We characterize the behavior of the Rough Heston model introduced by Jaisson and Rosenbaum (2016, Ann. Appl. Probab., 26, 2860–2882) in the small‐time, large‐time, and (i.e., ) limits. We show that the short‐maturity smile scales in qualitatively the same way as a general rough stochastic volatility model , 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 El Euch and Rosenbaum (2019, Math. Financ., 29, 3–38), 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 and 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 which is useful for calibration purposes. We later derive a formal saddle point approximation for call options in the Forde and Zhang (2017) large deviations regime which goes to higher order than previous works for rough models. Our higher‐order expansion captures the effect of both drift terms, and at leading order is of qualitatively the same form as the higher‐order expansion for a general model which appears in Friz et al. (2018, Math. Financ., 28, 962–988). 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 Forde and Jacquier (2011, Finance Stoch., 15, 755–780). Finally, using Lévy's convergence theorem, we show that the log stock price tends weakly to a nonsymmetric random variable as (i.e., ) whose moment generating function (MGF) is also the solution to the Rough Heston VIE with , and we show that tends weakly to a nonsymmetric random variable as , which leads to a nonflat nonsymmetric asymptotic smile in the Edgeworth regime, where the log‐moneyness as , and we compute this asymptotic smile numerically. We also show that the third moment of the log stock price tends to a finite constant as (in contrast to the Rough Bergomi model discussed in Forde et al. (2020, Preprint) where the skew flattens or blows up) and the process converges on pathspace to a random tempered distribution which has the same law as the hyper‐rough Heston model, discussed in Jusselin and Rosenbaum (2020, Math. Finance, 30, 1309–1336) and Abi Jaber (2019, Ann. Appl. Probab., 29, 3155–3200).

中文翻译：

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Rough Heston模型的小时间，大时间和渐近性

我们通过定性和Jaisson罗森鲍姆（2016年，推出了粗糙赫斯顿模型的行为安。申请Probab，26在小的时候，大的时间，而且，2860年至2882年）（即）的限制。我们显示，短期到期微笑的定性方式与一般的粗糙随机波动率模型相同，并且速率函数等于对同一Volterra积分方程（VIE）的简单转换，即Fenchel-Legendre转换出现在El Euch and Rosenbaum（2019，Math。Financ。，29，3–38），但移除了漂移和均值回归项。要解决这个VIE满足这意味着我们只需要解决这个方程的力矩值时空缩放特性和因此可以使用亚当斯（Adams）方案或幂级数来高效地计算费率函数，并且我们在对数货币变量中计算渐近隐含波动率的幂级数，该幂序列产生了隐含的斜率和凸度的易于表达的表达式，可用于校准目的。我们随后在Forde和Zhang（2017）大偏差方案中得出看涨期权的形式化鞍点近似值，其阶数比之前的粗糙模型高。我们的高阶展开式同时捕获了两个漂移项的影响，并且定性的形式与Friz等人提出的通用模型的高阶展开式在质量上相同。（2018，Math。Financ。，28，962–988）。在大的成熟机制限制渐近微笑经由VIE的固定点的稳定性分析获得的，并且是与用于在弗勒和雅基耶（2011年，标准模型赫斯顿金融斯托赫。，15，755-780 ）。最后，使用李维（Lévy）的收敛定理，我们证明了对数股票价格趋于弱于非对称随机变量，例如（），其矩生成函数（MGF）也是Rough Heston VIE的解，并且表明了弱趋势。到非对称随机变量as ，这导致Edgeworth体制中的非平坦非对称渐近微笑，其中对数货币为，并且我们通过数值计算此渐近微笑。我们还表明，原木股票价格的第三时刻趋向于一个有限常数，因为（与Forde等人（2020年，预印本）中讨论的Rough Bergomi模型的偏斜趋于平缓或爆炸）相反，并且过程收敛于pathspace到具有相同的法律为随机分布的脾气超粗赫斯顿模型，在Jusselin和罗森鲍姆（2020，讨论数学，金融，30，1309年至1336年）和阿比·贾比尔（2019年，安。申请Probab，29，3155-3200）。