Gourieroux-Monfort (2007) Econometric Specification of Stochastic Discount Factor Models

Summary

Gourieroux and Monfort (2007) propose a unified framework for derivative pricing in incomplete discrete-time markets by restricting the stochastic discount factor (SDF) to the exponential-affine (Esscher) class: $M_{t,t+1} = \exp(a_t' r_{t+1} + b_t)$. This convention resolves the identification problem (infinitely many SDFs are no-arbitrage compatible) and uniquely pins down the risk-neutral distribution when the only state variables are traded-asset returns. The framework nests Black-Scholes, GARCH option pricing (Duan 1995), the variance-gamma model (Madan et al. 1998), and affine term structure models (Cox-Ingersoll-Ross, CIR). A semi-parametric pricing algorithm is developed for models with path-dependent drift/volatility but unspecified error distribution.

Key Claims

• The incompleteness problem: In discrete time, markets are generically incomplete — there exist infinitely many SDFs compatible with no-arbitrage (Harrison-Kreps 1979). Pricing an illiquid derivative requires selecting one; this selection is a convention (an identifiability restriction), not identified by the data. The choice is consequential because different SDFs give different derivative prices.

• Exponential-affine convention (Esscher transform): The paper proposes restricting the SDF to: $$M_{t,t+1} = \exp(a_t' r_{t+1} + b_t)$$ where $r_{t+1}$ is the vector of log-returns on traded assets and $a_t, b_t$ are path-dependent (functions of the return history $r_t$). This class:

  • Guarantees positivity (hence validity as a pricing kernel)
  • Is equivalent to the Esscher transform from actuarial science (Esscher 1932; Gerber-Shiu 1994)
  • Nests the major equilibrium SDF specifications: Consumption Capital Asset Pricing Model (CCAPM; power utility), constant absolute risk aversion (CARA) utility, Epstein-Zin recursive utility, and the discrete-time approximation of stochastic volatility models (Hull-White 1987, Heston 1993)

• Proposition 1 (uniqueness; Section 3): For a zero riskfree rate and exponential-affine SDF with state variable $r_{t+1}$, there exists a generically unique admissible risk-neutral distribution with conditional log-Laplace transform: $$\log E^Q[\exp(u' r_{t+1}) | r_t] = c_t(a_t + u;\theta) - c_t(a_t;\theta)$$ where $c_t(u;\theta) = \log E[\exp(u'r_{t+1})|r_t]$ is the conditional log-Laplace (moment generating) transform of the historical distribution, and $a_t$ is the unique solution to: $$c_t(a_t + e_j;\theta) = c_t(a_t;\theta), \quad j = 1,\ldots,J$$ (one constraint per traded asset). The key result: a single convention (exponential-affine SDF) is sufficient to complete the market and pin down all derivative prices, without specifying a full equilibrium model.

• Conditionally Gaussian case (Section 4.1): When $r_{t+1}|r_t \sim N(\mu_t(\theta), \Sigma_t(\theta))$: $$a_t = -\Sigma_t(\theta)^{-1}\left[\mu_t(\theta) + \tfrac{1}{2}\text{vdiag}(\Sigma_t(\theta))\right]$$ The risk-neutral distribution is also Gaussian with the same conditional covariance — only the mean shifts. Nests Duan (1995) GARCH option pricing and Black-Scholes. The Jensen-term correction $\frac{1}{2}\text{vdiag}(\Sigma_t)$ distinguishes the result from the Hansen-Jagannathan (1997) affine SDF approach.

• Variance-gamma model (Section 4.2): The time-deformation parameter $\nu_t$ is invariant under the risk correction; only the drift $\mu_t$ and scale $\sigma_t^2$ change. Both historical and risk-neutral distributions belong to the variance-gamma family.

