Arrow-Debreu Securities and State Prices¶

The previous page proved the First Fundamental Theorem of Asset Pricing in finite states: a market is arbitrage-free if and only if its prices satisfy \(\mathbf{P} = \mathbf{X}\boldsymbol{\phi}\) for some \(\boldsymbol{\phi} \gg 0\). We now interpret that algebraic equation. The components \(\phi_s\) of \(\boldsymbol{\phi}\) are the prices of elementary state-contingent claims called Arrow–Debreu securities, introduced by Kenneth Arrow and Gérard Debreu in the 1950s. Rescaling \(\boldsymbol{\phi}\) produces a probability measure \(\mathbb{Q}\), the risk-neutral measure, under which every asset price is the discounted expected payoff; dividing by physical probabilities produces the stochastic discount factor that links asset pricing to consumption-based macroeconomic models. This page is the canonical home for these three interpretations and for the Breeden–Litzenberger extraction of state prices from option markets.

Notation (recall)¶

Recall (see § One-Period Market Model): the market has \(S\) states \(\omega_1, \ldots, \omega_S\) with physical probabilities \(p_s > 0\), \(N\) traded assets with price vector \(\mathbf{P} \in \mathbb{R}^N\), and payoff matrix \(\mathbf{X} \in \mathbb{R}^{N \times S}\). A risk-free bond pays \(1 + r_f\) in every state. Throughout this page we assume no arbitrage, so by the FTAP proved in § Arbitrage and dominance there exists \(\boldsymbol{\phi} \gg 0\) with \(\mathbf{P} = \mathbf{X}\boldsymbol{\phi}\).

Arrow-Debreu Securities¶

Definition¶

Formally, the Arrow-Debreu security for state \(s\) has the payoff vector:

If we have \(S\) states, there are \(S\) possible Arrow-Debreu securities \(\mathbf{e}_1, \mathbf{e}_2, \ldots, \mathbf{e}_S\), which together form the identity matrix \(\mathbf{I}_S\).

Intuition¶

Arrow-Debreu securities isolate and "price" individual states of the world. They decompose uncertainty into its most elementary components. Any asset with an arbitrary payoff profile can be viewed as a portfolio of Arrow-Debreu securities:

This makes Arrow-Debreu securities the "building blocks" of all financial claims.

State Prices¶

Definition¶

The vector of state prices is:

Pricing Any Asset with State Prices¶

If state prices exist, the price of any asset \(j\) with payoffs \((X_{j1}, X_{j2}, \ldots, X_{jS})\) is simply:

This is the fundamental pricing equation. It says that an asset's price equals the sum of its payoffs in each state, weighted by the state prices.

In matrix form for all \(N\) assets:

where \(\mathbf{P} = (P_1, P_2, \ldots, P_N)^\top\) is the vector of asset prices.

Two Properties of \(\boldsymbol{\phi}\)¶

Under no-arbitrage, the FTAP gives positivity: \(\phi_s > 0\) for every \(s\). Economically, every state is "valuable" — investors are willing to pay a positive price for insurance against any possible state.

The risk-free bond pays \(1 + r_f\) in every state at price \(1\). Applying \(P = \boldsymbol{\phi}^\top \mathbf{X}_j\) to this row gives the bond identity:

so the state prices sum to the discount factor \(\beta\). In particular \(\sum_s \phi_s < 1\) when \(r_f > 0\). This identity is the bridge from \(\boldsymbol{\phi}\) to the risk-neutral measure below.

State Prices and Risk-Neutral Pricing¶

Constructing the Risk-Neutral Measure¶

State prices naturally give rise to risk-neutral probabilities. Define:

Since \(\phi_s > 0\) for all \(s\) and \(\sum_s q_s = 1\), the \(q_s\) form a valid probability measure \(\mathbb{Q}\), called the risk-neutral measure (or equivalent martingale measure).

Risk-Neutral Pricing Formula¶

Under \(\mathbb{Q}\), the fundamental pricing equation becomes:

Comparing Physical and Risk-Neutral Probabilities¶

Risk-neutral probabilities overweight bad states (where marginal utility is high) and underweight good states relative to physical probabilities. This reflects the market's aggregate risk aversion.

