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

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

Stochastic processes and financial applications19 citations

TL;DR

This paper establishes a spectral theory for Markovian asset pricing models with continuous-time Markov processes and positive semimartingale stochastic discount factors, proving the uniqueness of a recurrent positive eigenfunction. It extends the Hansen-Scheinkman factorization and Ross's Recovery Theorem to general Borel right processes, 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 (2013) 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 under general continuous-time Markov processes with positive semimartingale stochastic discount factors.
• To establish the uniqueness of a positive eigenfunction that ensures recurrence of the state process under a new probability measure.
• To extend the Hansen-Scheinkman factorization and Ross's Recovery Theorem from discrete finite-state chains to general Borel right processes.
• To derive long-maturity asymptotics of the pricing operator under the recurrence condition.
• To provide sufficient conditions and explicit examples for the existence of recurrent eigenfunctions in affine, quadratic, and jump-diffusion models.

Proposed method

• Uses the theory of Borel right processes (BRPs) to model continuous-time Markov processes with general state spaces.
• Applies the concept of multiplicative functional semimartingales to represent the stochastic discount factor as a function of the state process.
• Establishes the existence and uniqueness of a positive eigenfunction of the pricing operator under the condition that the process is recurrent under the measure induced by this eigenfunction.
• Employs spectral theory for Markovian pricing operators to analyze long-maturity asymptotics and factorization structure.
• Derives sufficient conditions for the existence of recurrent eigenfunctions in affine and quadratic diffusion models, including those with jumps.
• Uses the eigenfunction to define a new probability measure under which the state process is recurrent, enabling recovery of physical measures from risk-neutral dynamics.

Experimental results

Research questions

• RQ1 Under what conditions does a unique positive eigenfunction exist for a Markovian pricing operator such that the underlying process is recurrent under the associated measure?
• RQ2 Can the Hansen-Scheinkman factorization of the stochastic discount factor be uniquely extended to continuous-time Markov processes beyond discrete finite-state chains?
• RQ3 How can Ross's Recovery Theorem be generalized from discrete, finite, irreducible Markov chains to general Borel right processes with general state spaces?
• RQ4 What are the long-maturity asymptotic properties of the pricing operator under the recurrence condition?
• RQ5 What conditions ensure the existence of a recurrent eigenfunction in affine, quadratic, and jump-diffusion models?

Key findings

• The positive eigenfunction of the pricing operator is uniquely determined when the underlying Markov process is recurrent under the measure induced by this eigenfunction.
• The Hansen-Scheinkman factorization of the stochastic discount factor is uniquely characterized by the recurrent eigenfunction, extending its validity beyond discrete models.
• Ross's Recovery Theorem is generalized to continuous-time, general-state-space Markov processes, enabling the recovery of the physical measure from risk-neutral dynamics.
• The long-maturity asymptotics of the pricing operator are characterized by the dominant eigenvalue and eigenfunction, with convergence rates determined by the spectral gap.
• In affine and quadratic diffusion models, sufficient conditions for the existence of a recurrent eigenfunction are derived, and explicit examples are constructed that avoid explosive or trapped dynamics such as the short rate diverging or hitting a zero lower bound without escape.
• The recurrence assumption ensures economically stable dynamics by excluding scenarios where the short rate becomes unbounded or trapped, thus providing a structural foundation for reliable recovery and pricing.