Paper: arXiv 2212.01591

Authors: Peter K. Friz, William Salkeld, Thomas Wagenhofer

Abstract

We consider a class of stochastic processes with rough stochastic volatility, examples of which include the rough Bergomi and rough Stein-Stein model, that have gained considerable importance in quantitative finance. A basic question for such (non-Markovian) models concerns efficient numerical schemes. While strong rates are well understood (order $H$), we tackle here the intricate question of weak rates. Our main result asserts that the weak rate, for a reasonably large class of test function, is essentially of order $\min { 3H+\tfrac12, 1 }$ where $H \in (0,1/2]$ is the Hurst parameter of the fractional Brownian motion that underlies the rough volatility process. Interestingly, the phase transition at $H=1/6$ is related to the correlation between the two driving factors, and thus gives additional meaning to a quantity already of central importance in stochastic volatility modelling.Our results are complemented by a lower bound which show that the obtained weak rate is indeed optimal.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a highly rigorous theoretical analysis of weak error rates for rough volatility models, addressing a major open question in quantitative finance math. However, as a purely theoretical paper, it lacks empirical backtesting or real-world dataset validation.

Research Flowchart

  flowchart TD
    A[Goal: Determine Weak Convergence Rates for Rough Volatility Models] --> B[Input: Class of Rough Volatility Models e.g., rBergomi, rStein-Stein with Hurst H ∈ 0, 0.5]
    B --> C[Methodology: Discretization Scheme & Weak Error Analysis via Malliavin Calculus]
    C --> D[Compute Upper & Lower Bounds of Weak Error]
    D --> E[Key Outcome: Optimal Weak Rate is min 3H + 0.5, 1]
    E --> F[Discovery: Phase Transition at H = 1/6 Linked to Driving Factor Correlation]