The Stochastic Discount Factor
In this note I generalize the notion of stochastic discount factor by showing that it emerges naturally as a consequence of the law of one price. The absence of arbitrage opportunities implies the existence of a strictly positive discount factor. Thus, in complete markets there is a unique strictly positive stochastic discount factor. Original sources of the analysis can be found in Hansen and Richard (1987), Hansen and Jagannathan (1991) and Hansen and Jagannathan (1997).
The Set of Traded Payoffs
In this economy, not all payoffs are necessarily traded unless the market is complete. We denote by X the linear subspace of traded payoffs spanned by \{x_{1}, x_{2}, \ldots, x_{N}\}, where N \leq S and all payoffs are assumed to be linearly independent. Denote by \mathbf{x}' = (x_{1}, x_{2}, \ldots, x_{N}) \tag{1} a vector containing all the basis payoffs. In the following, we assume that the Gram matrix \operatorname{E}(\mathbf{x} \mathbf{x}') = \sum_{s=1}^{S} q(s) \mathbf{x}(s) \mathbf{x}'(s) is invertible.
We can create other payoffs by buying or selling our N original assets. Any x \in X can be expressed as: x = \sum_{i = 1}^{N} a_{i} x_{i}, \tag{2} for a_{i} \in \mathbb{R}, 1 \leq i \leq N. Finally, we denote by \pi_{i} the price of asset i for 1 \leq i \leq N, and by \boldsymbol{\pi}' = (\pi_{1}, \pi_{2}, \ldots, \pi_{n}) a vector containing the prices. of the N basis payoffs.
The Law of One Price
Now, we would like to know if there is a way to create a pricing functional p: X \rightarrow \mathbb{R} that gives the price of any traded payoff. Clearly, we have that p(x_{i}) = \pi_{i} for 1 \leq i \leq N, and our intuition tells us that the price of any other asset should be given in terms of the other asset prices. If this was not the case, this would be an arbitrage.
Example 1 Suppose that p(x) = 1 and p(y) = 2. There is also an asset z = 3x + 4y such that p(z) = 12. Is there an arbitrage opportunity?
Of course! We could buy 3 units of x and 4 units of y and bundle them as z. The cost of the bundle is $11, but we can sell it for $12, generating a riskless profit of $1 per trade. Since there is no shortage of these securities, we could continue doing it until the prices of x and y go up and/or the price of z goes down.
In order to avoid these type of situations, we will assume the following.
Assumption 1 (The Law of One Price) Suppose that x_{i} \in X and a_{i} \in \mathbb{R} for i \in 1, 2, \ldots, N \leq S. If x = \sum_{i = 1}^{N} a_{i} x_{i} \in X, then p(x) = \sum_{i = 1}^{N} a_{i} p(x_{i}). \tag{3}
In competitive markets, the law of one price (LOOP) guarantees that the price of a basket of stocks is equal to the sum of the prices of its constituents. This logic is at the heart of how Exchange-Traded Funds (ETF) operate, as the next example shows.
Example 2 An ETF is a type of investment fund that is traded on stock exchanges, similar to individual stocks. ETFs hold a diversified portfolio of assets, such as stocks, bonds, or commodities, which provides investors with broad exposure to specific markets or investment strategies.
ETF arbitrage is the mechanism that helps keep the market price of an ETF in line with its Net Asset Value (NAV). Authorized Participants (APs), typically large financial institutions, have the ability to create or redeem ETF shares in large blocks called creation units.
When the ETF market price is higher than the NAV, APs can buy the underlying securities of the ETF in the open market and then deliver them to the ETF issuer in exchange for new ETF shares. The AP can then sell these ETF shares at the higher market price, making a profit. This buying of underlying securities pushes their prices up, while the selling of new ETF shares pushes the ETF price down, bringing the two prices closer together.
When the ETF market price is lower than the NAV, APs can buy ETF shares in the open market and deliver them to the ETF issuer in exchange for the underlying securities. The AP can then sell these underlying securities at the higher NAV price, making a profit. This buying of ETF shares pushes their price up, while the selling of the underlying securities pushes their prices down, again bringing the two prices closer together.
This creation and redemption process happens continuously and helps to keep the ETF price in line with the NAV. The arbitrage opportunities are typically small but sufficient for APs to engage in the process for profit, ensuring that the ETF price does not deviate significantly from its NAV.
