Short time behavior of the ATM implied skew in the ADO-Heston model

Abstract

In the current literature, there exists an opinion that Markovian approximations of rough volatility models are able to catch the behavior of the at-the-money implied volatility skew S(T ) at small maturities T for vanilla options, while for forward started options this is not the case. To analyze this, similar to [P. Carr, A. Itkin, 2019], we construct another Markovian approximation of the rough Heston-like volatility model -the ADO-Heston model. By using the Dobric-Ojeda process instead of the fractional Brownian motion in the rough variance process, we came up to an inhomogeneous Markovian model where the characteristic function (CF) solves an unsteady three-dimensional inhomogeneous partial differential equation (PDE) with some coefficients being function of the Hurst exponent H. It then turned out that the characteristic function of the log-price and S(T ) in this model can be represented in a semi-analytical form. Since we were basically interested in the behavior of the S(T ) at T → 0, instead of using numerical inverse transforms, we further simplified the representation of the CF for vanillas by making a particular choice of the market price of risk and looking at the asymptotic of the CF at T → 0. It was observed that the behavior of thus obtained S(T ) ∝ a(H)T b•(H−1/2) , b = const was not exactly the same as in rough volatility models since b ̸ = 1, but to be close enough for all practical values of T > 0. Thus, we concluded that the proposed Markovian model was able to replicate the behavior of the corresponding rough volatility model for the range of T inherent to the options' market. Similar analysis was provided for the forward starting options, where the S(t, s, T ) for the forward started options exhibited a similar behavior for any s > t when T → s.