**FM 5091/5092 Computation, Algorithms and Coding in Finance**

**FM 5011/5012 Mathematical Background for Finance**

A theoretical sequence that focuses on graduate level mathematics and statistics that builds a solid foundation for modeling and using financial data.

**FM 5011:** This course covers the basics of probability and measure theory useful in stochastic calculus. Its purpose is to develop many of the advanced mathematical tools that are necessary for the understanding of stochastic calculus and the derivation of the Black Scholes option pricing formula. Topics will include: sample spaces, Lebesgue measure and Lebesgue integral, limit theorems, martingales, elements of stochastic processes (example: Brownian motion), stochastic integration and Ito’s lemma, stochastic differential equations, some numerical approximations (example: Euler and Milstein), the derivation of the Black-Scholes option pricing formula.

**FM 5012:** The objective of this course is to introduce core ideas behind statistical methods and optimization techniques, with a special focus on their application in financial mathematics. Topics include univariate and multivariate random variables, distributions, time series analysis - especially ARMA and GARCH models, univariate and multivariate regressions and optimization – theory and applications.

**FM 5021/5022 Mathematical Theory Applied in Finance**

**FM 5021:**Linear contracts: forwards, futures and swaps. Arbitrage. Cash and carry arguments. Introduction to the valuation of options, binomial tree approach, delta-hedging argument in discrete time. and Monte Carlo simulation. Basic properties of the Brownian Motion, stochastic integral and Itō's lemma. Black-Scholes-Merton model. Delta-hedging argument in continuous time. Greek letters. Volatility smiles.

**FM 5022:** Review of Black-Scholes, Greeks and shortcomings, Value at risk, Principal component analysis, Introduction to time series applications to volatility estimation: ARCH, GARCH, Exotic Options, Stochastic and local volatility models, Equivalent martingale measure approach, Interest rate derivatives, standard market models, 1-factor and 2-factor models of the short rate, Heath-Jarrow-Morton model, LIBOR market model.

**FM 5031/5032 Practitioners Course**

This practicum sequence features four modules that are taught by financial industry practitioners. Each module is an independent mini-course that exposes students to various aspects of financial practice.

**Fixed Income Mathematics and Risk Management:** In this module you will learn about fixed income markets and the variety of instruments traded. Using Risk Management as a motivator, we explore various interest rate derivatives and the underlying mathematics necessary to understand and manage them. You will learn how to construct yield curves, measure interest rate risk, and calibrate stochastic interest rate models. The final project will be an Asset Liability Management application and will involve Monte Carlo simulation.

**Quantitative Risk Management:** The objective of this course is to provide a grounding in applied probability and statistics as it relates to the measurement of financial risk. The material is mainly organized around the text "Quantitative Risk Management, Concepts, Techniques, and Tools (Revised Edition)" by Alexander McNeil, Rüdiger Frey, and Paul Embrechts. Quizzes and assignments motivate the acquisition of vocabulary, financial and mathematical concepts, and scientific computing techniques. Projects provide exposure to the practice of professional research..

**Copula Models and Markov chain Monte Carlo (MCMC):** In this module we will review foundations of Bayesian statistical models and Monte Carlo methods leading to the construction of Metropolis-Hastings algorithm and Gibbs sampling. Copula models will be introduced and reassessed in the context of stochastic simulation. Case studies in credit risk analysis and economic capital evaluation will utilize the material learned in two prior modules in 5031. The mid-term simulation project will be presented by small groups in a classroom environment to a panel of industry representatives.

**Volatility Data Analysis: **Volatility Data Analysis: In this module, students will apply their knowledge of options theory and statistics to complete large data analysis projects. The analyses will focus on equity volatility, and will utilize the historical options data set provided by our partners at Delta Neutral. We will compare various volatility forecasting methodologies and also analyze the replication of variance swaps with vanilla options. Readings for the module will be sourced from industry white papers.

**FM 5990 - Topics in Financial Mathematics: Financial Data Analysis and Visualization Using Python**

The first part of the course is an introduction to financial data analysis and data visualization in Python. We use Jupyter notebooks and cover the numpy, pandas, and matplotlib packages. The second part of the course covers various supervised and unsupervised machine learning techniques including regression, k-nearest neighbors, principal components analysis, and k- means. This course is project-based and pass/fail. All exercises and projects use real-world financial data.

Projects and assignments will be turned in through Github, a framework for version control and collaboration.

**Note: This course is optional and offered periodically.**