumps, and stochastic volatility, leaving a genuinely non-unique price and reintroducing preference/risk-premium dependence.
P-vs-Q identification. Separating physical from risk-neutral dynamics (hence the market price of risk $\lambda_1$ ) is empirically fragile; estimates are often insignificant and sample-sensitive even when no-arbitrage is imposed.