Principal Eigenfunction of a Pricing Operator

Definition

Given a one-parameter family of pricing operators $\{{\mathbb{M}}_t{\}}_{t\ge 0}$ acting on functions of an underlying Markov state $X$, a principal eigenfunction is a strictly positive function $\varphi$ together with a real eigenvalue $\rho$ satisfying

${\mathbb{M}}_t\varphi (x)=e^{\rho t}\varphi (x)$

for every $t\ge 0$. The pair $(\rho ,\varphi )$ is called a principal eigenpair. The eigenvalue $\rho$ measures the long-run continuously-compounded growth (or, when negative, decay) rate of the semigroup; $\varphi$ encodes its limiting state-dependence.

Intuition

If you scale the pricing operator by $e^{-\rho t}$ to remove the deterministic exponential trend, the rescaled semigroup converges to a rank-one operator whose range is spanned by $\varphi$. Intuitively, $\varphi$ is the “long-run shape function”: no matter what payoff $\psi$ you start with, after a long horizon ${\mathbb{M}}_t\psi \approx (\text{const})\cdot e^{\rho t}\cdot \varphi (x)$. This is the asset-pricing analog of a Perron-Frobenius eigenvector for nonnegative matrices: the dominant eigenvalue governs long-run growth, and the dominant eigenvector governs long-run direction.

Formal notation

Hansen and Scheinkman (2009) use the operator pair

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

where $\mathbb{A}$ is the extended generator of the multiplicative semigroup obtained by differentiating ${\mathbb{M}}_t$ at $t=0$. The eigenvalue equation in generator form is the more practical one, since it usually reduces to a PDE or a finite linear system.

Variants

• Affine-state closed form (Hansen-Scheinkman Example 6.2): When the state is a Feller-square-root + Ornstein-Uhlenbeck process and the multiplicative functional is exponential-affine, $\varphi (x)=\exp (c_f{x}_f+c_o{x}_o)$ and $(c_f,c_o,\rho )$ solve a quadratic + linear system. The quadratic equation has two roots; only one yields a stochastically stable distortion.
• Markov-chain version: For a finite-state chain with intensity matrix $\mathbb{U}$ and multiplicative functional generator $\mathbb{A}$, the principal eigenfunction is the strictly positive eigenvector of $\mathbb{A}$ with the largest real-part eigenvalue (classical Perron-Frobenius).
• Smallest admissible eigenvalue: When several positive eigenfunctions exist, the relevant one (Proposition 7.2 of Hansen-Scheinkman) corresponds to the smallest eigenvalue whose associated martingale induces a stationary distorted process.

Comparison

• vs Subdominant eigenvalues: The subdominant eigenvalues control the rate of approach to the long-run limit, but they do not appear in the long-run yield $\rho$ itself. Hansen-Scheinkman explicitly call out the use of subdominant eigenvalues for refined approximations as an open direction.
• vs Local risk-neutral pricing: Local risk-neutral pricing characterizes the $t\to 0$ end of the term structure of risk prices; the principal eigenpair characterizes the $t\to \infty$ end.

When to use

• Computing long-horizon discount rates and cash-flow risk prices in nonlinear continuous-time Markov asset-pricing models.
• Decomposing a stochastic discount factor into permanent (martingale) and transient components for term-structure analysis.
• Constructing the appropriate change of measure under which a Markov process is stationary, for use in long-horizon Monte Carlo.

Known limitations

• No general numerical algorithm: closed forms exist for affine and a few other special cases. For generic nonlinear models, computing $(\rho ,\varphi )$ remains a hard problem.
• Multiplicity: more than one positive eigenfunction can exist, and selecting the stochastically stable one requires checking the drift of the associated distorted process.
• The whole framework presumes a Markov state summarizes valuation-relevant information.

Open problems

• General-purpose numerical algorithms for $(\rho ,\varphi )$ in non-affine, nonlinear, possibly regime-switching environments.
• Sharp characterization of the convergence rate $\eta$ at which the rescaled semigroup approaches its long-run limit.
• Use of subdominant eigenpairs to construct corrections that improve finite-horizon approximations.

Key papers

• hansen-scheinkman-2009-long-term-risk-operator-approach — foundational existence/uniqueness, multiplicative factorization, and applications to long-run risk-return trade-offs.

My understanding

In the CRE asset pricing project, the asymptotic limit $A\to -\infty$ in the Riccati recursion of compute_quadratic_pricing_factors_msvar is the project’s numerical realization of the principal eigenvalue equation in exponential-quadratic coefficient form: at long horizons the recursion’s $A_t$ slope is exactly $\rho$. The “Apartment is the slow asset (needs $T_{\stackrel{ˉ}{}}\approx 50K$ at the new MAP)” gotcha is a statement about the convergence rate $\eta$ — the slow asset has a small spectral gap between dominant and subdominant eigenvalues. Any pricing approximation that replaces a long Riccati run by a closed-form geometric tail (the ${\eta }_{H}r/(1-r)$ operator floor) is implicitly betting that a single dominant eigenvalue captures the relevant long-run dynamics; the leading Exp 12 follow-up hypothesis N3 is exactly that this single-rate assumption is too coarse for some assets.