Linear short rate model with several delays

This paper introduces a short rate model in continuous time that adds one or more memory (delay) components to the Merton model [R.C. Merton, A dynamic general equilibrium model of the asset market and its application to the pricing of the capital structure of the firm, Working Paper 497, Sloan School of Management, MIT, Cambridge, 1970; R.C. Merton, Theory of rational option pricing, Bell J. Econ. Manage. Sci. 4 (1973), pp. 141–183.] or the Vasiček model [Vasiček, An equilibrium characterization of the term structure, J. Financ. Econ. 5 (1977), pp. 177–188.] for the short rate. The distribution of the short rate in this model is normal, with the mean depending on past values of the short rate, and a limiting distribution exists for certain values of the parameters. The zero coupon bond price is an affine function of the short rate, whose coefficients satisfy a system of delay differential equations. This system can be solved analytically, obtaining a closed formula. An analytical expression for the instantaneous forward rate is given: it satisfies the risk neutral dynamics of the Heath-Jarrow-Morton model. Formulae for both forward looking and backward looking caplets on overnight risk free rates are presented. Finally, the proposed model is calibrated against forward looking caplets on SONIA rates and the United States yield curve.