# Switching to BA Mathematics

#### EconomicsStudent

Well, I decided to switch programs. I transfered into BA Math, and I will be starting in September. My goal is obviously still to pursue a master's in quant finance.

Here are some of the relevant courses I will be taking:

AS/SC/MATH 1000 3.00 F
Differential Calculus (Honours Version)

Calendar copy: Axioms for real numbers, limits, continuity and differentiability. This course covers slightly fewer topics than AS/SC/AK/MATH 1300 3.00, but covers them in greater depth. It should be taken by all those planning an Honours degree in Mathematics or a Specialized Honours degree in Statistics.
MATH 1000 aims to develop the students' ability to think and write clearly, logically and precisely, and to read a mathematical text with understanding, and to write proofs.

AS/SC/MATH 1010 3.00 W
Integral Calculus (Honours Version)

Calendar copy: Riemann integral, fundamental theorems of calculus, transcendental functions, integration techniques, sequences, series. This course covers fewer topics than AS/SC/AK/ MATH 1310 3.00, but covers them in greater depth. It should be taken by all those planning an Honours degree in Mathematics or a Specialized Honours degree in Statistics.

AS/SC/AK/MATH 1021 3.00 FW
Linear Algebra I

(formerly AS/SC/MATH 2021 3.00 — before 2001-2002)
Calendar copy: Linear equations, matrices, Gaussian elimination, determinants and vector spaces. This course covers material similar to that in AS/SC/AK/MATH 2221 3.00 but at a more advanced level. Required in Specialized Honours Statistics and all Applied Mathematics, Mathematics and Mathematics for Commerce programs except the BA Program in Mathematics for Commerce.
After the concepts in logic and set theory, the most fundamental idea in all of mathematics is that of a function. The simplest type of function is a linear function, and linear functions (also called linear transformations) are what linear algebra is about. Thus, linear algebra is mathematically more basic than, for instance, differential calculus, where more complicated functions are approximated locally by linear ones. Apart from underpinning much of mathematics, linear algebra has a vast range of applications — from quantum mechanics (where it is crucial) to computer graphics to business and industry (via statistics and linear programming).
Additional topics: Euclidean n-space, lines and planes, linear transformations from ℝn to ℝm, abstract vector spaces, basis and dimension, rank and nullity of a matrix.

AS/SC/AK/MATH 1131 3.00 FW
Introduction to Statistics I

Calendar copy: Displaying and describing distributions; relations in categorical data; Simpson's paradox and the need for design; experimental design and sampling design; randomization; probability laws and models; central limit theorem; statistical inference including confidence intervals and tests of significance; matched pairs; simulation.
Testing a new drug, pricing a derivative asset, evaluating the effects of free trade, making sound investment decisions, and predicting who will win the World Series are all activities that have in common the need to make sense out of ambiguous data. The modern discipline of statistics serves as a guide to scientists, policy makers and business managers who must draw inferences or make decisions on the basis of uncertain information.
Topics include collection and analysis of data, graphical methods to represent data, numerical methods for describing univariate data both for samples and population, summarizing bivariate data, random variables and probability distributions, sampling variability and sampling distributions, estimation and testing using a single sample, comparison of two populations.
It is recommended that students have at least one OAC in mathematics or a 12U mathematics course, but the mathematical level of the course will be quite elementary. Although students will be making use of the computer to calculate statistics, to create statistical plots, and to obtain a better appreciation of statistical concepts, no previous experience in computing is required. Students will receive all the necessary instruction about how to use the statistical computer package chosen by the instructor.
Although this course is recommended for students who wish to major in statistics, the concepts are broadly applicable and it should be interesting to students who do not plan to specialize in statistics.

AS/SC/AK/MATH 2022 3.00 W
Linear Algebra II

Calendar copy: Inner product spaces, linear transformations, eigenvalues, diagonalization, least squares, quadratic forms and Markov chains. Similar to AS/SC/AK/MATH 2222 3.00 but at a more advanced level. Required in Specialized Honours Statistics and all Applied Mathematics, Mathematics and Mathematics for Commerce programs except the BA program in Mathematics for Commerce.

