Summary
Sundaresan surveys 30 years of continuous-time finance (1969–1999), organized around two eras: the foundational breakthroughs of 1969–1980 (Merton's portfolio theory and option pricing, Harrison-Kreps martingale representation, Cox-Ingersoll-Ross (CIR)/Vasicek term structure) and the post-1980 expansion into richer dynamics and formal estimation. The unifying mathematical thread is the Harrison-Kreps (1979) / Harrison-Pliska (1981) no-arbitrage equivalence, which established that absence of arbitrage ↔ existence of an equivalent martingale measure (EMM). The review covers affine jump-diffusion (AJD) models as the tractable workhorse for term structure and derivatives, six methods for estimating continuous-time Markov models, and partial resolutions of the equity premium puzzle via habit formation.
Key Claims
- Harrison-Kreps (1979) martingale representation is the unifying foundational theorem: in a complete, frictionless market, no-arbitrage ↔ existence of an equivalent martingale measure; asset pricing reduces to Q-expectation discounting under this measure.
- The affine jump-diffusion (AJD) framework (Duffie-Kan 1996) is the tractable workhorse for term structure and derivatives pricing: bond yields are affine in state variables, the characteristic function has exponential-affine form, and closed-form (or near-closed-form) prices are available for bonds and options.
- The CIR (1985b) square-root model is the canonical single-factor AJD term structure model; multi-factor extensions (Duffie-Kan 1996) preserve the affine-yield property while allowing richer volatility and risk-premium dynamics.
- Stochastic volatility models alone (Heston 1993) explain the volatility smile across strikes but fail to match its term structure (the rate at which the smile flattens with maturity); adding jumps in returns partially corrects this, but neither specification alone fits all dimensions simultaneously.
- Six estimation methods for continuous-time Markov models: (1) Generalized Method of Moments (GMM) using infinitesimal generator eigenfunctions (Hansen-Scheinkman 1995); (2) Simulated Method of Moments (SMM) on simulated paths (Duffie-Singleton 1993); (3) Efficient Method of Moments/indirect inference matching Semi-Nonparametric (SNP) score functions (Gallant-Tauchen 1996); (4) Maximum Likelihood (ML) via closed-form transition density approximation (Aït-Sahalia 1999); (5) characteristic function (CF)-based GMM for AJD (Singleton 2001); (6) Markov Chain Monte Carlo (MCMC)/Bayesian (Jacquier-Polson-Rossi 1994).
- Habit formation (Sundaresan 1989; Constantinides 1990) generates time-varying risk aversion that can partially explain the observed equity premium without implausibly high Constant Relative Risk Aversion (CRRA); the habit stock modifies the Stochastic Discount Factor (SDF) so that effective risk aversion rises in bad times when consumption is close to the habit level.
- Two open problems as of 2000: endogenous financial contracts are analytically tractable only for single-state-variable economies; extending to ≥2 state variables remains unsolved. Liquidity, market microstructure, and contagion are not satisfactorily handled by standard continuous-time methods.
Concepts Introduced or Extended
Entities Mentioned
Quotes
"The Harrison-Kreps (1979) and Harrison-Pliska (1981) results establish that the absence of arbitrage in a securities market is equivalent to the existence of an equivalent martingale measure."
"The Duffie-Kan (1996) affine yield factor model… provides tractable closed-form solutions for bond prices as exponential-affine functions of a vector of state variables."
"Habit formation models of Sundaresan (1989) and Constantinides (1990) imply time-varying risk aversion that can potentially explain the equity premium puzzle."
My Take
An authoritative presidential address survey that serves as a bridge between 1970s foundational theory and late-1990s estimation practice. The comparative review of six estimation methods is useful as a decision guide: GMM/SMM are distribution-free but moment-selection-dependent; ML via Aït-Sahalia density expansion is elegant but relies on series approximation accuracy; CF-GMM is tailored to AJD but requires the characteristic function in closed form; MCMC is the most flexible but computationally demanding. The habit formation discussion is brief relative to the space devoted to term structure and derivatives. The volatility smile vs. smile term structure failure of pure Stochastic Volatility (SV) (without jumps) is presented clearly. Open problems listed remain largely open in 2026.