Small-time, large-time and $H\to 0$ asymptotics for the Rough Heston model

We characterize the behaviour of the Rough Heston model introduced by Jaisson\&Rosenbaum \cite{JR16} in the small-time, large-time and $\alpha \to 1/2$ (i.e. $H\to 0$) limits. We show that the short-maturity smile scales in qualitatively the same way as a general rough stochastic volatility model (cf.\ \cite{FZ17}, \cite{FGP18a} et al.), and the rate function is equal to the Fenchel-Legendre transform of a simple transformation of the solution to the same Volterra integral equation (VIE) that appears in \cite{ER19}, but with the drift and mean reversion terms removed. The solution to this VIE satisfies a space-time scaling property which means we only need to solve this equation for the moment values of $p=1$ and $p=-1$ so the rate function can be efficiently computed using an Adams scheme or a power series, and we compute a power series in the log-moneyness variable for the asymptotic implied volatility which yields tractable expressions for the implied vol skew and convexity. The limiting asymptotic smile in the large-maturity regime is obtained via a stability analysis of the fixed points of the VIE, and is the same as for the standard Heston model in \cite{FJ11}. Finally, using L\'{e}vy's convergence theorem, we show that the log stock price $X_t$ tends weakly to a non-symmetric random variable $X^{(1/2)}_t$ as $\alpha \to 1/2$ (i.e. $H\to 0$) whose mgf is also the solution to the Rough Heston VIE with $\alpha=1/2$, and we show that $X^{(1/2)}_t/\sqrt{t}$ tends weakly to a non-symmetric random variable as $t\to 0$, which leads to a non-flat non-symmetric asymptotic smile in the Edgeworth regime. We also show that the third moment of the log stock price tends to a finite constant as $H\to 0$ (in contrast to the Rough Bergomi model discussed in \cite{FFGS20} where the skew flattens or blows up) and the $V$ process converges on pathspace to a random tempered distribution.