Semigroup Approach to Asset Pricing

Definition

Asset valuation is represented as a one-parameter family of linear operators $\{{\mathbb{M}}_t{\}}_{t\ge 0}$ acting on payoffs that depend on a Markov state $X$:

${\mathbb{M}}_t\psi (x)=E[M_t\psi (X_t)\mid X_0=x],$

where $M$ is a multiplicative functional ($M_0=1,M_{t+u}=M_u({\theta }_t)M_t$). The multiplicative property combined with the Markov property of $X$ implies the semigroup identity

${\mathbb{M}}_0=I,{\mathbb{M}}_{t+u}={\mathbb{M}}_t{\mathbb{M}}_u,$

which is equivalent to the law of one price for valuation across intermediate trading dates. Differentiating ${\mathbb{M}}_t$ at $t=0$ defines an (extended) generator $\mathbb{A}$ that captures the local instantaneous behavior of the operator and that turns long-horizon questions about ${\mathbb{M}}_t$ into eigenvalue/PDE questions about $\mathbb{A}$.

Intuition

A semigroup is “the natural language for time-consistent pricing.” Pricing-by-iteration (price today = expected discounted price tomorrow) is exactly the composition rule ${\mathbb{M}}_{t+u}={\mathbb{M}}_t{\mathbb{M}}_u$, and absence of arbitrage at intermediate dates is exactly the multiplicative property of the SDF. Once you frame valuation this way, all the machinery of operator semigroups (generators, spectra, perturbation, change-of-measure) becomes available, and you can ask sharp questions about long-horizon limits that have no analog in single-period pricing.

Formal notation

A continuous-time Markov process $X$ on $D_0\subset {\mathbb{R}}^n$ generates pricing operators

${\mathbb{M}}_t\psi (x)=E[M_t\psi (X_t)\mid X_0=x]$

with extended generator

$\mathbb{A}\varphi ={\lim }_{t\to 0^{+}}\frac{{\mathbb{M}}_t\varphi -\varphi }{t}$

(defined on a suitable domain). For exponential-affine setups, $\mathbb{A}$ is a second-order differential operator plus a jump term, and the eigenvalue problem $\mathbb{A}\varphi =\rho \varphi$ becomes a system of ODEs/algebraic equations on the coefficients of $\log \varphi$.

Variants

• Garman (1984) semigroups: First semigroup formulation of asset pricing; allowed non-Markov environments via “evolution semigroups.” Hansen-Scheinkman specialize to the Markov case for tractability.
• Markov-chain semigroup: For a finite-state chain with intensity matrix $\mathbb{U},{\mathbb{M}}_t=e^{t\mathbb{A}}$ where $\mathbb{A}$ is built from $\mathbb{U}$ and the multiplicative functional generator. Long-run dominance reduces to standard Perron-Frobenius theory.
• Affine-state semigroup: For Feller-square-root + Ornstein-Uhlenbeck states with exponential-affine multiplicative functionals (Hansen-Scheinkman Example 6.2), the generator action on $\exp (c^{\prime }x)$ is affine in $x$, so $(\rho ,c)$ solve a closed-form quadratic + linear system.
• Regime-switching extension: When the Markov state contains a discrete regime indicator (as in the CRE project’s monetary-policy × wage-rigidity 4-state compound regime), the semigroup acts on regime-conditional payoff functions and the eigenvalue problem becomes block-structured.

Comparison

• vs Stochastic differential equations: SDE formulations characterize local evolution; the semigroup formulation characterizes operators and is the natural setting for studying horizon-indexed objects.
• vs Risk-neutral measure: Standard risk-neutral pricing changes measure once and prices forward; the semigroup approach indexes operators by horizon and lets the spectral structure of the operator family carry the long-horizon information.
• vs Backward induction trees: Trees evaluate ${\mathbb{M}}_t\psi$ at one $t$ at a time; the semigroup view treats the whole family $\{{\mathbb{M}}_t\}$ as the object of study, which is what enables principal-eigenvalue analysis.

When to use

• Long-horizon asset pricing where the term-structure of risk prices is the object of interest.
• Constructing distorted probability measures via change-of-numeraire / change-of-measure operations.
• Connecting local (instantaneous) and global (long-horizon) risk-return trade-offs in a single operator framework.
• Spectral / eigenvalue analysis of pricing operators (term structure of bond yields, long-run growth-portfolio analysis).

Known limitations

• Requires a Markov state assumption: genuinely non-Markov environments need either Garman-style evolution semigroups (which sacrifice tractability) or auxiliary state augmentation.
• Generator-based methods need the generator’s domain to be characterized, which can be technical for jump-diffusion processes.
• Numerical computation of operator spectra in nonlinear settings is a hard problem with no general-purpose solution.

Open problems

• Numerical algorithms for the principal eigenpair in non-affine, non-finite-state environments.
• Extension to multiplicative functionals built from infinite-activity Lévy processes (Hansen-Scheinkman conclusions, open problem).
• Quantitative use of subdominant eigenvalues for refined finite-horizon approximations.

Key papers

• hansen-scheinkman-2009-long-term-risk-operator-approach — full operator-theoretic treatment, principal eigenvalue/eigenfunction characterization, multiplicative factorization, existence and uniqueness, long-run dominance.

My understanding

The CRE asset pricing project’s operator pipeline (compute_quadratic_pricing_factors_msvar → get_Q_quadratic_pricing_factors → approximate_Q_cond_x) is a finite-dimensional realization of the Hansen-Scheinkman semigroup approach: each step composes pricing operators across regimes and horizons, and the Riccati recursion at the heart of compute_quadratic_pricing_factors_msvar is the discrete-time analog of the affine generator action on $\exp (c^{\prime }x)$. The reason Apartment is the “slow asset” requiring $T_{\stackrel{ˉ}{}}\approx 50K$ is exactly that the spectral gap between the dominant and subdominant eigenvalues of its pricing operator is small, so the semigroup takes many iterations to project onto the principal eigenfunction. Operator-floor null hypotheses such as N3 (the geometric tail ${\eta }_{H}r/(1-r)$ with a single global $r$) are bets that a single exponential-rate term dominates the semigroup decomposition; the Hansen-Scheinkman framework is the right place to ask whether that bet is justified.