Stochastic Discount Factor

Definition

The stochastic discount factor (SDF), also called the pricing kernel or state-price deflator, is a positive random variable $M_t$ such that for any asset paying cash flow $X_t$ at date t:

$P_0 = \sum_{t} E[M_t \cdot X_t]$

$M_t$ encodes both time-discounting and risk adjustment: states where $M_t$ is large (typically low-consumption, high-marginal-utility states) are discounted less, so assets paying more in those states have higher prices.

Key Ideas

Existence: Under the single-price law of markets and nonsatiation, a positive SDF $Y_t$ exists for all dates and states (Rubinstein 1976, Theorem 1). It is unique only when markets are complete (securities span all states).
CPRA and the power kernel: Under constant proportional risk aversion (CPRA) with coefficient $b$ and weak aggregation, $Y_t = R_{Mt}^{-b}$ where $R_{Mt}$ is the gross market return (Rubinstein 1976, Theorem 4). This links the unobservable SDF to an observable market statistic.
Risk premium: From the pricing equation, $E(r_t) = r_F - r_F \cdot \text{Cov}(r_t, M_t)/E(M_t)$ . Assets negatively correlated with $M_t$ (paying well when the market is high / $M_t$ is low) carry a positive risk premium.
Log-CAPM: Under CPRA, for returns jointly lognormal with $R_{Mt}$ : $\ln E(r_i) = \ln r_F + b \cdot \text{Cov}(\ln r_i, \ln r_{Mt})$ . The parameter $b$ can be estimated: $b = [E(\ln r_M) - \ln r_F] / \text{Var}(\ln r_M)$ when market returns are lognormal.
No-arbitrage equivalence (Harrison-Kreps 1979; Harrison-Pliska 1981): Absence of arbitrage in a securities market ↔ existence of an equivalent martingale measure (EMM) $Q$ under which all discounted asset prices are martingales. The SDF is the Radon-Nikodym derivative $dQ/dP$ scaled by the riskless discount factor. This is the unifying foundation of continuous-time finance (Sundaresan 2000).
Habit formation (Sundaresan 1989; Constantinides 1990): When the utility function depends on a habit stock $H_t$ (a moving average of past consumption), effective risk aversion at date $t$ equals $-C_t U_{CC}/(C_t - H_t) \cdot U_C$ , which rises sharply as $C_t \to H_t$ in bad times. The resulting SDF has higher volatility in recessions, which can partially resolve the equity premium puzzle without implausibly high unconditional CRRA.

How It Works

Rubinstein (1976) Framework

Theorem 1 establishes: given assumptions (i) single-price law and (ii) nonsatiation, there exists a positive random variable $Y_t$ such that for any security:

$P_0 = \sum_{t} \frac{E(X_t) + \text{Cov}(X_t, Y_t)/E(Y_t)}{R_{Ft}}$

Setting $M_t = Y_t / [E(Y_t) \cdot R_{Ft}]$ recovers the SDF representation $P_0 = \sum_t E[M_t X_t]$ .

Under CPRA (Theorem 4), the average investor maximizes time-additive power utility, and the first-order conditions yield $Y_t = R_{Mt}^{-b}$ . The proof uses the result that under CPRA, the average propensity to consume is independent of wealth, so per capita consumption is proportional to per capita wealth, making $Y_t$ a function of the market return alone.

Connection to Option Pricing

The normalized kernel $Z'_t = Y_t R_F^{-1} / E(Y_t)$ is the risk-neutral probability kernel. When $S_t$ and $Y_t = R_{Mt}^{-b}$ are jointly lognormal, integration over the bivariate lognormal density yields the Black-Scholes formula — see Black-Scholes Option Pricing.

Stein–Rubinstein Lemma

The key analytic tool for computing SDF-based prices under lognormality:

$\text{Cov}(x, g(y)) = E[g'(y)] \cdot \text{Cov}(x, y) \quad (x, y \text{ bivariate normal, } g \text{ differentiable})$

First proved in Rubinstein's 1971 dissertation; independently discovered by Stein (1973). Used to evaluate expectations of the form $E[S_t \cdot f(Y_t)]$ that appear in option pricing and Intertemporal Capital Asset Pricing Model (ICAPM) derivations.

Why It Matters

Unifies asset pricing theories: Capital Asset Pricing Model (CAPM), ICAPM, Arbitrage Pricing Theory (APT), and consumption-CAPM all correspond to specific parameterizations of $M_t$ . A single framework prices all securities consistently.
Empirically testable: Under CPRA, $M_t = R_{Mt}^{-b}$ — testable using market return data and an estimate of $b$ . The equity premium puzzle (Hansen-Jagannathan 1991) arises because matching observed risk premia requires an implausibly large $b$ .
Foundation for derivatives: Any no-arbitrage formula (Black-Scholes, Hull-White, Heston) can be written as $E[M_t \cdot \text{payoff}]$ for an appropriate $M_t$ . The distributional assumptions on $M_t$ determine the formula's form.

Open Questions

The equity premium puzzle: empirically calibrated equity premia require $b \approx 20$ – $50$ under CPRA, inconsistent with survey estimates of risk aversion ( $\approx 2$ – $4$ ). Habit formation, recursive utility, rare disasters, and parameter uncertainty (Jobert-Platania-Rogers 2006) are proposed resolutions; see Equity Premium Puzzle.
With incomplete markets, $M_t$ is not unique; the minimum-variance SDF (Hansen-Jagannathan) bounds the ratio of the equity premium to volatility without specifying $M_t$ .
Identifying $M_t$ nonparametrically from option prices (implied risk-neutral distribution) versus from consumption data (structural SDF) gives inconsistent estimates — the "SDF puzzle."