Date: From the 25th to the 29th of November

Time and Location: Monday and Tuesday talks are scheduled at 10:30 in Room 407-H of the Institute of Mathematics and Statistics (IME-UFF). Friday will be of more general interest. These talks are schedule starting at 10:30 and the IME-UFF Auditorium -- Ground Floor of Building G.

This will be a series of talks that will present different viewpoints on these subjects -- with an emphasis on Mathematical Finance. This Workshop is open to all interested in these Mathematics (pure or applied) and on these specific applications.

This talk concerns the calibration of Dupire's model in the presence of jumps.

We present a detailed analysis and implementation of a splitting strategy to identify simultaneously

the local-volatility surface and the jump-size distribution from quoted European prices.

The underlying model consists of a jump-diffusion driven asset with time and price dependent volatility.

Our approach uses a forward Dupire-type partial-integro-differential equation for the option prices

to produce a parameter-to-solution map. The ill-posed inverse problem for such map is then solved by

means of a Tikhonov-type convex regularization. We present numerical examples that substantiate

the local-volatility surface and the jump-size distribution from quoted European prices.

The underlying model consists of a jump-diffusion driven asset with time and price dependent volatility.

Our approach uses a forward Dupire-type partial-integro-differential equation for the option prices

to produce a parameter-to-solution map. The ill-posed inverse problem for such map is then solved by

means of a Tikhonov-type convex regularization. We present numerical examples that substantiate

the robustness of the method both for synthetic and real data. This is joint work with Vinicius Albani (UFSC).

**10:30**

We review the history of volatility modeling, from Black-Scholes to Local Volatility to Stochastic Volatility to Stochastic Local Volatility to Path-Dependent Volatility. We explain the benefits and limitations of each model class and motivate their successive introductions. In particular, we focus on path-dependent volatility models, which have drawn little attention so far. Like the local volatility model, they are complete and can fit exactly the market smile of the underlying asset. Like stochastic volatility models, they can produce rich joint dynamics of spot and implied volatility. Path-dependent volatility models also capture prominent historical patterns of volatility, such as volatility depending on the recent trend of the underlying asset. We give examples and show many graphs to demonstrate their great capabilities.

**Short Bio:**

Julien is a senior quantitative analyst in the Quantitative Research group at Bloomberg L.P., New York. He is also an adjunct professor in the Department of Mathematics at Columbia University and at the Courant Institute of Mathematical Sciences, NYU. Before joining Bloomberg, Julien worked in the Global Markets Quantitative Research team at Societe Generale in Paris for six years (2006-2012), and was an adjunct professor at Universite Paris 7 and Ecole des ponts. He co-authored the book Nonlinear Option Pricing (Chapman & Hall, CRC Financial Mathematics Series, 2014) with Pierre Henry-Labordere. His main research interests include volatility and correlation modeling, nonlinear option pricing, and numerical probabilistic methods. Julien holds a Ph.D. in Probability Theory and Statistics from Ecole des ponts. He graduated from Ecole Polytechnique (Paris), Universite Paris 6, and Ecole des ponts. A big football fan, Julien has also developed a strong interest in sports analytics, and has published several articles on the FIFA World Cup, the UEFA Champions League, and the UEFA Euro in top-tier newspapers such as The New York Times, Le Monde, and El Pais, including a new, fairer draw method for the FIFA World Cup.

**11:30**

**A Short Introduction to mean-field games**

**Roberto Velho (IMPA)**

**Abstract:**

Mean-field games (MFG) is a recent area that emerged both from mathematical and engineering fields. Its ideas come from statistical mechanics and its application range from city planning to economics, to finance. The main idea is to provide an approximation for the description of a system with many particles/agents/players evolving in time while in equilibrium under a non-collaborative differential game. The ingredients are (stochastic) optimal control and Fokker-Planck equations. Surprisingly, many PDEs can be regarded as a kind of MFG. The goal of this talk is to introduce the notions to formulate a mean-field game problem, describe modeling through some examples and connect it with other mathematical problems. We will focus on the ideas of the construction rather than technicalities.