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The fact that the market for ETFs is so liquid and works flawlessly reassures us that LOOP is a reasonable axiom to start working from. The law of one price implies the price functional defined in (3) is a continuous linear functional. By the Riesz Representation Theorem, there exists a unique x^{*} \in X such that p(x) = \operatorname{E}(x^{*} x) for all x \in X.
Since p(x_{i}) = \pi_{i} for each basis asset, we have \operatorname{E}(m \mathbf{x}) = \boldsymbol{\pi} for any valid SDF m. Substituting into the Projection onto a Subspace formula (with y = m, M = X, and G = \operatorname{E}(\mathbf{x}\mathbf{x}')) identifies x^{*} as the projection of any SDF m onto X.
Property 1 Under LOOP, the unique SDF in X is the projection of any valid SDF m onto X, given by x^{*} = \boldsymbol{\pi}' \operatorname{E}(\mathbf{x} \mathbf{x}')^{-1} \mathbf{x} \in X. \tag{4}
To verify directly, take any x = \mathbf{x}' \mathbf{a} \in X and compute \begin{aligned} \operatorname{E}(x^{*} x) & = \operatorname{E}(\boldsymbol{\pi}' \operatorname{E}(\mathbf{x} \mathbf{x}')^{-1} \mathbf{x} \mathbf{x}' \mathbf{a}) \\ & = \boldsymbol{\pi}' \operatorname{E}(\mathbf{x} \mathbf{x}')^{-1} \operatorname{E}(\mathbf{x} \mathbf{x}') \mathbf{a} \\ & = \boldsymbol{\pi}' \mathbf{a} = p(x), \end{aligned} confirming that x^{*} prices all assets correctly.
Any other SDF m will price the assets correctly, so that \operatorname{E}((m - x^{*}) x) = \operatorname{E}(m x) - \operatorname{E}(x^{*} x) = p(x) - p(x) = 0. This shows that we can create new SDFs by combining x^{*} with any vector e orthogonal to \mathcal{X}. In other words, all the SDFs that price assets correctly in X can be written as: m = x^{*} + e, where e \perp x for all x \in X.
The Principle of No-Arbitrage
The law of one price disciplines prices across portfolios: equal payoffs must have equal prices. But LOOP says nothing about the relationship between prices and the sign of payoffs. A pricing functional can be perfectly linear — satisfying LOOP — and yet assign a non-positive price to a payoff that is non-negative in every state. An investor facing such an opportunity would want to hold an unbounded position: buying costs nothing (or generates an immediate cash inflow) while the payoff can only be zero or positive. In competitive markets, unbounded demand is inconsistent with equilibrium clearing, so such opportunities cannot persist.
Assumption 2 (Principle of No-Arbitrage) The price of a payoff that is non-negative in all states and strictly positive in at least one state of the world must be positive.
To see that PNA is strictly stronger than LOOP, consider a market with a single risky asset x_1 that pays $1 in state 1 and $0 in state 2. Any pricing functional of the form p(a x_1) = a \pi_1 is linear and satisfies LOOP regardless of the sign of \pi_1. But if \pi_1 \leq 0, buying x_1 costs nothing (or generates income) while delivering a non-negative payoff — a free lunch. LOOP is silent on this because there is no portfolio inconsistency; the violation is purely about the sign of the price relative to the direction of the payoff.
More generally, PNA adds the requirement that prices respect dominance: a payoff that weakly improves on zero in every state, and strictly improves in at least one, must carry a strictly positive price. This is the minimal condition consistent with the behavior of a rational investor who strictly prefers more consumption to less in every state of the world.
The principle of no-arbitrage (PNA) is a strictly stronger assumption than LOOP, as the next property shows.
Property 2 \text{PNA} \Rightarrow \text{LOOP}.
Proof
We will prove this claim by contradiction. Assume that PNA holds but not LOOP. A violation of LOOP implies that the price of a zero payoff is not zero; without loss of generality, assume p_{0} = p(0) > 0. Take any payoff x^{+} \in X that is non-negative and non-zero, with price p > 0. Form a portfolio that buys one unit of x^{+} and sells n > \frac{p}{p_{0}} units of the zero payoff. The cost of that portfolio is \pi = p - n p_{0} < p - \frac{p}{p_{0}} p_{0} = 0, but its payoff is identical to x^{+}, which is non-negative and non-zero — a violation of PNA. This contradiction establishes the result.The reverse implication does not hold: a market can satisfy LOOP while still admitting free lunches, as the single-asset example above illustrates. PNA therefore places a genuinely additional restriction on the pricing functional beyond linearity.
