Heat Kernel Interest Rate Models with Time-Inhomogeneous Markov Processes

TL;DR

This paper introduces a heat kernel approach for stochastic pricing kernels using time-inhomogeneous Markov processes, enabling analytical bond pricing and interest rate modeling within an information-based asset pricing framework.

Contribution

It develops a novel weighted heat kernel method driven by time-inhomogeneous Markov processes for modeling stochastic pricing kernels and interest rates.

Findings

Provides analytical formulas for bond prices.

Derives explicit interest rate expressions.

Addresses pricing of fixed-income derivatives.

Abstract

We consider a heat kernel approach for the development of stochastic pricing kernels. The kernels are constructed by positive propagators, which are driven by time-inhomogeneous Markov processes. We multiply such a propagator with a positive, time-dependent and decreasing weight function, and integrate the product over time. The result is a so-called weighted heat kernel that by construction is a supermartingale with respect to the filtration generated by the time-inhomogeneous Markov processes. As an application, we show how this framework naturally fits the information-based asset pricing framework where time-inhomogeneous Markov processes are utilized to model partial information about random economic factors. We present examples of pricing kernel models which lead to analytical formulae for bond prices along with explicit expressions for the associated interest rate and market price of risk. Furthermore, we also address the pricing of fixed-income derivatives within this framework.