Abstract We provide explicit small-time formulae for the at-the-money implied volatility, skew, and curvature in a large class of models, including rough volatility models and their multi-factor versions. Our general setup encompasses both European options on a stock and VIX options, thereby providing new insights on their joint calibration. The tools used are essentially based on Malliavin calculus for Gaussian processes. We develop a detailed theoretical and numerical analysis of the two-factor rough Bergomi model and provide insights on the interplay between the different parameters for joint SPX–VIX smile calibration.

Antoine Jacquier

Aitor Muguruza

Alexandre Pannier

Advances in Applied Probability

THE ADVANCES IN APPLIED PROBABILITY TRUST 由 APPLIED PROBABILITY TRUST 出版了 40 多年，是《应用概率杂志》的配套出版物。它包含应用概率学家感兴趣的数学和科学论文，重点是广泛学科的应用，包括生物科学、运筹学、电信、计算机科学、工程、流行病学、金融数学、物理和社会科学以及任何使用随机建模的字段。提交给 APPLIED PROBABILITY 代表的提交内容可能会根据主编的判断出现在 JOURNAL OF APPLIED PROBABILITY 或 ADVANCES IN APPLIED PROBABILITY 中。通常，较短的论文出现在 JOURNAL 中，较长的论文出现在 ADVANCES 中。

Advances in Applied Probability, published by the Applied Probability Trust for over 40 years, serves as a companion publication to the Journal of Applied Probability. It features mathematical and scientific papers of interest to applied probabilists, focusing on a wide range of applications across disciplines including biological sciences, operations research, telecommunications, computer science, engineering, epidemiology, financial mathematics, physics, and social sciences, as well as any field that utilizes stochastic modeling. Submissions to the Journal of Applied Probability may appear in either the Journal of Applied Probability or Advances in Applied Probability, at the discretion of the editor-in-chief. Typically, shorter papers are published in the Journal, while longer papers are published in Advances.

《Advances in Applied Probability》是由应用概率信托（Applied Probability Trust）出版的期刊，已有40多年的历史，是《应用概率杂志》（Journal of Applied Probability）的配套出版物。该期刊包含应用概率学家感兴趣的数学和科学论文，重点关注广泛学科的应用，包括生物科学、运筹学、电信、计算机科学、工程、流行病学、金融数学、物理和社会科学等领域，以及任何使用随机建模的领域。提交给《应用概率杂志》的稿件，可能会根据主编的判断，出现在《应用概率杂志》或《Advances in Applied Probability》中。通常，较短的论文会发表在《应用概率杂志》中，而较长的论文则会发表在《Advances in Applied Probability》中。

Category	Quartile
数学	4
数学, 统计学与概率论	3

Rough multi-factor volatility for SPX and VIX options

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