The Volterra Stein-Stein model with stochastic interest rates

2025-03-03

AI Summary

This paper addresses the fragmentation between Markovian and non-Markovian frameworks in financial derivative pricing and hedging, along with the scarcity of analytical solutions. We propose a unified framework wherein correlated Gaussian Volterra processes jointly drive volatility and interest rates, yielding the first Volterra extension of the Stein–Stein model incorporating stochastic interest rates. Despite its non-Markovian nature, the model retains strong analytical tractability: it admits closed-form expressions for zero-coupon bonds and interest rate caps/floors, and delivers a semi-explicit characteristic function for the log-forward index—derived via Fredholm resolvents and determinants. Methodologically, we integrate Volterra stochastic calculus, Fredholm theory, and infinite-dimensional Riccati equations to enable efficient pricing and calibration. Empirical analysis demonstrates that the model accurately captures the “humped” term structure of ATM volatility and the concave log-moneyness slope observed in S&P 500 options.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Calibration & Uncertainty Quantification
• Cognitive Modeling & Cognitive Systems: Other Foundations of Cognitive Modeling & Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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 well-known 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 log-log 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.

Problem

• Modeling stochastic volatility and interest rates with correlated Gaussian Volterra processes
• Deriving pricing formulas for financial derivatives using Fredholm resolvents
• Capturing empirical market features like implied volatility term structures

Innovation

• Volterra Stein-Stein model with stochastic interest rates
• Analytical tractability for pricing and hedging
• Fast pricing via Fourier methods