The ATM implied skew in the ADO-Heston model

TL;DR

This paper introduces a Markovian approximation of the rough Heston model, called the ADO-Heston model, deriving its characteristic function and demonstrating its ability to replicate the market implied skew behavior for small maturities.

Contribution

The paper develops a new Markovian approximation of the rough Heston model with a closed-form characteristic function, capturing key features of implied skew behavior.

Findings

The ADO-Heston model approximates the implied skew curve ${ m S}(T)$ with a power law involving the Hurst exponent.

The model can reproduce the small-maturity skew behavior observed in markets.

Forward starting options' implied skew can blow up as maturity approaches the start date.

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.