The ATM implied skew in the ADO-Heston model

ArXiv ID: 2309.15044 “View on arXiv”

Authors: Unknown

Abstract

In this paper similar to [“P. Carr, A. Itkin, 2019”] we construct another Markovian approximation of the rough Heston-like volatility model - the ADO-Heston model. The characteristic function (CF) of the model is derived under both risk-neutral and real measures which is an unsteady three-dimensional PDE with some coefficients being functions of the time $t$ and the Hurst exponent $H$. To replicate known behavior of the market implied skew we proceed with a wise choice of the market price of risk, and then find a closed form expression for the CF of the log-price and the ATM implied skew. Based on the provided example, we claim that the ADO-Heston model (which is a pure diffusion model but with a stochastic mean-reversion speed of the variance process, or a Markovian approximation of the rough Heston model) is able (approximately) to reproduce the known behavior of the vanilla implied skew at small $T$. We conclude that the behavior of our implied volatility skew curve ${"\cal S"}(T) \propto a(H) T^{“b\cdot (H-1/2)”}, , b = const$, is not exactly same as in rough volatility models since $b \ne 1$, but seems to be close enough for all practical values of $T$. Thus, the proposed Markovian model is able to replicate some properties of the corresponding rough volatility model. Similar analysis is provided for the forward starting options where we found that the ATM implied skew for the forward starting options can blow-up for any $s > t$ when $T \to s$. This result, however, contradicts to the observation of [“E. Alos, D.G. Lorite, 2021”] that Markovian approximation is not able to catch this behavior, so remains the question on which one is closer to reality.

Keywords: Rough Heston Model, Markovian Approximation, Characteristic Function, Implied Volatility Skew, Partial Differential Equations (PDEs), Equities

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper is mathematically dense, featuring a complex three-dimensional PDE, characteristic functions, and asymptotic analysis, while the empirical evaluation is limited to theoretical examples without backtests or implementation details.

  flowchart TD
    A["Research Goal<br>Model ATM Implied Skew<br>using Markovian Heston Approx."] --> B
    subgraph B ["Methodology"]
        B1["Construct ADO-Heston Model<br>Stochastic Mean-Reversion Speed"]
        B2["Derive 3D PDE &<br>Characteristic Function"]
        B3["Calibrate Market Price of Risk<br>to replicate skew behavior"]
    end
    B --> C{"Computational Process"}
    C --> D["Solve PDE for<br>Closed Form CF"]
    D --> E["Calculate ATM Implied Skew<br>for Vanillas & Forward Starters"]
    E --> F
    subgraph F ["Key Findings/Outcomes"]
        F1["Skew Curve: S(T) ∝ T^{"b*(H-1/2)"}<br>b ≠ 1 (vs. Rough Heston b=1)"]
        F2["Replicates rough volatility properties<br>in a Markovian setting"]
        F3["Forward Starter Skew blows up<br>contradicting previous literature"]
    end