• Semi-parametric pricing (Section 4.3): When $r_{t+1} = \mu_t + \sigma_t\epsilon_{t+1}$ with parametric drift/volatility but unspecified error distribution $F_\epsilon$, a 5-step algorithm:

  1. Calibration: Estimate $\theta = (\mu_t, \sigma_t)$ by quasi-maximum-likelihood estimation (QMLE; Gaussian quasi-log-likelihood)
  2. Error distribution: Compute residuals $\hat{\epsilon}_t$; the empirical distribution $\hat{P} = (T-1)^{-1}\sum_t \delta_{\hat{\epsilon}_t}$ consistently estimates $F_\epsilon$
  3. Risk correction: Solve $\sum_t \exp(\hat{s}_t\hat{a}_t\hat{\epsilon}_t)[\exp(\hat{m}_t + \hat{s}_t\hat{\epsilon}_t) - 1] = 0$ for $\hat{a}_t$
  4. SDF estimation: $\hat{M}_{t,t+1} = \exp[\hat{a}_t(r_{t+1} - \hat{\mu}_t)] / \hat{c}_T(\hat{s}_t\hat{a}_t)$
  5. Pricing: Price of derivative $g$ is $\hat{C}_T(g) = \frac{\sum_{t} g(\hat{\mu}_T + \hat{\sigma}_T\hat{\epsilon}_t)\exp(\hat{a}_T\hat{\sigma}_T\hat{\epsilon}_t)}{\sum_{t}\exp(\hat{a}_T\hat{\sigma}_T\hat{\epsilon}_t)}$

• Proposition 3 (residual incompleteness; Section 5): When the investor's information also includes unobservable factors $f_t$ and real-sector variables $x_t$, plus a stochastic riskfree rate $r^f_{t+1}$, the exponential-affine convention no longer pins down a unique risk-neutral distribution. The residual indeterminacy has dimension $K + L + 1$ (= #latent factors + #real variables + 1 for the future riskfree rate). Risk premia on non-traded state variables ($D_t$) can be freely chosen.

• CIR extension (Section 5.5, Proposition 4): Using an autoregressive gamma (AR-Gamma) process for the riskfree rate — the discrete-time analogue of Cox-Ingersoll-Ross (CIR 1985) — multi-period bond derivative prices satisfy closed-form recursive equations: $$C_t(u,h) = \exp[a(h,u)r^f_{t+1} + b(h,u)]$$ where $a(h,u)$ and $b(h,u)$ satisfy a rational recursive equation. Standard CIR = special case with zero risk premium ($a_0 = 0$). Closed-form solution given by ratios of geometric series in $a(h,u)/g_1$ and $a(h,u)/g_2$ where $g_1, g_2$ are roots of a second-degree polynomial.

• Information set and time-horizon dependence: The exponential-affine SDF property is NOT preserved under (i) marginalisation over a subset of assets, or (ii) time aggregation to a longer horizon. The convention must be applied at the correct information set and time unit.

Concepts Introduced or Extended

• Stochastic Discount Factor
• Affine Term Structure Model
• Black-Scholes Option Pricing
• Risk-Neutral Measure

Entities Mentioned

• Christian Gourieroux
• Alain Monfort

Quotes

"Since the market is incomplete in discrete time, there exists a multiplicity of [stochastic discount factors] that are compatible with the valuation formula… In our framework the conventions consist in restricting a priori the set of admissible stochastic discount factors." (pp. 509–510)

"The convention is simply an identifiability restriction. However the selection of the identifiability restriction is not innocuous since the pricing of other financial assets involves the whole sdf, not only its identifiable components." (p. 511, fn. 2)

My Take

The paper's key contribution is framing the incomplete-market pricing problem as a statistical identification problem: the SDF is not point-identified by traded asset prices, so any pricing convention is an identifying restriction in the same sense that a normalization is in a structural vector autoregression (VAR). The exponential-affine/Esscher choice is natural because (a) it preserves positivity, (b) it is implied by most equilibrium models, and (c) it gives closed-form results under Gaussianity. The semi-parametric algorithm in Section 4.3 is an elegant way to separate the parametric (drift/volatility) and nonparametric (error distribution) components while still recovering a usable risk-neutral distribution. The extension to stochastic interest rates (Proposition 4) is technically clean. The main gap is empirical: the paper is entirely theoretical, and the semi-parametric algorithm's performance relative to parametric competitors (Black-Scholes, GARCH option pricing) is not evaluated. The connection to Affine Term Structure Models via the AR-Gamma process deserves further development (see Gourieroux-Monfort-Polimenis 2002a).