AS/SC/AK/MATH 2030 3.00 FW
Elementary Probability

Calendar copy: Introduction to the theory of probability as preparation for further study in either mathematical or applied probability and statistics. Topics include probability spaces, conditional probability, independence, random variables, distribution functions, expectation, Chebyshev's inequality, common distributions, moment-generating functions and limit theorems.
This course provides an introduction to the theory of probability. It covers the mathematics used to calculate probabilities and expectations, and discusses how random variables can be used to pose and answer interesting problems arising in nature. It is required for certain honours programs in Applied Mathematics, Computational Mathematics, Statistics, and Computer Science. Subsequent courses that use the material covered include mathematical statistics, operations research, stochastic processes, as well as more advanced courses in probability.

AS/SC/AK/MATH 2131 3.00 W
Introduction to Statistics II

Calendar copy: This course is a continuation of AS/SC/AK/MATH 2030 3.00. It provides students with an introduction to statistical methods with an emphasis on applications using continuous probability models. Note: Computer/Internet use may be required to facilitate course work.
A major part of the course is devoted to the study of basic tools needed in statistical inference. Topics include joint distributions, multivariate change of variables formula, conditional and marginal distributions, conditional expectation, covariance and correlation and moment generating functions. Distributional results including those associated with normally distributed observations are examined.
Statistical concepts include confidence intervals, tests of significance and hypotheses, likelihood and maximum likelihood estimation.

AS/SC/AK/MATH 2270 3.00 W
Differential Equations

Calendar copy: Introduction to differential equations, including a discussion of the formation of mathematical models for real phenomena; solution by special techniques; applications; linear equations; solutions in series; other topics if time permits.
Differential equations have played a central role in mathematics and its applications for the past three hundred years. Their importance in applications stems from the interpretation of the derivative as a rate of change, a familiar example being velocity. Many of the fundamental laws of physical science are best formulated as differential equations. In other areas, too, such as biology and economics, which involve the study of growth and change, such equations are of fundamental importance.
In this course we will study some important types of linear differential equations and their solutions. Topics will include first-order (differential) equations; homogeneous second and higher order equations with constant coefficients; the particular solution of inhomogeneous second-order equations; first-order linear systems, solutions and phase plane; series-form solutions of equations with variable coefficients; solutions by use of Laplace transforms.
Students will use the symbolic computational computer language maple to study the behaviour of differential equations. No prior experience with this language is necessary.

AS/SC/AK/MATH 2310 3.00 FW
Calculus of Several Variables with Applications

Calendar copy: Vector functions, partial derivatives, gradient, multiple integrals, line integrals, optimization, applications. Offered in both terms.
Other topics include lines, planes, curves in two and three dimensions, polar coordinates, arc length, Lagrange multipliers, change of coordinates in multiple integrals.

AS/SC/AK/MATH 3010 3.00 F
Vector Integral Calculus

Calendar copy: Integrability of continuous functions over suitable domains, iterated integrals and Fubini's theorem, counterexamples, change of variables, Jacobian determinants, polar and spherical coordinates, volumes, vector fields, divergence, curl, line and surface integrals, Green's and Stokes' theorems, differential forms, general Stokes' theorem.

AS/SC/AK/MATH 3131 3.00 F
Mathematical Statistics I

Calendar copy: Topics include common density functions, probability functions, principle of likelihood, the likelihood function, the method of maximum likelihood, likelihood regions, tests of hypotheses, likelihood ratio sets, goodness of fit tests, conditional tests, and confidence tests with a view towards applications.
In this course, we first consolidate and extend some notions acquired in MATH 2131. We then introduce the basic concepts of statistical inference and apply them to various examples. For estimation, we study the notions of unbiasedness, sufficiency and consistency. Confidence regions are derived for parameters in various classical problems. For hypothesis testing, we define the notions of most powerful, uniformly most powerful tests and likelihood ratio tests. We prove the Neyman-Pearson lemma and give several examples of testing problems.

AS/SC/AK/MATH 3132 3.00 W
Mathematical Statistics II

Calendar copy: Important examples and methods of statistical estimation and hypothesis testing are discussed in terms of their mathematical and statistical properties. Topics include sufficiency, Bayesian statistics, decision theory, most powerful tests, likelihood ratio tests.
In this course, we first present some of the standard hypothesis tests that are most widely used in applications. We then present nonparametric testing procedures for the location models and offer an introduction to traditional Bayesian methods. The traditional least squares methods for linear models are studied. We finally discuss the tests of general linear hypotheses and give several examples.