A more important consequence of PNA is that it is equivalent to the existence of a strictly positive SDF. Recall that any SDF m satisfying LOOP prices assets via p(x) = \operatorname{E}(mx), but m need not be positive in every state. If m(s) \leq 0 for some state s, then an Arrow-Debreu security e_s — which pays $1 in state s and $0 elsewhere — would receive price p(e_s) = \operatorname{E}(m e_s) = q(s) m(s) \leq 0, a free asset with a non-negative payoff. Ruling this out for every state is precisely the condition m > 0.
Property 3 (No-Arbitrage and Strictly Positive SDFs) \text{PNA} \Leftrightarrow \exists m > 0.
A proof is provided in Section 6.1. The economic content of this equivalence is that no-arbitrage is exactly the condition that the market assigns a positive shadow value to wealth in every state — no state is “throwaway.” In complete markets (N = S), PNA pins down a unique strictly positive SDF, since X = L leaves no room for the orthogonal residual e in the decomposition m = x^{*} + e.
Arrow-Debreu Securities
Arrow-Debreu securities are the most elementary financial contracts imaginable: security e_s pays $1 if state s occurs and $0 in every other state. They are the atomic building blocks of the payoff space — every contingent claim can be decomposed into a portfolio of Arrow-Debreu securities, one for each state of the world.
Formally, define the random variable e_{s} by e_{s}(i) = \begin{cases} 1 & \text{if } i = s, \\ 0 & \text{otherwise,} \end{cases} for each s \in \mathcal{S}. Since the payoff space L has dimension S, the collection \{e_1, e_2, \ldots, e_S\} forms a basis for L. Any payoff x \in L admits the unique decomposition x = x(1)\, e_{1} + x(2)\, e_{2} + \cdots + x(S)\, e_{S}, \tag{5} since in state s the right-hand side evaluates to x(s) \cdot 1 = x(s).
State prices. The price of e_s is called the state price for state s and is denoted \psi(s). It represents the cost today of receiving $1 contingent on state s occurring. By the law of one price, the price of any payoff x follows immediately from the spanning representation: p(x) = \sum_{s=1}^{S} x(s)\, \psi(s) = \boldsymbol{\psi}' \mathbf{x}, where \boldsymbol{\psi} = (\psi(1), \ldots, \psi(S))' collects all state prices. This is the state-price representation of the pricing functional: prices are just inner products with the state-price vector.
Connection to the SDF. The state-price vector and the SDF are related by \psi(s) = q(s)\, m(s), since p(e_s) = \operatorname{E}(m\, e_s) = \sum_{i} q(i) m(i) e_s(i) = q(s) m(s). The positivity condition m > 0 is therefore equivalent to \psi(s) > 0 for all s: no-arbitrage requires every state to carry a strictly positive price.
Complete markets. If all S Arrow-Debreu securities are traded — that is, N = S and X = L — the market is complete. In this case the state-price vector \boldsymbol{\psi} is uniquely determined by the S asset prices, and so is the SDF: m(s) = \psi(s)/q(s) for each s. Incomplete markets (N < S) admit multiple valid state-price vectors and hence multiple SDFs, all of which price the N traded assets correctly but assign different values to non-traded payoffs.
Maximum Sharpe Ratio
The Sharpe ratio of asset i measures its excess expected return per unit of return volatility: \mathit{SR}^{i} = \frac{\operatorname{E}(R^{i}) - R^{f}}{\sigma(R^{i})}, where R^{f} is the return on a risk-free asset. It is the most widely used measure of risk-adjusted performance. In this section we show that any valid SDF places a universal upper bound on the Sharpe ratio of every traded asset.
The risk-free rate. A risk-free asset delivers a constant return R^{f} in every state. Applying the pricing equation \operatorname{E}(mR) = 1 to this asset gives \operatorname{E}(m R^{f}) = R^{f} \operatorname{E}(m) = 1, \quad \text{so} \quad R^{f} = \frac{1}{\operatorname{E}(m)}. The risk-free rate is the reciprocal of the expected SDF. A higher average discount factor (more patient or more risk-averse investors) implies a lower risk-free rate.
