Paper: arXiv 2402.16428

Authors: Alet Roux, Álvaro Guinea Juliá

Abstract

This paper introduces a short rate model in continuous time that adds one or more memory (delay) components to the Merton model (Merton 1970, 1973) or the Vasiček model (Vasiček 1977) 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.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical framework extending classic short rate models (Vasicek/Merton) to incorporate multiple memory delays, deriving explicit closed-form solutions for zero-coupon bonds and caplets. It complements the strong theoretical analysis with empirical calibrations to SONIA derivatives and US yield curve data.

Research Flowchart

  flowchart TD
    A[Research Goal: Extend short rate models<br>Merton/Vasiček with multiple delays] --> B[Formulate Continuous-Time Model<br>using Delay Differential Equations]
    B --> C[Derive Analytical Solutions:<br>Zero-coupon bonds, Forward rates & Caplets]
    D[(Data Inputs:<br>SONIA Caplets & US Yield Curve)] --> E[Model Calibration & Empirical Testing]
    C --> E
    E --> F[Outcomes: Affine bond prices, HJM alignment,<br>and successful market calibration]

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