Stochastic Discount Factor (Pricing Kernel)¶

The stochastic discount factor (SDF), also called the pricing kernel, connects state prices to physical probabilities:

The SDF allows us to write the pricing equation using physical probabilities:

Relationships Summary¶

The three representations — state prices, risk-neutral probabilities, and the SDF — are equivalent ways to enforce no-arbitrage:

Completeness and Uniqueness of State Prices¶

Recall (see § Portfolios and payoffs): the market is complete iff \(\operatorname{rank}(\mathbf{X}) = S\), i.e. every payoff vector \(\mathbf{c} \in \mathbb{R}^S\) is replicable.

The proof and the pricing-interval consequences in the incomplete case are the subject of the Second FTAP: see § Complete Markets and Uniqueness.

Extracting State Prices from Market Data¶

From Observed Prices¶

Given \(N\) assets with price vector \(\mathbf{P}\) and payoff matrix \(\mathbf{X}\), state prices solve:

• If \(N = S\) and \(\mathbf{X}\) has full rank: \(\boldsymbol{\phi} = \mathbf{X}^{-1} \mathbf{P}\) (unique solution).
• If \(N > S\): overdetermined system; use least squares or check consistency.
• If \(N < S\): underdetermined; infinitely many solutions (incomplete market).

Numerical Example¶

From Option Prices: The Breeden-Litzenberger Result¶

In continuous-state settings, state prices can be extracted from European option prices. If \(C(K)\) denotes the price of a European call option with strike \(K\) on an asset with terminal price \(S_T\), and the state price density is \(\phi(s)\), then:

Differentiating twice with respect to \(K\):

This is the Breeden-Litzenberger formula. It shows that the curvature of the call price function with respect to the strike price reveals the state price density (and hence the risk-neutral density).

Connections to Other Topics¶

Summary¶

Exercises¶

Exercise 1. Consider a two-state economy with states \(\Omega = \{\omega_1, \omega_2\}\). A risk-free bond has price \(P_{\text{rf}} = 0.90\) and pays $1 in both states. A stock has price \(P_2 = 60\) and pays $80 in state \(\omega_1\) and $50 in state \(\omega_2\). Compute the state prices \(\phi_1\) and \(\phi_2\), the risk-neutral probabilities \(q_1\) and \(q_2\), and the stochastic discount factor values \(m_1\) and \(m_2\) assuming physical probabilities \(p_1 = 0.5\) and \(p_2 = 0.5\).

Exercise 2. Prove that if no-arbitrage holds and a risk-free bond exists paying $1 in every state, then the sum of the state prices satisfies

where \(r_f\) is the risk-free rate. What does this imply about the relationship between state prices and risk-neutral probabilities?

Exercise 3. In a three-state economy with \(\Omega = \{\omega_1, \omega_2, \omega_3\}\), the payoff matrix and price vector are

Solve \(\mathbf{X}\,\boldsymbol{\phi} = \mathbf{P}\) for the state price vector \(\boldsymbol{\phi}\). Verify that all components are strictly positive and compute the implied risk-free rate.

Exercise 4. Show that the stochastic discount factor \(m_s = \phi_s / p_s\) satisfies

when a risk-free bond exists. Interpret this result in terms of the relationship between the SDF and the time value of money.

Exercise 5. A market has two states and two traded assets (a bond and a stock). The state prices are \(\phi_1 = 0.35\) and \(\phi_2 = 0.60\). A new derivative is introduced with payoff \(\Phi(\omega_1) = 10\) and \(\Phi(\omega_2) = 3\). Using the fundamental pricing equation, compute the no-arbitrage price of this derivative. Then verify your answer using the risk-neutral pricing formula \(P = \frac{1}{1+r_f}\mathbb{E}^{\mathbb{Q}}[\Phi]\).

Exercise 6. Explain why the Breeden-Litzenberger formula \(\phi(K) = e^{r_f T} \frac{\partial^2 C}{\partial K^2}\) implies that the call price \(C(K)\) must be a convex function of the strike price \(K\) in an arbitrage-free market. What would a violation of convexity imply about state prices?