A Feynman-Kac type formula for a fixed delay CIR model. Issue 4 (4th July 2019)

• Record Type: Journal Article
• Authors: Flore, Federico; Nappo, Giovanna
• Is Part Of: Stochastic analysis and applications. Volume 37:Issue 4(2019)
• Page Start: 550
• Page End: 573
• Publication Date: 2019-07-04
• Subjects: Stochastic delay differential equations -- interest rate model -- equivalent martingale measure -- generalized Bessel-square processes; Primary 91G30 -- Secondary 60H30; Stochastic analysis -- Periodicals; 519.2205
• DOI: 10.1080/07362994.2019.1592691
• Languages: English
• ISSNs: 0736-2994

Abstract

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 differential equations.