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.

A Feynman-Kac type formula for a fixed delay CIR model / Flore, Federico; Nappo, Giovanna. - In: STOCHASTIC ANALYSIS AND APPLICATIONS. - ISSN 0736-2994. - ELETTRONICO. - (2019). [10.1080/07362994.2019.1592691]

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

Giovanna Nappo
2019

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.
2019
Stocastic delay differential equations; generalized exponential-affine formula; bond price; interest rate model; equivalent martingale measure; generalized Bessel-square processes
01 Pubblicazione su rivista::01a Articolo in rivista
A Feynman-Kac type formula for a fixed delay CIR model / Flore, Federico; Nappo, Giovanna. - In: STOCHASTIC ANALYSIS AND APPLICATIONS. - ISSN 0736-2994. - ELETTRONICO. - (2019). [10.1080/07362994.2019.1592691]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1112698
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