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

Alet Roux and \'Alvaro Guinea Juli\'a

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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.

Date: 2024-02
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