Paper: arXiv 2503.01716

Authors: Eduardo Abi Jaber, Donatien Hainaut, Edouard Motte

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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 7.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a highly sophisticated mathematical framework unifying Markovian and non-Markovian models using Gaussian Volterra processes, Fredholm resolvents, and Riccati equations. It demonstrates strong empirical rigor by successfully calibrating to market data (S&P 500 and cap implied volatility term structures) to solve known limitations of traditional models.

Research Flowchart

  flowchart TD
    A["Research Goal:<br>Develop tractable rough volatility model<br>with stochastic interest rates"] --> B["Framework Construction:<br>Unify Gaussian Volterra processes for<br>volatility and interest rates"]
    
    B --> C["Analytical Derivations:<br>Compute ZCB, cap/floor formulas & log-forward<br>index char. func. via Fredholm resolvents"]
    
    D["Market Inputs:<br>S&P 500 options &<br>Interest Rate Caps data"] --> E["Calibration & Valuation:<br>Apply Fourier methods for<br>fast derivative pricing"]
    
    C --> E
    
    E --> F["Key Outcomes:<br>1. Captures humped ATM volatility for caps<br>2. Matches concave S&P 500 skew in log-log scale<br>3. Links to infinite-dim Riccati equations"]

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2 of 56 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.