🤖 AI Summary
This paper addresses the memoryless limitation of classical short-rate models (e.g., Vasicek) by proposing an extension based on linear stochastic delay differential equations (SDDEs). Methodologically, it integrates affine term structure theory, SDDE analysis, and probabilistic asymptotic techniques. The contributions are threefold: (i) it derives, for the first time, an exact closed-form solution for zero-coupon bond prices under a delayed short-rate model and proves that the price is an affine function of the current short rate; (ii) it rigorously establishes that the short rate follows a history-dependent normal distribution; and (iii) it proves the existence and uniqueness of both a stationary distribution and a limiting distribution for the delay model. These results extend the theoretical scope of affine models and provide a rigorous foundation—along with analytical tools—for pricing interest-rate derivatives incorporating memory effects.
📝 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 Vasiv{c}ek model (Vasiv{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.