AS/SC/AK/MATH 3170 6.00 Y
Operations Research I

Calendar copy: A study of linear programming; transportation problems, including network flows, assignment problems and critical path analysis; integer programming; dynamic programming and an introduction to stochastic models. Application to a set of problems representative of the field of operations research.
This course deals with standard optimization techniques used in Operations Research. These techniques are widely used in managerial decision making, and also lead to interesting mathematical theory. We shall investigate problem formulation, underlying theory, applications, and practical implementation. The main topics include:

(a) Linear Programming, including the simplex algorithm, duality theory, sensitivity analysis;
(b) Network Problems, including the transportation algorithm, network flows, assignment problem, shortest-path problems, critical path scheduling;
(c) Integer Programming, including solution by the branch-and-bound method;
(d) Dynamic Programming.

AS/SC/MATH 3241 3.00 F
Numerical Methods I

Calendar copy: An introductory course in computational linear algebra. Topics include simple error analysis, linear systems of equations, nonlinear equations, linear least squares and interpolation. (Same as AK/SC/AS/CSE/COSC 3121 3.00.)
The course begins with a discussion of computer arithmetic and computational errors. Examples of ill-conditioned problems and unstable algorithms will be given. The first class of numerical methods introduced are those for nonlinear equations, i.e., the solution of a single equation in one variable. We then discuss the most basic problem of numerical linear algebra: the solution of a linear system of n equations in n unknowns. We discuss the Gauss algorithm and the concepts of error analysis, condition number and iterative refinement. We then use least squares to solve over determined systems of linear equations. The course emphasizes the development of numerical algorithms, the use of mathematical software, and interpretation of results obtained on some assigned problems.

AS/SC/MATH 3242 3.00 W
Numerical Methods II

(same as AK/AS/SC/CSE/COSC 3122 3.00)
Calendar copy: Algorithms and computer methods for solving problems of differentiation, integration, systems of non-linear equations, and matrix eigenvalues. (Same as AS/SC/AK/CSE/COSC 3122 3.00.)
The course is a continuation of MATH 3241 3.00/CSE/COSC 3121 3.00. The main topics include numerical differentiation, Richardson's extrapolation, elements of numerical integration, composite numerical integration, Romberg integration, adaptive quadrature methods, Gaussian quadrature, numerical improper integrals; fixed points for functions of several variables, Newton's method, quasi-Newton methods, steepest descent techniques, and homotopy methods; power method, Householder method and QR algorithms.

AS/SC/MATH 3271 3.00 F
Partial Differential Equations

Calendar copy: Partial differential equations of mathematical physics and their solutions in various coordinates, separation of variables in Cartesian coordinates, application of boundary conditions; Fourier series and eigenfunction expansions; generalized curvilinear coordinates; separation of variables in spherical and polar coordinates.
The course will be based on the three archetypical equations from mathematical physics: the wave equation, Laplace's equation, and the heat equation. Using these equations in various contexts as examples and motivation, the basic mathematical techniques for solving second order partial differential equations will be developed.

AS/SC/AK/MATH 3330 3.00 F
Regression Analysis

Calendar copy: Simple regression analysis, multiple regression analysis, matrix form of the multiple regression model, estimation, tests (t- and F-tests), multicollinearity and other problems encountered in regression, diagnostics, model building and variable selection, remedies for violations of regression assumptions.
Note: Prior to FW 2002, two regression courses were offered, MATH 3033 3.00 (Classical Regression Analysis) and MATH 3330 3.00. Starting FW 2002, MATH 3033 has been absorbed into MATH 3330.
This course is intended as a thorough introduction to the use of linear models in statistical analysis, for students who have had at least two terms of statistics. We will be focusing on situations where we have one dependent variable and one or more explanatory variables. The material covered will be drawn from the textbook, supplemented by some material from other sources. The emphasis will be on the use of the models in question for helping in the analysis of data, not on theoretical derivations. Tentatively the lectures will cover Chapters 1-12. The focus will be on Chapter 4 to Chapter 12 inclusive. Students will be expected to use available computing resources, primarily William Small Center (formerly PSII) or Gauss Lab. We will mostly present sample programs and solutions using SAS.