Excess returns. For any risky asset i, the pricing equation expands as 1 = \operatorname{E}(m R^{i}) = \operatorname{E}(m)\operatorname{E}(R^{i}) + \operatorname{Cov}(m, R^{i}). Substituting \operatorname{E}(m) = 1/R^{f} and rearranging: \operatorname{E}(R^{i}) - R^{f} = -\frac{\operatorname{Cov}(m, R^{i})}{\operatorname{E}(m)} = -R^{f} \operatorname{Cov}(m, R^{i}). The excess return of any asset equals -R^{f} times its covariance with the SDF. Assets that tend to pay off when the SDF is high — that is, when the marginal value of wealth is high, typically in bad times — are valuable as insurance and therefore command lower excess returns. Assets that pay off when the SDF is low (good times) offer no insurance and must compensate investors with higher expected returns.
The Sharpe ratio bound. Writing the covariance in terms of the correlation \rho(m, R^{i}): \begin{aligned} \operatorname{E}(R^{i}) - R^{f} & = -\frac{\operatorname{Cov}(m, R^{i})}{\operatorname{E}(m)} \\ & = -\frac{\rho(m, R^{i})\, \sigma(m)\, \sigma(R^{i})}{\operatorname{E}(m)}. \end{aligned} Dividing by \sigma(R^{i}) and taking absolute values, the Cauchy-Schwartz inequality gives |\rho(m, R^{i})| \leq 1, so \left| \frac{\operatorname{E}(R^{i}) - R^{f}}{\sigma(R^{i})} \right| \leq \frac{\sigma(m)}{\operatorname{E}(m)}. \tag{6}
This is the Hansen-Jagannathan bound (Hansen and Jagannathan 1991): the absolute Sharpe ratio of any traded asset is bounded above by the coefficient of variation \sigma(m)/\operatorname{E}(m) of the SDF. The bound holds for every valid SDF m and every asset i. Equality holds when |\rho(m, R^{i})| = 1, i.e., when the return is perfectly (negatively) correlated with the SDF.
Economic content. Since \sigma(m)/\operatorname{E}(m) = R^{f}\,\sigma(m), a large observed Sharpe ratio forces any valid SDF to be highly volatile relative to its mean. The U.S. equity market has historically delivered a Sharpe ratio of roughly 0.5 per year, implying \sigma(m)/\operatorname{E}(m) \geq 0.5. Standard consumption-based models with power utility and observed aggregate consumption growth produce SDFs that are far too smooth to satisfy this bound unless the coefficient of relative risk aversion is implausibly large. This tension is the equity premium puzzle of Mehra and Prescott (1985), and the Hansen-Jagannathan bound provides its sharpest quantitative statement.
Appendix
Proof of Property 3
The proof relies on the following geometric result.
Lemma 1 (Separating Hyperplane Theorem) If A and B are non-empty disjoint convex subsets of \mathbb{R}^{n}, there exists a non-zero vector \phi \in \mathbb{R}^{n} and a scalar \alpha \in \mathbb{R} such that \phi' a \leq \alpha \leq \phi' b \quad \text{for all } a \in A,\; b \in B.
Road map. The direction (\Leftarrow) is direct: if m > 0, every non-negative non-zero payoff receives a positive price, ruling out arbitrage. For (\Rightarrow), we argue by contradiction. We reformulate the problem as finding a strictly positive state-price vector that correctly prices all assets. Assume no such vector exists, so the plane of valid state prices and the positive orthant are disjoint. The Separating Hyperplane Theorem then places a hyperplane between them, producing a separating normal vector. We show this normal is the payoff vector of some traded portfolio, and that the three properties forced by the separation — non-negative payoff, non-zero payoff, and non-positive cost — make that portfolio an arbitrage, contradicting PNA.
(\Leftarrow): Suppose m > 0 and let x \in X satisfy x \geq 0 and x \neq 0. Since x is non-negative but not identically zero, there is at least one state s^{*} with x(s^{*}) > 0. In the sum p(x) = \operatorname{E}(mx) = \sum_{s=1}^{S} q(s)\, m(s)\, x(s), every term is non-negative (because q(s) > 0, m(s) > 0, and x(s) \geq 0), and the term at s = s^{*} is strictly positive. Hence p(x) > 0, which is exactly PNA.
(\Rightarrow): Suppose PNA holds. Since PNA \Rightarrow LOOP (by Property 2), the SDF x^{*} exists by Property 1. We must show that some SDF is strictly positive in every state.
