Closed form solution to zero coupon bond using a linear stochastic delay differential equation

Alet Roux; Álvaro Guinea Julia

doi:10.48550/arXiv.2402.16428

Closed form solution to zero coupon bond using a linear stochastic delay differential equation

Alet Roux, Álvaro Guinea Julia^*

^*Corresponding author for this work

Mathematics

Research output: Working paper › Preprint

Abstract

We present a short rate model that satisfies a stochastic delay differential equation. The model can be considered a delayed version of the Merton model (Merton 1970, 1973) or the Vasi\v{c}ek model (Vasi\v{c}ek 1977). Using the same technique as the one used by Flore and Nappo (2019), we show that the bond price is an affine function of the short rate, whose coefficients satisfy a system of delay differential equations. We give an analytical solution to this system of delay differential equations, obtaining a closed formula for the zero coupon bond price. Under this model, we can show that the distribution of the short rate is a normal distribution whose mean depends on past values of the short rate. Based on the results of K\"uchler and Mensch (1992), we prove the existence of stationary and limiting distributions.

Original language	English
Publisher	arXiv
DOIs	https://doi.org/10.48550/arXiv.2402.16428
Publication status	Published - 26 Feb 2024

Keywords

q-fin.MF
math.PR
91G30, 60G44

Access to Document

10.48550/arXiv.2402.16428

2402.16428v1

Cite this

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title = "Closed form solution to zero coupon bond using a linear stochastic delay differential equation",

abstract = "We present a short rate model that satisfies a stochastic delay differential equation. The model can be considered a delayed version of the Merton model (Merton 1970, 1973) or the Vasi\v{c}ek model (Vasi\v{c}ek 1977). Using the same technique as the one used by Flore and Nappo (2019), we show that the bond price is an affine function of the short rate, whose coefficients satisfy a system of delay differential equations. We give an analytical solution to this system of delay differential equations, obtaining a closed formula for the zero coupon bond price. Under this model, we can show that the distribution of the short rate is a normal distribution whose mean depends on past values of the short rate. Based on the results of K\{"}uchler and Mensch (1992), we prove the existence of stationary and limiting distributions.",

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language = "English",

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T1 - Closed form solution to zero coupon bond using a linear stochastic delay differential equation

AU - Roux, Alet

AU - Guinea Julia, Álvaro

PY - 2024/2/26

Y1 - 2024/2/26

N2 - We present a short rate model that satisfies a stochastic delay differential equation. The model can be considered a delayed version of the Merton model (Merton 1970, 1973) or the Vasi\v{c}ek model (Vasi\v{c}ek 1977). Using the same technique as the one used by Flore and Nappo (2019), we show that the bond price is an affine function of the short rate, whose coefficients satisfy a system of delay differential equations. We give an analytical solution to this system of delay differential equations, obtaining a closed formula for the zero coupon bond price. Under this model, we can show that the distribution of the short rate is a normal distribution whose mean depends on past values of the short rate. Based on the results of K\"uchler and Mensch (1992), we prove the existence of stationary and limiting distributions.

AB - We present a short rate model that satisfies a stochastic delay differential equation. The model can be considered a delayed version of the Merton model (Merton 1970, 1973) or the Vasi\v{c}ek model (Vasi\v{c}ek 1977). Using the same technique as the one used by Flore and Nappo (2019), we show that the bond price is an affine function of the short rate, whose coefficients satisfy a system of delay differential equations. We give an analytical solution to this system of delay differential equations, obtaining a closed formula for the zero coupon bond price. Under this model, we can show that the distribution of the short rate is a normal distribution whose mean depends on past values of the short rate. Based on the results of K\"uchler and Mensch (1992), we prove the existence of stationary and limiting distributions.

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KW - math.PR

KW - 91G30, 60G44

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DO - 10.48550/arXiv.2402.16428

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