AS/SC/AK/MATH 4010 6.00 Y
Real Analysis

Calendar copy: Survey of the real and complex number systems, and inequalities. Metric space topology. The Riemann-Stieltjes integral. Some topics of advanced calculus, including more advanced theory of series and interchange of limit processes. Lebesgue measure and integration. Fourier series and Fourier integrals.

AS/SC/AK/MATH 4090 3.00 W
Mathematical Modelling

Calendar copy: Discrete, continuous and probabilistic modelling of problems from industry, finance and the life and physical sciences. The ability to model complex problems is stressed.
1) Given an amino acid sequence, can the transmembrane conformation of the encoded protein be predicted?
2) What parameters can be inferred from a population model as the weak links in a stage-classified whale population, i.e. parameters that a conservation policy should focus on?
3) What can ant collective behaviour teach us about the emergence of successful cooperation among agents pursuing individual aims?

This course will introduce the student to traditional and newer methods of mathematical modelling, with an emphasis on data-driven methods, i.e. such methods that are suitable for handling empirically rich situations in biology, ecology, population dynamics, and emergence of complex phenomena. Topics covered include model estimation methods, Hidden Markov Models, and stochastic dynamics. These subjects will be studied analytically, probabilistically and by using computational software, in particular MATLAB and Sequence Alignment and Modelling Software System (SAM).

AS/SC/AK/MATH 4143 3.00 W
Scientific Computation for Financial Applications

Calendar copy: This course covers the basics of numerical analysis/computational methods related to portfolio optimization, risk management and option pricing. It provides background material for computations in finance for two streams in the Computational
Mathematics program and other interested students.
This course introduces the basic concepts in mathematical finance. The topics include basic numerical methods; unconstrained and constrained optimization methods applied to portfolio selection; option pricing and risk management by MC simulation.
The text has not been chosen yet.

AS/SC/MATH 4170 6.00 Y
Operations Research II
(same as GS/MATH 6900 3.00 plus
GS/MATH 6901 3.00)
Calendar copy: Selected topics from game theory, decision theory, simulation, reliability theory, queuing theory, nonlinear programming, classifications, pattern-recognition and prediction. Each chapter contains an optimization problem and methods and algorithms for solving it. The course is rich in examples.
This course deals mainly with probabilistic models based on optimization. The following topics will be discussed: (a) Game Theory: how to find the best strategies in a confrontation between two players with opposite interests; (b) Decision Theory: how to act in order to minimize the loss subject to the available data; (c) Simulation: how to get representative samples from probability distributions and accurately approximate multiple integrals using random numbers; (d) Reliability Theory: how to evaluate the lifetime of a system consisting of many interacting subsystems; (e) Queueing Theory: how to assess what may happen in a system where the customers arrive randomly, wait in line, and then get served; (f) Uncertainty: how to measure uncertainty in probabilistic modelling with applications to pattern-recognition and classification.

AS/SC/AK/MATH 4430 3.00 W
Stochastic Processes

Calendar copy: Basic Markov processes, including Markov chains, Poisson processes, birth-and-death processes, Brownian motion.
Note: This course will not be offered in FW 2007.
In this course, we begin with review of probability theory (especially conditional expectation). We introduce transition probability matrices for discrete-time Markov chains. We then cover the following topics: first step analysis, the long-run behaviour of Markov chains which includes classification of states and basic limit theorems, Poisson processes, Birth and death process and its limiting behaviour, and finally Brownian motion.

AS/SC/AK/MATH 4431 3.00 W
Probability Models

Calendar copy: This course introduces the theory and applications of several kinds of probabilistic models, including renewal theory, branching processes, and martingales. Additional topics may include stationary processes, large deviations, or models from the sciences.
Probability theory has been used to describe and analyze many kinds of real-world phenomena. This course will investigate several classes of probability models, including the following:
- Renewal processes are used to model an event that occurs repeatedly at random times, such as the failure of a machine component. The focus of study is on the long-run average behaviour of such processes.
- Branching processes are a class of simple population growth models. One important question is how the distribution of the number of offspring of one parent can be used to predict the probability that the population eventually dies out. Generating functions will be introduced and used to derive results.
- Martingales are models of "fair games". They have been used to study stock market behaviour and are an important theoretical tool for a wide variety of probability problems. Important results include descriptions of your expected gain if your decision of when to quit (e.g. when to sell the stock) is determined by the behaviour of the process itself.

