[Paper Review] Positive Eigenfunctions of Markovian Pricing Operators: Hansen-Scheinkman Factorization, Ross Recovery and Long-Term Pricing

Likuan Qin, Vadim Linetsky|arXiv (Cornell University)|Nov 12, 2014

Stochastic processes and financial applications63 references17 citations

TL;DR

This paper establishes a spectral theory for Markovian asset pricing models with continuous-time Borel right processes and positive semimartingale stochastic discount factors. It proves the uniqueness of the recurrent positive eigenfunction, extends the Hansen-Scheinkman factorization and Ross Recovery Theorem to general recurrent diffusions, and derives long-maturity asymptotics, ensuring economically stable dynamics by ruling out explosive or trapped short-rate behaviors.

ABSTRACT

This paper develops a spectral theory of Markovian asset pricing models where the underlying economic uncertainty follows a continuous-time Markov process X with a general state space (Borel right process (BRP)) and the stochastic discount factor (SDF) is a positive semimartingale multiplicative functional of X. A key result is the uniqueness theorem for a positive eigenfunction of the pricing operator such that X is recurrent under a new probability measure associated with this eigenfunction (recurrent eigenfunction). As economic applications, we prove uniqueness of the Hansen and Scheinkman (2009) factorization of the Markovian SDF corresponding to the recurrent eigenfunction, extend the Recovery Theorem of Ross (2015) from discrete time, finite state irreducible Markov chains to recurrent BRPs, and obtain the long maturity asymptotics of the pricing operator. When an asset pricing model is specified by given risk-neutral probabilities together with a short rate function of the Markovian state, we give sufficient conditions for existence of a recurrent eigenfunction and provide explicit examples in a number of important financial models, including affine and quadratic diffusion models and an affine model with jumps. These examples show that the recurrence assumption, in addition to fixing uniqueness, rules out unstable economic dynamics, such as the short rate asymptotically going to infinity or to a zero lower bound trap without possibility of escaping.

Motivation & Objective

• To develop a spectral theory for Markovian asset pricing models with general continuous-time Borel right processes and positive semimartingale stochastic discount factors.
• To establish the uniqueness of a positive eigenfunction under which the underlying Markov process is recurrent, ensuring economically meaningful dynamics.
• To extend the Hansen-Scheinkman factorization of the stochastic discount factor to general recurrent Markov processes beyond discrete-time finite-state chains.
• To generalize Ross's Recovery Theorem from discrete to continuous-time, general-state-space Markov processes.
• To derive the long-maturity asymptotic behavior of the pricing operator under the recurrent eigen-measure.

Proposed method

• Formalizes the pricing operator as a semigroup acting on Borel-measurable payoffs, with the stochastic discount factor as a positive multiplicative functional of the Markov process.
• Introduces the concept of an eigen-measure $\mathbb{Q}^\pi$ associated with a positive eigenfunction $\pi$, under which the eigen-security acts as a numeraire.
• Applies Girsanov's theorem to transform the dynamics of the state process under the eigen-measure, ensuring mean-reverting and recurrent behavior when the eigenvalue condition is satisfied.
• Uses the continuous-time algebraic Riccati equation (CTARE) to solve for eigenfunctions in affine and quadratic diffusion models with and without jumps.
• Employs Itô's formula and martingale representation to derive the dynamics of the $\mathbb{Q}^\pi$-local martingale $\tilde{M}^\pi_t$, ensuring the validity of the change of measure.
• Applies results from stochastic processes theory—such as the Feller property, irreducibility, and recurrence criteria (Tweedie’s and Schilling’s theorems)—to verify recurrence under the eigen-measure.

Experimental results

Research questions

• RQ1 Under what conditions does a unique positive eigenfunction exist such that the Markov process is recurrent under the associated eigen-measure?
• RQ2 How can the Hansen-Scheinkman factorization of the stochastic discount factor be extended to continuous-time, general-state-space Markov processes?
• RQ3 Can Ross’s Recovery Theorem be generalized from discrete-time finite-state Markov chains to continuous-time recurrent diffusions?
• RQ4 What are the long-maturity asymptotics of the pricing operator under the recurrent eigen-measure?
• RQ5 What conditions ensure that the short rate does not explode or get trapped at the zero lower bound in affine and quadratic diffusion models?

Key findings

• The recurrent positive eigenfunction is unique under the condition that the Markov process is recurrent under the eigen-measure, which ensures stability and avoids explosive or trapped dynamics.
• The Hansen-Scheinkman factorization is uniquely characterized by the recurrent eigenfunction, extending its validity beyond symmetric or square-integrable settings.
• The Ross Recovery Theorem is extended to continuous-time, general-state-space Borel right processes, allowing the recovery of real-world probabilities from risk-neutral dynamics.
• In affine and quadratic diffusion models, the existence of a recurrent eigenfunction is guaranteed if the associated CTARE has a stabilizing solution, with explicit conditions on the drift and diffusion matrices.
• For the JCIR process with jumps, recurrence is established via positive transition density and the existence of a stationary distribution, confirming topological recurrence and satisfying the recurrence criteria.
• The long-maturity asymptotics of the pricing operator are derived, showing convergence to the projection onto the recurrent eigenfunction, which governs long-term risk-neutral pricing.