Rough volatility

Stochastic modelling is concerned with finding the right balance between consistency (with the features of reality one wants to capture) and tractability (the capacity to analyse and get outputs). Modeling volatility is a crucial aspect of financial mathematics for a good model enhances our understanding of the behaviour of market prices. The long tradition of Markovian models is challenged by the emergence of rough volatility models, which solve three crucial consistency issues of the previous generation. Namely, they incorporate memory of the past states, they mimic the low regularity of the time-series under the historical measure, and they reproduce the short-time behaviour of option prices under the risk-neutral measure (see figure below). This class of models thus reconciles under the same umbrella the two main motivations of quantitative finance: predicting future trends and managing risks. However, in the realm of stochastic Volterra equations where these models live, the rules of traditional stochastic calculus no longer prevail. Hence, the rise of the rough volatility paradigm represents a tremendous opportunity, not only as a pathway to better understand the markets, but also as a motivation to design new techniques for a promising class of stochastic processes.