The Stochastic Discount Factor

Lorenzo Naranjo

Fall 2026

Introduction

The Main Idea

• The stochastic discount factor emerges naturally as a consequence of the law of one price.
  • We do not need to make assumptions on preferences
  • Mathematically, is a consequence of the Riesz representation theorem on Euclidean spaces.
• The absence of arbitrage opportunities implies the existence of a strictly positive discount factor.
• In complete markets there is a unique strictly positive stochastic discount factor.
• The presentation follows Chapter 4 in Cochrane (2009) closely.
  • 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

The 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.

Creating New Payoffs

• 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.
• We denote by \pi_{i} the price of asset i for 1 \leq i \leq N, and by \pmb{\pi} = (\pi_{1}, \pi_{2}, \ldots, \pi_{n})' a vector containing the prices of the N basis payoffs.

The Law of One Price

Creating a Pricing Functional

• We want 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.
• 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.

The Law of One Price (LOOP)

• 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 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.

Implications of LOOP

• The law of one price implies the price functional defined in (3) is a continuous linear functional, and the Riesz representation theorem implies the existence of a stochastic discount factor x^{*} that is also a payoff in X.
• Therefore, it must be the case that x^{*} = \sum_{i = 1}^{N} c_{i} x_{i} = \mathbf{c}' \mathbf{x}, such that p(x_{i}) = \operatorname{E}(x^{*} x_{i}) for 1 \leq i \leq N.

Solving for the SDF

• We should be able to price all assets using x^{*} \pmb{\pi}' = \operatorname{E}(x^{*} \mathbf{x}') = \operatorname{E}(\mathbf{c}' \mathbf{x} \mathbf{x}') = \mathbf{c}' \operatorname{E}(\mathbf{x} \mathbf{x}'), or \mathbf{c}' = \pmb{\pi}' \operatorname{E}(\mathbf{x} \mathbf{x}')^{-1}.
• Thus, x^{*} = \pmb{\pi}' \operatorname{E}(\mathbf{x} \mathbf{x}')^{-1} \mathbf{x} \tag{4} is a valid discount factor.

Creating Other SDFs

• If the market is complete, there is only one SDF given by x^{*}.
• If the market is incomplete, it means that there are vectors in L orthogonal to all vectors in X.
• If there is another SDF m that prices the assets correctly, it must be the case 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 X. m = x^{*} + e, where e \mathrel\bot x for all x \in X.

An Alternative SDF

• If a risk-free asset is available, we can try to find an SDF such that m = \operatorname{E}(m) + (\mathbf{x} - \operatorname{E}(\mathbf{x}))' \mathbf{c}.
• Again, we must have that \begin{aligned} \pmb{\pi}' & = \operatorname{E}(m \mathbf{x}') = \operatorname{E}((\operatorname{E}(m) + \mathbf{c}' (\mathbf{x} - \operatorname{E}(\mathbf{x}))) \mathbf{x}') \\ & = \operatorname{E}(m) \operatorname{E}(\mathbf{x}') + \mathbf{c}' \mathbf{V}, \end{aligned} where \mathbf{V} denotes the covariance matrix of \mathbf{x}.

Solving for the SDF

• Solving for \mathbf{c} yields \mathbf{c}' = (\pmb{\pi}' - \operatorname{E}(m) \operatorname{E}(\mathbf{x}')) \mathbf{V}^{-1}, so that m = \operatorname{E}(m) + (\mathbf{x} - \operatorname{E}(\mathbf{x}))' \mathbf{V}^{-1} (\pmb{\pi} - \operatorname{E}(m) \operatorname{E}(\mathbf{x})). \tag{5}
• The previous expression is particularly useful when using zero-cost portfolios.
  • In (4), if \pmb{\pi} is a vector of zeros, then x^{*} = 0.

Using Returns as Payoffs

• Denote by \mathbf{R}^{e} = \mathbf{R} - \pmb{\iota} R^{f}, where \pmb{\iota} is a conformal vector of ones.
• We can rewrite (6) as m^{*} = \frac{1}{R^{f}} \left(1 - (\mathbf{R} - \operatorname{E}(\mathbf{R}))' \mathbf{V}^{-1} \operatorname{E}(\mathbf{R}^{e}) \right). \tag{6}
• We will see later that the weights of a portfolio that achieves the highest Sharpe ratio are given by \mathbf{w} = \mathbf{V}^{-1} \operatorname{E}(\mathbf{R}^{e}). \tag{7}
  • Note that 1 - \pmb{\iota}' \mathbf{V}^{-1} \operatorname{E}(\mathbf{R}^{e}) is invested in the risk-free asset.

