Rough volatility, CGMY jumps with a finite history and the Rough Heston model – small-time asymptotics in the regime

A small-time Edgeworth expansion is established for near-the-money European options under a general rough stochastic volatility (RSV) model driven by a Riemann-Liouville (RL) process plus an additional generalised tempered stable Lévy process with $Y\in (1,2) $Y∈(1,2) when $H \in (1- \frac {1}{2} Y,2(1- \frac {1}{2} Y)\wedge \frac {1}{2} ) $H∈(1−12Y,2(1−12Y)∧12) (this relaxes the more complicated and restrictive condition which appeared in an earlier version of the article.), in the regime where log-moneyness $\log \frac {K}{S_0}\sim z\sqrt {t} $log⁡KS0∼zt as $t\to 0 $t→0 for z fixed, conditioned on a finite volatility history. This can be viewed as a more practical variant of Theorem 3.1 in Fukasawa [Short-time at-the-money skew and rough fractional volatility. Quant. Finance, 2017, 17(2), 189–198] (Fukasawa [Short-time at-the-money skew and rough fractional volatility. Quant. Finance, 2017, 17(2), 189–198] does not allow for jumps or a finite history and uses the somewhat opaque Muravlev representation for fractional Brownian motion), or if we turn off the rough stochastic volatility, the expansion is a variant of the main result in Mijatovic and Tankov [A new look at short-term implied volatility in asset price models with jumps. Math. Finance, 2016, 26(1), 149–183] and Theorem 3.2 in Figueroa-López et al. [Third-order short-time expansions for close-to-the-money option prices under the CGMY model. J. Appl. Math. Finance, 2017, 24, 547–574]. The $z\sqrt {t} $zt regime is directly applicable to FX options where options are typically quoted in terms of delta (.10,.25 and .50) not absolute strikes, and we also compute a new prediction formula for the Riemann-Liouville process, which allows us to express the history term for the Edgeworth expansion in a more useable form in terms of the volatility process itself. We later relax the assumption of bounded volatility, and we also compute a formal small-time expansion for implied volatility in the Rough Heston model in the same regime (without jumps) which includes an additional at-the-money, convexity and fourth order correction term, and we outline how one can go to even higher order in the three separate cases $H>\frac {1}{6} $H>16, $H=\frac {1}{6} $H=16 and $H