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Maestría en Finanzas Cuantitativas


The program curriculum seeks to reconcile theoretical training with practical tools needed to successfully meet current challenges and requirements in the area of quantitative finance. It also seeks to foster the spirit of innovation and research in this area at the regional level.

Applicants are strongly advised to take the pre-master course Introduction to stochastic modeling, especially those who have not taken any intermediate probability and stochastic processes in their previous studies. This course aims to formally review some concepts of probability and present an introduction to the theory of stochastic processes. It has a duration of 32 hours (2 credits) and is offered as an intensive course months before the beginning of the first semester. The course will be graded as approved or not approved.


First Semester

During the first semester the student acquires fundamental knowledge common to different problems of quantitative finance. Special emphasis is placed on general aspects of financial markets, financial economics and derivatives and on mathematical tools and statistics such as stochastic calculation and advanced econometrics.

Code 14610002 - Credits 5

The course goal is to review the theoretical foundation of modern financial economics and basic concepts in finance. The course will review the basic financial instruments and the main models in portfolio theory, valuation of assets and financial derivatives.

Contents: Time value of money, arbitrage and short sale. Market rates: Libor, OIS, DTF, IBR. Cost of carry, Carry trade. Colombian capital market. Price, yield, and volatility of term structure of interest rates. Bond portfolio management. Sovereign and corporate Bonds. Stylized facts of the Colombian market. Introduction to stock valuation. Corporate financing. CAPM. Arbitrage pricing theory. Efficient market hypothesis empirical evidence. Fama-French model. Decisions under uncertainty and risk aversion. Capital allocation in risky assets. Optimal portfolios. Mean-variance analysis. Efficient portfolios. Efficient frontier and tangent portfolio. Active portfolio management: Black-Litterman. International investments. Derivatives: futures, forwards and swaps. Introduction to options. Valuation using binomial trees. Derivatives market in Colombia.

Code 14610003 - Credits 4

Econometrics seeks to measure and empirically verify the hypotheses proposed by economic theory. This course covers the main concepts and methods of econometrics and mathematical statistics with a level aimed at students of advanced training. Throughout the course, examples of implementation of the methods will be presented with the help of the R statistical package.

Contents: Review of basic concepts and methods: linear algebra, probability and distributions. Introduction to the R statistical package. Introduction to econometrics: modeling, identification and estimation. Estimation methods. Asymptotic theory. Linear regression model. Ordinary Least Squares (OLS) estimation; OLS Geometry. Partitioned regression and restricted regression. Goodness of fit and analysis of variance. Hypothesis tests. Linear model generalizations: Maximum Likelihood Estimation (ML). Score function and information matrix. Asymptotic properties: consistency, normality and efficiency. Statistical inference based on MV. Heteroscedasticity and the generalized regression model: Estimation by MCO. Generalized least squares (GLS). Feasible generalized least squares (FGLS).

Weighted generalized least squares (WGLS). Tests to detect heteroscedasticity. Estimation with Instrumental Variables: Models with latent variable and measurement errors. Explanatory variable omitted. Models of simultaneous equations. Consistent estimation and estimation by least squares in two stages (2EMC). Generalized Method of Moments (GMM): Definition. Estimates based on GMM. Identification. Asymptotic properties. Hypothesis tests based on GMM.

Code 14610001 - Credits 5

The stochastic calculus is a fundamental part of the modern theory of financial markets. Mathematical tools such as stochastic integration and stochastic differential equations allow a continuous time modeling of asset prices and other variables in financial markets. The objective of this course is to provide the student with the necessary skills to properly use this type of tools.

Contents: Brownian motion. Quadratic variation and variance of stochastic processes. Itô Integration. Itô formula. Stochastic differential equations. Principles of reflection. Distribution of maximums and minimums. Changes of measure. Martingale representation theorem. Girsanov's theorem. Kolmogorov equations. Feynman-Kac formula. Lévy processes: Infinite divisibility, definition of Lévy processes. Jumps measure, Levy measure. Lévy-Itô decomposition. Lévy–Khintchine representation.

