Convergence rates of large-time sensitivities with the Hansen--Scheinkman decomposition

TL;DR

This paper analyzes the large-time behavior of sensitivities of cash flows in finance using Hansen--Scheinkman decomposition, providing explicit convergence rates and applications to popular market models.

Contribution

It introduces a PDE approach incorporating Hansen--Scheinkman decomposition to derive detailed convergence rates of sensitivities in large-time limits.

Findings

Explicit convergence rates for sensitivities are derived.

Applications to utility maximization, risk measures, and bond pricing are demonstrated.

Explicit results for CIR, 3/2, and CEV models are provided.

Abstract

This paper investigates the large-time asymptotic behavior of the sensitivities of cash flows. In quantitative finance, the price of a cash flow is expressed in terms of a pricing operator of a Markov diffusion process. We study the extent to which the pricing operator is affected by small changes of the underlying Markov diffusion. The main idea is a partial differential equation (PDE) representation of the pricing operator by incorporating the Hansen--Scheinkman decomposition method. The sensitivities of the cash flows and their large-time convergence rates can be represented via simple expressions in terms of eigenvalues and eigenfunctions of the pricing operator. Furthermore, compared to the work of Park (Finance Stoch. 4:773-825, 2018), more detailed convergence rates are provided. In addition, we discuss the application of our results to three practical problems: utility maximization, entropic risk measures, and bond prices. Finally, as examples, explicit results for several market models such as the Cox--Ingersoll--Ross (CIR) model, 3/2 model and constant elasticity of variance (CEV) model are presented.