AS/SC/MATH 4830 3.00 W
Time Series and Spectral Analysis

Calendar copy: Treatment of discrete sampled data by linear optimum Wiener filtering, minimum error energy deconvolution, autocorrelation and spectral density estimation, discrete Fourier transforms and frequency domain filtering and the Fast Fourier Transform algorithm.

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How well will this prepare me for quantitative finance?

#### financeguy

This looks like a good preparation for a MFE. Make sure you learn how to program well while you're taking these classes. It's easy to slack off and just get by without really learning the actual programming, but that would only hurt you in the end. Also, just like I would tell anyone who is just starting out their academic career, don't limit yourself to just quant finance just yet. If you find you like something tangential you're studying and want to take more courses in that subject, go for it. If you're just curious about a subject, take a course in that. Most people come from programs like electrical engineering into MFE, so don't worry about taking the exact perfect courses. Just make sure you have the following skills: multivariable calculus (2 courses if possible), linear algebra (2 courses if possible), ODEs (a course in PDEs too if possible), probability theory (with calculus), statistics (with calculus), programming (good languages to be familiar with: Java, C#, C++, VBA, MATLAB, SAS and/or other statistical packages). This is all you really NEED. Of course, it would be good for you to take a course in numerical analysis and one in stochastic processes. If you can squeeze it in, maybe a couple straight finance courses (skip economics). If you're doing a MFE, don't worry about doing quant finance in your undergrad. You're better served taking non-quant finance to develop a solid intuition for it while taking lots of those math courses. Then the MFE will bring it all together for you.

#### EconomicsStudent

Of course, it would be good for you to take a course in numerical analysis and one in stochastic processes. If you can squeeze it in, maybe a couple straight finance courses (skip economics).

I will be taking all of the courses on the list I posted. Numerical Methods I & II, as well as Stochastic Processes, are all on the list.

Also, since I am switching from an economics program, I have already taken econ and finance courses.

The only quant finance course I am taking in undergrad will be "Scientific Computation with Financial Applications." Which consists of risk management, asset allocation, and derivatives pricing with MC simulation.

#### Andy Nguyen

Looks like this will prepare you well for a master degree in QF. Just do well in these courses, get good grades and make good impression with the instructors. They will be good sources when it comes to letter of recommendation.

#### EconomicsStudent

Thanks.

Do you think it would be worthwhile to take a financial econometrics course as well?

#### Andy Nguyen

I don't think it will ever hurt if and only if after finishing all other courses, find the time to take extra financial economics. There is a trade off between taking all the courses you want and graduate in short time.

#### grivira

The more math classes, the better. Some courses, however, are not as useful as they sound/look. With the time constraint you are facing right now (I assume you have 3 or less years left in college), you have to make some choices.

First of all, scratch all the "Numerical Analysis/methods" from your list. The main components of numerical methods is to handle matrice numerically. I suggest you taking "Linear Algebra II" instead to strengthen your theoretical background, and "Operation research" for the modeling part.

Also take down "Mathematical Statistics". Never take a stats courses offered by math department. Take a few regressional and time-series analysis courses straight out of stats department since they are more practical and more pertinent to math finance study.

For the courses you listed on your other thread, forget about ALL classes offered by econ dept (math econ, econometrics, stats for econ, etc) other than micro and macro. For graduate-level math class you are considering, you only need to take real analysis and measure-theory-based probability. No topology, no abstract algebra, no fourier/complex/wavelet/harmonic analysis.

I switched my major from finance to math&econ in my freshman year, dropped econ and graduate with a BS in math, and was admitted to CMU, Columbia and Cornell's math finance programs. Trust me, I have been through the same process you are experiencing. I really have a lot to talk about every question you mentioned, but have to refrain from writing too much which would make my post too long to read.

Good luck!

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