As its main objective, this lecture seeks to have among the professionals who enroll in their postgraduate studies at the Universidad del Rosario, to ponder on the tools and basic competences of learning, which they shall put into practice in their professional life. As such, this institutional subject provides the fundamental tools for graduate students to be able to detect strengths or weaknesses and build their personal academic project from the Rosarista project. In the framework of this class, the activities will be developed and transferred to the virtual platform of institutional learning, so it will be an entirely virtual subject. In this space, students will be able to browse online for content such as multimedia products, booklets and evaluation activities, freely, without restrictions and under the guidance of a virtual tutor. The subject is divided into the following study or learning modules: Historical Tour; Academic Support Tools; History and Institutional Project.

Empirical studies in finance

Multifactorial (linear) models are the main instrument used by academics and professionals in finance to analyze the relationships between financial variables and economic fundamentals, the validation of pricing and hedging models, and the formulation of models in risk management. The objective of this course is to understand the financial specification, estimation and inference and some of the main uses of multifactorial models from the academic point of view and / or professionals in finance.

Contents: Factor models and their role in finance. Main factor models: CAPM, Arbitrage Pricing Theory (APT) and extensions (multifactor models). Applications: Portfolio optimization, Credit Risk, Affine term structure models. Estimation methods for factor models: Ordinary least squares, geometric interpretation, ridge regression. Principal component analysis. Empirical Finance: Some factor models for portfolio optimization and asset valuation. CAPM Estimating. Asset pricing anomalies (size, value-growth, momentum, liquidity). Fama-MacBeth regression, Fama-French factor model. Event studies. Portfolio optimization with Bloomberg and factor models.

Second Semester

The second semester courses are designed to allow students to acquire numerical and programming skills in the R and Python languages ​​and combine them with the tools acquired in the first semester to efficiently solve general problems of quantitative finance: construction of performance curves, optimization of portfolios, simulation, valuation of financial derivatives, modeling of time series, empirical finance, among others.

Code 14610004 - Credits 5

From an essentially theoretical perspective, the course provides the necessary skills to understand and properly use the stochastic models of financial markets in continuous time, and its application to valuation and hedging of derivatives and other structured financial instruments.

Contents: Stylized facts and empirical evidence of financial returns. Black-Scholes model (BS). Strategies of negotiation in continuous time. Completeness and assessment by replication. Equivalent martingale measures (MME) and risk neutral valuation. Forward price. Black-Scholes formula. Implied volatility and risk-neutral density. Breeden-Litzenberger formula. Valuation of dividend-paying stock options: Continuous dividends and discrete proportional dividends. Black formula (version 1) for options on forwards. FX options. Garman-Kohlhagen formula. Risk-neutral valuation, existence of MME and absence of arbitrage d-dim BS model. 1st and 2nd fundamental theorems of mathematical finance. Currency options. Exchange option and margrabe formula. Derivatives on assets denominated in foreign currency. Measurements of domestic and foreign martingales. Quantos. Risk-neutral valuation in incomplete markets.

Valuation in jump models via Fourier transform. Markovian negotiation strategies. Valuation and hedging using BSM-EDP. Barrier and look-back options. Sensitivity analysis and Greeks of BS formula. Market price of risk. Delta, Gamma and Vega hedging strategies. Discrete dividends. Asian options. Robustness of BS model and pricing equation in incomplete markets. Futures vs. Forwards. Forward-future spread. Black's formula (version 2). BS formula for stochastic interest rates. 1-factor affine short rate models. Valuation of zero-coupon bonds. Valuation of options on bonds and interest rate derivatives in 1-factor Gaussian models.

Code 14610005 - Credits 5

The purpose of the course is to understand and properly apply the specification, estimation and evaluation of time series models, volatility, correlation and valuation of financial assets, with special emphasis on the verification of the statistical and financial assumptions implicit in the techniques and the implementation of practice with the help of the R statistical package.

Contents: Introduction to time series. The classic decomposition models. Auto-regressive process order 1, stationarity and ergodicity. Hilbert spaces, projections. Wold decomposition. Specification of ARMA models: definition, causality, invertibility. ACVF, ACF, PACF, sample and population. Estimation.

