ECTS : 6
Description du contenu de l'enseignement :
Rough volatility models were introduced by Gatheral and Rosenbaum in 2014. This new paradigm in financial engineering has had a considerable success ever since. In particular, it sheds new light on the interplay between high frequency data (and the microstructural properties of prices) and the more traditional stochastic volatility diffusion models used for risk hedging at coarse scales. In this course, we will adopt the point of view of statistical modelling: how to build a price model consistent with data together with broad guiding financial principles (like e.g. absence of arbitrage) that has some robustness across time scales ? We will follow the theory of Gatheral and Rosenbaum from a statistical perspective and discuss the following topics: point processes (in particular Hawkes processes), scaling limits for semimartingales, maximum likelihood estimation, nonparametric estimation for stochastic processes, fractional Brownian motion, simulation of rough Heston models and a bit of stochastic control. It is not necessary to have a background in statistics (we will cover all the required tools) but it is mandatory to have some proficiency in stochastic calculus (continuous semimartingales, stochastic integration, Itô formula, Girsanov theorem and the basics of SDEs)
Bibliographie, lectures recommandées :
Textbooks :
- Aït-Sahalia, Y. and Jacod, J. (2014). High-frequency financial econometrics. Princeton University Press.
- Ibragimov, I. A. and Has’ Minskii, R. Z. (2013). Statistical estimation: asymptotic theory, volume 16. Springer Science & Business Media.
- Revuz, D., and Yor, M. (2013). Continuous martingales and Brownian motion (Vol. 293). Springer Science & Business Media.
Research papers :
- Bacry, E., Delattre, S., Hoffmann, M., and Muzy, J.-F. (2013b).
Some limit theorems for Hawkes processes and application to financial statistics. Stochastic Processes and their Applications, 123(7):2475–2499.
- El Euch, O., Fukasawa, M., and Rosenbaum, M. (2018a). The microstructural foundations of leverage effect and rough volatility.
Finance and Stochastics, 22(2):241–280.
- Gatheral, J., Jaisson, T., and Rosenbaum, M. (2018). Volatility is rough. Quantitative Finance, 18(6):933–949.
- Hawkes, A. G. (2018). Hawkes processes and their applications to finance: a review.
Quantitative Finance, 18(2):193–198.
- Jaisson, T., Rosenbaum, M., et al. (2015). Limit theorems for nearly unstable hawkes processes.
The Annals of Applied Probability, 25(2):600–631.
- Jusselin, P. and Rosenbaum, M. (2020). No-arbitrage implies power-law market impact and rough volatility. Mathematical Finance, 30(4):1309–1336.