We characterize the asymptotic small-time and large-time implied volatility smile for the rough Heston model introduced by El Euch, Jaissonand Rosenbaum .0pt%DIFAUXCMD in the small-time, large-time and \alpha }\to \to  We show that the asymptotic short-maturity smile scales in qualitatively the same way as a general rough stochastic volatility model , and is characterized by0pt%DIFAUXCMD , \mbox{%DIFAUXCMD FGP18a }0pt%DIFAUXCMD et al.), and the rate function is equal to } the Fenchel-Legendre transform of the solution a Volterra integral equation (VIE) \hspace{0pt}%DIFAUXCMD , but with the drift and mean reversion terms removed} . The solution of this VIE satisfies a space-time scaling property which simplifies its computation. We corroborate our results numerically with Monte Carlo simulations. We also compute a power series in the log-moneyness variable for the asymptotic implied volatility , which yields tractable expressions for the  vol skew and convexity , thus being useful for calibration purposes. We also derive formal asymptotics for the small-time moderate deviations regime and a formal saddlepoint approximation for call options in the large deviations regime. This goes to higher order than previous works for rough models, and in particular captures the effect of the mean reversion term. In the large maturity case, the limiting asymptotic smile turns out to be the same as for the standard Heston model , for which there is a well known closed-form formula in terms of the SVI parametrization}\hspace{0pt}%DIFAUXCMD . 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 \to _t/\sqrt{t} tends weakly to a non-symmetric random variable as t}\to \to  }\hspace{0pt}%DIFAUXCMD where the skew flattens or blows up) and the V process converges on pathspace to a random tempered distribution} .
We characterize the behaviour of the Rough Heston model introduced by Jaisson\&Rosenbaum \mbox{%DIFAUXCMD JR160pt%DIFAUXCMD 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.\ \mbox{%DIFAUXCMD FZ170pt%DIFAUXCMD , \mbox{%DIFAUXCMD FGP18a }0pt%DIFAUXCMD 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 \mbox{%DIFAUXCMD ER19\hspace{0pt}%DIFAUXCMD , 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 \mbox{%DIFAUXCMD \cite{FJ11}\hspace{0pt}%DIFAUXCMD . 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 \mbox{%DIFAUXCMD \cite{FFGS20 }\hspace{0pt}%DIFAUXCMD where the skew flattens or blows up) and the V process converges on pathspace to a random tempered distribution} .
R
asymptotic small-time and large-time implied volatility smile for the rough
behaviour of the Rough
R
El Euch, Jaissonand Rosenbaum .
Jaisson\&Rosenbaum \mbox{%DIFAUXCMD JR16
A
1/2 (i.e. H
A
0) limits.
D
asymptotic
R
, and is characterized by
(cf.\ \mbox{%DIFAUXCMD FZ17
R
the solution a
a simple transformation of the solution to the same
A
that appears in \mbox{%DIFAUXCMD ER19
R
of
to
R
simplifies its computation. We corroborate our results numerically with Monte Carlo simulations. We also
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
D
,
A
implied
R
, thus being useful for calibration purposes. We also derive formal asymptotics for the small-time moderate deviations regime and a formal saddlepoint approximation for call options in the large deviations regime. This goes to higher order than previous works for rough models, and in particular captures the effect of the mean reversion term. In the large maturity case, the limiting asymptotic smile turns out to be the
. 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
R
, for which there is a well known closed-form formula in terms of the SVI parametrization
in \mbox{%DIFAUXCMD \cite{FJ11
A
1/2 (i.e. H
A
0) whose mgf is also the solution to the Rough Heston VIE with \alpha=1/2, and we show that X^{(1/2)
A
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
A
0 (in contrast to the Rough Bergomi model discussed in \mbox{%DIFAUXCMD \cite{FFGS20