Computation of Greeks under rough Volterra stochastic volatility models using the Malliavin calculus approach

Paper: arXiv 2312.00405

Authors: Mishari Al-Foraih, Òscar Burés, Jan Pospíšil, Josep Vives

Abstract

Using Malliavin calculus techniques, we obtain formulas for computing Greeks under different rough Volterra stochastic volatility models. Due to the fact that underlying prices are not always square integrable, we extend the classical integration by parts formula to integrable but not necessarily square integrable functionals. First of all, we obtain formulas for general stochastic volatility (SV) models, concretely the Greeks Delta, Gamma, Rho, Vega and we introduce the Greek with respect to the roughness parameter. Then, the particular case of rough Volterra SV models is analyzed. Finally, three examples are treated in detail: the family of alpha-RFSV models, that includes rough versions of SABR and Bergomi models, a mixed alpha-RFSV model with two different Hurst parameters representing short (roughness) and long memory, and the rough Stein-Stein model. For different models and Greeks we show a numerical convergence of our formulas in Monte Carlo simulations and depict for example a dependence of the Greeks on the roughness parameter.

Complexity vs Empirical Score

• Math Complexity: 9.2/10
• Empirical Rigor: 4.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents sophisticated advanced mathematics, leveraging Malliavin calculus to extend integration by parts formulas for non-square-integrable functionals in rough Volterra models. While it contains strong theoretical novelty and Monte Carlo convergence demonstrations, it lacks empirical backtesting on real-world market data.

Research Flowchart

  flowchart TD
    A[Research Goal: Compute Greeks under Rough Volterra SV Models via Malliavin Calculus] --> B[Inputs: SV Models, Hurst Parameter α, Malliavin Operators & Extended IBP Formula]
    B --> C[Methodology: Extend Integration by Parts to Integrable non-Square-Integrable Functionals]
    C --> D[Analytical Derivation: General & Volterra Greek Formulas for Delta, Gamma, Rho, Vega, and Roughness]
    D --> E[Model Application: Test on alpha-RFSV, Mixed alpha-RFSV, and Rough Stein-Stein Models]
    E --> F[Outcomes: Monte Carlo Numerical Convergence & Sensitivity Analysis on Roughness Parameter]