Econometric and statistical methods as applied to finance. Topics include: Asset returns and efficient markets, linear time series and dynamics of returns, volatility models, multivariate time series, efficient portolios and CAPM, multifactor pricing models, portfolio allocation and risk assessment, intertemporal equilibrium models, present value models, simulation methods for financial derivatives, econometrics of continuous time finance.

Under the direction of a faculty member, each student carries out research and presents the results. Directed Research II has to be taken before the General Exam.

The course covers the pricing and hedging of advanced derivatives, including topics such as exotic options, greeks, interest rate and credit derivatives, as well as risk management. The course further covers basics of stochastic calculus necessary for finance. Designed for Masters students.

A mathematical introduction to convex, conic, and nonlinear optimization. Topics include convex analysis, duality, theorems of alternatives and infeasibility certificates, semidefinite programming, polynomial optimization, sum of squares relaxation, robust optimization, computational complexity in numerical optimization, and convex relaxations in combinatorial optimization. Applications drawn from operations research, dynamical systems, statistics, and economics.

A theoretical introduction to statistical machine learning for data science. It covers multiple regression, kernel learning, sparse regression, high dimensional statistics, sure independent screening, generalized linear models, covariance learning, factor models, principal component analysis, supervised and unsupervised learning, deep learning, and related topics such as community detection, item ranking, and matrix completion.These methods are illustrated using real world data sets and manipulation of the statistical software R.

This course is an introduction to stochastic calculus for continuous processes. The main topics covered are: construction of Brownian motion, continuous time martingales, Ito integral, localization, Ito calculus, stochastic differential equations. Girsanov theorem, martingale representation, Feynman-Kac formula. If time permits, a brief introduction to stochastic control will be given.

An introduction to analytical and computational methods common to problems in financial mathematics and mathematical economics. Aimed at Ph.D. students and advanced masters students who have studied stochastic calculus, the course focuses on applications of partial differential equations (PDEs) in models used in finance and economics. These include pricing financial derivatives, stochastic control problems and stochastic differential games, as well as mean field games. We discuss analytical, asymptotic, and numerical techniques for their solution.

We start this lecture by introducing some classical stochastic control problems, including optimal portfolio allocation, Merton utility maximization problem, real option, and contract theory. This introduction motivates us to study, after a short recall on stochastic calculus, some ways to solve stochastic control problems as well as optimal stopping problem. This leads us on a journey through the dynamic programming principle, the Hamilton Jacobi Bellman (HJB) equations, the notion of viscosity solution, up to the theory of BSDEs.

An introduction to the theory and practice of high frequency trading in modern electronic financial markets. We give an overview of the institutional landscape and basic empirical features of modern equity, futures, and fixed income markets. We discuss theoretical models for market making and price formation. Then we dig into detailed empirical aspects of market microstructure and how these can be used to construct effective trading strategies. Course work is a mixture of theoretical and data-driven problems. Programming environment is a mixture of the R statistical environment, with the Kdb database language.

This course explores cutting-edge aspects of transformers and large language models, which have revolutionized natural language processing and various other domains in artificial intelligence. Key topics include transformer architecture fundamentals, self-attention mechanisms and positional encodings, probabilistic foundations of language modeling and sequence prediction, pretraining strategies and transfer learning in language models, scaling laws and the implications of model size on performance, fine-tuning techniques for specific tasks and domains, and efficiency improvements and model compression techniques.