ARIMA R Applications. Construction of a model and applications of ARMA models. Sampling, data generating processes and characteristics of financial series. Univariate Volatility: definition and measurement. Parametric estimation (discrete models and models in continuous time SV). Non-Parametric Estimation (Filters and Realized Volatility). Monitoring and forecasting of volatility. Correlation Models: definition and types of models. GARCH Multivariate. Scalar BEKK. Conditional correlation, dynamic conditional and applications. Estimation of discrete diffusion models (Parametric Models). Maximum Inference Exact Likelihood. Pseudo Maximum Likelihood. Methods of approximation to the function of Maximum Likelihood: Estimation equations and GMM.

Code 14610006 - Credits 5

In quantitative finance, there are many problems not supporting analytical solutions and demand the application of numerical methods to approximate a solution. This course addresses the study and application of these methods for solving problems in finance. The program includes the presentation of basic tools of numerical analysis, and specific methods for solving financial problems. Throughout the course, specific applications are studied, and computational tools are developed to find solutions to valuation problems of financial instruments, estimation of yield curves, portfolio optimization, among others.

Contents: Basics of Python programming. Generation of random variables. Inverse transform, Box-Muller and the polar method of Marsaglia. Pseudo-random numbers and seeds. Halto, Faure Sequences and Sobol of quasi-random numbers. Euler-Maruyama method. Analysis of mean-variance of portfolios. Implied volatility, bootstrapping, non-analytic inverse transformation for the generation of random numbers. Roots of equations: bisection, fixed point, Newton-Raphson, secant and regula falsi. Integration methods: trapezoidal, Runge-Kutta, Simpson. Application to valuation of options. Fast Fourier and Fourier transforms. Errors, convergence and numerical stability. Monte-Carlo simulation, confidence intervals and valuation of European, American and Asian options. Extensions of Monte-Carlo methods. Antithetic variables. Control variate, moment matching, low discrepancy. Interpolation and extrapolation. Data adjustment and model calibration: Least squares, splines, Lagrange and Fourier. Discount curves. Binomial models and applications to valuation of American options. Convergence of CRR to BS. Finite Differences: Implicit and Explicit Schemes. Cholesky and LU decomposition. Iterative SOR Crank-Nicolson, PSOR, application to American and exotic options (Asian, barrier and lookback).

Third Semester

In the third semester, students are required to take a compulsory course of Quantitative Risk Management (2 credits - 24 hours) and must elect 4 electives (each of 2 credits) which allows them to deepen and specialize in more specific subjects, depending on their interests.

Code 14610007 - Credits 2

*Prerequisite: Financial Econometrics

This course seeks to guide students in the field of quantitative risk management, so that they can analyze the latest conceptual and methodological advances in the subject. The student is expected to become familiar and able to build models for the different types of financial risk: market, credit, counterparty, operational, liquidity, systemic risk, among others. At the end of the course the student must have sufficient skills and the necessary knowledge in quantitative risk management to formulate and implement a generic model of risk management, with special emphasis on market and credit risks; develop a technical document detailing the model to a non-specialized public and understand and interpret a text or specialized journal of financial risk analysis.

Contents: Why manage the risk? Regulation (Basel 1, 2 and 3). Introduction to the most relevant types of Risk: Market, Credit and Counterparty, Operational, Liquidity, Systemic. Loss distribution. Estimation and statistical Inference. Risk factors and risk maps. Risk Measures: Definition, Properties and statistical inference. Risk analysis. Interpretation and use of the models. Aggregation of risk. Distribution of Loads of Risk, provisions, reserves and economic capital. Market Risk: measures and main models, implementation and back testing. Credit Risk: Structural Models of Bankruptcy Risk, Credit Risk Portfolio Model, CreditVaR. Study Case: rural bank.

Credits 5


This expertise can be attended with a dedication of full or parttime. The program is designed for the student to complete the programin one year and half or two years and a half, in accordance with its flexibility to attend classes and the evolutionof their degree.

Total credits per semester
Credits Time intensity Hours of independent work Total hours
15 12 33 45