The Volterra Stein-Stein model with stochastic interest rates

Eduardo Abi Jaber (), Donatien Hainaut () and Edouard Motte ()
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Eduardo Abi Jaber: CMAP - Centre de Mathématiques Appliquées de l'Ecole polytechnique - Inria - Institut National de Recherche en Informatique et en Automatique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique
Donatien Hainaut: UCL - Université Catholique de Louvain = Catholic University of Louvain
Edouard Motte: UCL - Université Catholique de Louvain = Catholic University of Louvain

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Abstract: We introduce the Volterra Stein-Stein model with stochastic interest rates, where both volatility and interest rates are driven by correlated Gaussian Volterra processes. This framework unifies various wellknown Markovian and non-Markovian models while preserving analytical tractability for pricing and hedging financial derivatives. We derive explicit formulas for pricing zero-coupon bond and interest rate cap or floor, along with a semi-explicit expression for the characteristic function of the log-forward index using Fredholm resolvents and determinants. This allows for fast and efficient derivative pricing and calibration via Fourier methods. We calibrate our model to market data and observe that our framework is flexible enough to capture key empirical features, such as the humped-shaped term structure of ATM implied volatilities for cap options and the concave ATM implied volatility skew term structure (in loglog scale) of the S&P 500 options. Finally, we establish connections between our characteristic function formula and expressions that depend on infinite-dimensional Riccati equations, thereby making the link with conventional linear-quadratic models.

Keywords: volatility; interest rate; memory; Fredholm resolvents and determinants; Fourier pricing; Riccati equations; Gaussian Volterra processes (search for similar items in EconPapers)
Date: 2025-03-06
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