A Feynman-Kac type formula for a fixed delay CIR model

TL;DR

This paper introduces a fixed delay CIR model using stochastic delay differential equations, proves the existence of a unique solution, and derives a Feynman-Kac type formula for bond pricing with delay-dependent interest rates.

Contribution

It develops a novel fixed delay CIR model and provides a Feynman-Kac formula for bond pricing, extending classical models to include fixed delays in the interest rate dynamics.

Findings

Existence of a unique positive solution for the fixed delay CIR process.

Derivation of a generalized exponential-affine bond pricing formula.

Identification of the affine structure of forward rates with delay-dependent coefficients.

Abstract

Stochastic delay differential equations (SDDE's) have been used for financial modeling. In this article, we study a SDDE obtained by the equation of a CIR process, with an additional fixed delay term in drift; in particular, we prove that there exists a unique strong solution (positive and integrable) which we call fixed delay CIR process. Moreover, for the fixed delay CIR process, we derive a Feynman-Kac type formula, leading to a generalized exponential-affine formula, which is used to determine a bond pricing formula when the interest rate follows the delay's equation. It turns out that, for each maturity time T, the instantaneous forward rate is an affine function (with time dependent coefficients) of the rate process and of an auxiliary process (also depending on T). The coefficients satisfy a system of deterministic delay differential equations.