An SDF as a Traded Portfolio

• The SDF defined in (6) is proportional to the returns of the portfolio that achieves the highest Sharpe ratio!
  • This is a consequence of the Hansen and Jagannathan (1991) duality.
• We want to use portfolios that we know perform well in terms of high Sharpe ratio
  • Diversification matters.
• Any trading strategy that outperforms according to an existing model should be added
  • This is why we use multifactor models to measure portfolio performance such as the ones proposed by Fama and French (1993) and Fama and French (2015), for example.

Using Zero-Cost Portfolios

• Let \mathbf{z} be a vector of zero-cost portfolios such that \pmb{\pi} is a vector of zeros.
• If R^{f} \in X, we can re-write (6) as z^{*} = \frac{1}{R^{f}} \left(1 - (\mathbf{z} - \operatorname{E}(\mathbf{z}))' \mathbf{V}^{-1} \operatorname{E}(\mathbf{z})\right). \tag{8}
• In the previous expression, z^{*} = x^{*} as long R^{f} \in X, which is usually the case in empirical applications.

Estimating the SDF Dynamically

• Let’s not forget that equation (8) depends on time: z^{*}_{t+1} = \frac{1}{R^{f}_{t}} \left(1 - (\mathbf{z}_{t+1} - \operatorname{E}_{t}(\mathbf{z}_{t+1}))' \mathbf{V}_{t}^{-1} \operatorname{E}_{t}(\mathbf{z}_{t+1})\right). where \mathbf{V}_{t} = \operatorname{E}_{t}(\mathbf{z}_{t+1}\mathbf{z}_{t+1}') - \operatorname{E}_{t}(\mathbf{z}_{t+1}) \operatorname{E}_{t}(\mathbf{z}_{t+1}').
• Modern asset pricing boils down to:
  • Determining the zero-cost portfolios that achieve the highest Sharpe ratio at each point in time.
  • Getting a good estimate of \operatorname{E}_{t}(\mathbf{z}_{t+1}), both in terms of conditioning variables (predictors) anb techniques (machine learning).
  • Obtaining a precise estimate of \mathbf{V}_{t}.

The Principle of No-Arbitrage

Arbitrage Opportunities

• A violation to the law of one price is an arbitrage opportunity, but not all arbitrage opportunities are violations of the law of one price.
• There might be situations in which some investors manage to build a payoff that is positive in some states and zero in others.
• In competitive financial markets, the price of that payoff must be positive, otherwise the demand for that asset would be infinite.

Assumption 2 (Principle of No-Arbitrage) The price of a payoff that is positive in all states and strictly positive in at least one state of the world must be positive.

PNA Implies LOOP

• We have the following relationship between PNA and LOOP.

Property 1 \text{PNA} \Rightarrow \text{LOOP}.

• LOOP implies that the price of a zero payoff must be zero.
  • A violation of LOOP implies that p(0) > 0.
• PNA implies that the price of a payoff x^{+} \in X that is positive in some states of the world and zero otherwise must be positive.

Proof of Property 1

• We can prove this claim by contradiction by assuming that NA holds but not LOOP.
  
• A violation of LOOP implies that p_{0} = p(0) > 0.
• Say that we have a payoff x^{+} that is positive in some states of the world and zero otherwise, and whose price is 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 positive in some states and zero otherwise, a contradiction.

Strictly Positive Discount Factors

• A more important consequence of PNA is that it implies the existence of a strictly positive SDF, i.e., a SDF that is greater than zero in all states.
• The reverse is also true.
  • There are no arbitrage opportunities if there is one SDF that is strictly positive.

Property 2 \text{PNA} \Leftrightarrow \exists m > 0.

• The proof of \exists m > 0 \Rightarrow \text{PNA} can be done by contradiction.
• To prove \text{PNA} \Rightarrow \exists m > 0 requires the use of the Separating Hyperplane Theorem, see e.g. Duffie (2010).

References

References

Cochrane, John. 2009. Asset Pricing: Revised Edition. Princeton university press.

Duffie, Darrell. 2010. Dynamic Asset Pricing Theory. Princeton University Press.

Fama, Eugene F., and Kenneth R. French. 1993. “Common Risk Factors in the Returns on Stocks and Bonds.” Journal of Financial Economics 33 (1): 3–56.

Fama, Eugene F., and Kenneth R. French. 2015. “A Five-Factor Asset Pricing Model.” Journal of Financial Economics 116 (1): 1–22.

Hansen, Lars Peter, and Ravi Jagannathan. 1991. “Implications of Security Market Data for Models of Dynamic Economies.” Journal of Political Economy 99 (2): 225–62.

Hansen, Lars Peter, and Ravi Jagannathan. 1997. “Assessing Specification Errors in Stochastic Discount Factor Models.” Journal of Finance 52 (2): 557–90.

Hansen, Lars Peter, and Scott F Richard. 1987. “The Role of Conditioning Information in Deducing Testable Restrictions Implied by Dynamic Asset Pricing Models.” Econometrica 55 (3): 587–613.