Reformulation in terms of state prices. Define the state-price vector \psi \in \mathbb{R}^{S} by \psi(s) = q(s)\, m(s). Since q(s) > 0, we have m(s) > 0 if and only if \psi(s) > 0, so proving m > 0 is equivalent to finding \psi \gg 0. In terms of \psi, the pricing condition \pi_{i} = \operatorname{E}(m x_{i}) becomes a Euclidean dot product \pi_i = \psi'd_i, where d_{i} \in \mathbb{R}^{S} is the payoff vector of asset i. Stacking the N conditions into the S \times N payoff matrix D (whose i-th column is d_{i}), the pricing constraints become D'\psi = \boldsymbol{\pi}. Note that the traded payoff space is X = \operatorname{col}(D) = \{D\theta : \theta \in \mathbb{R}^{N}\}, so its orthogonal complement is X^{\perp} = \{e \in \mathbb{R}^{S} : D'e = 0\}. We therefore need to show that D'\psi = \boldsymbol{\pi} has a solution with \psi \gg 0.
Geometric setup. Define \Psi = \{\psi \in \mathbb{R}^{S} : D'\psi = \boldsymbol{\pi}\}, the flat plane of all state-price vectors consistent with observed prices. It is non-empty because \psi^{*}(s) = q(s)\, x^{*}(s) \in \Psi. The open positive orthant \mathbb{R}^{S}_{++} is the set of strictly positive state prices. Our goal is to show that \Psi \cap \mathbb{R}^{S}_{++} \neq \emptyset under PNA.
Contradiction. Suppose \Psi \cap \mathbb{R}^{S}_{++} = \emptyset. Since both sets are non-empty and convex, the Separating Hyperplane Theorem yields a non-zero \phi \in \mathbb{R}^{S} and \alpha \in \mathbb{R} with \phi'\psi \leq \alpha \leq \phi' y \quad \text{for all } \psi \in \Psi,\; y \in \mathbb{R}^{S}_{++}.
Step 1: \phi \geq 0 and \alpha \leq 0, by the cone property of \mathbb{R}^{S}_{++}. The positive orthant is a cone: y \in \mathbb{R}^{S}_{++} implies ty \in \mathbb{R}^{S}_{++} for any t > 0. Fix any state s and let e_{s} be the s-th standard basis vector. Then te_{s} \in \mathbb{R}^{S}_{++} for all t > 0, so the separation requires t\phi(s) = \phi'(te_{s}) \geq \alpha. Taking t \to 0^{+} gives \alpha \leq 0. Taking t \to \infty shows that \phi(s) < 0 would force t\phi(s) \to -\infty, eventually falling below \alpha, so \phi(s) \geq 0. Since s was arbitrary, \phi \geq 0 and \alpha \leq 0.
Step 2: \phi is the payoff of some portfolio. The plane \Psi extends infinitely in every direction belonging to X^{\perp}, the orthogonal complement of the traded payoff space. Indeed, any e \in X^{\perp} satisfies D'e = 0 by definition. For any such e and any \psi_{0} \in \Psi, D'(\psi_{0} + te) = D'\psi_{0} + t\underbrace{D'e}_{=\,0} = \boldsymbol{\pi}, so the entire line \psi_{0} + te lies in \Psi for every t \in \mathbb{R}. The separation requires \phi'(\psi_{0}+te) = \phi'\psi_{0} + t\,\phi'e \leq \alpha for all t \in \mathbb{R}. A linear function of t that is bounded above for all t must be constant, forcing \phi'e = 0. Since this holds for every e \in X^{\perp}, we have \phi \in (X^{\perp})^{\perp} = X. Therefore there exists a portfolio \theta \in \mathbb{R}^{N} with \phi = D\theta, i.e., \phi is the payoff vector of portfolio \theta. Substituting \phi = D\theta, for any \psi \in \Psi: \phi'\psi = (D\theta)'\psi = \theta'(D'\psi) = \theta'\boldsymbol{\pi} = \boldsymbol{\pi}'\theta, confirming \alpha = \boldsymbol{\pi}'\theta.
Step 3: Arbitrage. The portfolio \theta satisfies:
- Non-negative payoff in every state: D\theta = \phi \geq 0.
- Strictly positive payoff in at least one state: \phi \neq 0 (since \phi is a non-zero separating vector).
- Non-positive cost: \boldsymbol{\pi}'\theta = \alpha \leq 0.
This is precisely an arbitrage, contradicting PNA.
Conclusion. The assumption \Psi \cap \mathbb{R}^{S}_{++} = \emptyset leads to an arbitrage, so it must be false. Hence there exists \psi \gg 0 with D'\psi = \boldsymbol{\pi}, and the SDF m(s) = \psi(s)/q(s) > 0 is strictly positive in every state.