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.

My PhD thesis results (available here)

After establishing theorems in general frameworks, we apply our asymptotic techniques to shed new light on the benefits and limitations of rough volatility model. If, in the last chapter of the thesis, we take some altitude to recover Markovianity in an infinite-dimensional space, the common thread is the use of agnostic tools, in particular asymptotic methods, which remain efficient in the absence of this property. Explicit expressions often become available when passing to the limit, and they can be used to compute quick approximations of the quantities of interest, gain insights on the impact of the model parameters, improve numerical schemes, or compare the asymptotic behaviour with Markovian models. In option pricing, these can lead to a more efficient model choice and faster calibration to market data: concrete small-time formulae for implied volatilities are provided in Chapters 2 and 4. In the background of Chapters 3 and 4, we also rely upon Malliavin calculus (also known as stochastic calculus of variations). Suited to the analysis on Gaussian spaces, this theory is indifferent to Markovianity, hence the universal representations it provides are precious.

The skew (first derivative with respect to strike) of the SPX implied volatility exhibits this power-law shape elegantly captured by rough volatility models.