Paper: arXiv 2303.07222

Authors: Eduardo Abi Jaber, Nathan De Carvalho

Abstract

We reconcile rough volatility models and jump models using a class of reversionary Heston models with fast mean reversions and large vol-of-vols. Starting from hyper-rough Heston models with a Hurst index $H \in (-1/2,1/2)$, we derive a Markovian approximating class of one dimensional reversionary Heston-type models. Such proxies encode a trade-off between an exploding vol-of-vol and a fast mean-reversion speed controlled by a reversionary time-scale $ε>0$ and an unconstrained parameter $H \in \mathbb R$. Sending $ε$ to 0 yields convergence of the reversionary Heston model towards different explicit asymptotic regimes based on the value of the parameter H. In particular, for $H \leq -1/2$, the reversionary Heston model converges to a class of Lévy jump processes of Normal Inverse Gaussian type. Numerical illustrations show that the reversionary Heston model is capable of generating at-the-money skews similar to the ones generated by rough, hyper-rough and jump models.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 4.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly rigorous mathematical bridge between rough volatility models and jump processes via asymptotic analysis of reversionary Heston models. While mathematically sophisticated and novel, the evaluation relies predominantly on theoretical asymptotics and basic numerical illustrations rather than extensive empirical backtesting.

Research Flowchart

  flowchart TD
    Goal["Research Goal: Reconcile Rough Volatility<br>& Jump Models via Heston Proxies"] --> Input["Input: Hyper-Rough Heston Model<br>with Hurst Index H ∈ (-1/2, 1/2)"]
    
    Input --> Step1["Derive Markovian Approximating Class:<br>1D Reversionary Heston Model"]
    
    Step1 --> Step2["Apply Asymptotic Limit (ε → 0):<br>Fast Mean-Reversion & Exploding Vol-of-Vol"]
    
    Step2 --> Branch{"Parameter H Regime"}
    
    Branch -- "H ≤ -1/2" --> Outcome1["Convergence to Lévy Jump Processes<br>(Normal Inverse Gaussian Type)"]
    Branch -- "H > -1/2" --> Outcome2["Convergence to Rough / Hyper-Rough<br>Volatility Regimes"]
    
    Outcome1 --> Final["Key Finding: Reversionary Heston Matches<br>ATM Skews of Rough & Jump Models"]
    Outcome2 --> Final