Aggressive Curriculum:
Computational Finance (B.S.) + Economics (B.S.)
Jerome
Combes-Knoke is pursuing a major in both Computational Finance (B.S)
and Economics (B.S.). He was one of ten students selected to
participate in the highly-competitive CMU Computational Finance, an interdepartmental blend providing depth in the triad
of business, mathematics, and computer science. Below are the
courses completed to date. Despite the course load, he is
on the "Dean's List with High Honors."
MATHEMATICS:
21-120 Differential and
Integral Calculus (10 units). Functions, limits,
derivatives, logarithmic, exponential, and trigonometric
functions, inverse functions; L’Hospital’s Rule, curve
sketching, Mean Value Theorem, related rates, linear and
quadratic approximations, maximum-minimum problems, inverse
functions, definite and indefinite integrals, and hyperbolic
functions; applications of integration, integration by
substitution and by parts.
21-122 Integration,
Differential Equations, and Approximation (10 units).
Integration by trigonometric substitution and partial fractions;
arc length; improper integrals; Simpson’s and Trapezoidal
Rules for numerical integration; separable differential
equations, first order linear differential equations,
homogeneous second order linear differential equations with
constant coefficients, series solution, Newton’s method,
Taylor’s Theorem including a discussion of the remainder,
sequences, series, power series.
21-126 Introduction to
Mathematical Software (3 units). This course provides
an introduction to the use of several software packages, which
are useful to mathematics students. Among the packages are
Maple and Mathematica for symbolic computing, TeX
and LaTeX for mathematical documents, and Matlab
for numerical computing. The course will also introduce the
mathematical facilities built into spreadsheets such as Excel.
The aim of the course is to provide the student with some basic
skills in the use of this software without attempting complete
coverage. A deeper knowledge of the software will be easy to
obtain after completing this course.
21-241 Matrix Algebra
(9 units). Vectors and matrices, the solution of linear systems
of equations, vector spaces and subspaces, orthogonality,
determinants, real and complex eigenvalues and eigenvectors,
linear transformations.
21-256 Multivariate Analysis
and Approximation (9 units). Taylor’s Theorem;
geometric sequences and series and their applications in
compound interest; vectors and matrices, lines, and planes;
partial derivatives, directional derivatives, gradient, chain
rule, maximum-minimum problems, Lagrange multipliers and
the Kuhn-Tucker Theorem.
21-260 Differential
Equations (9 units). Ordinary differential equations:
first and second order equations, applications, Laplace
transforms; partial differential equations: partial
derivatives, separation of variables, Fourier series;
systems of ordinary differential equations; applications.
21-270 Introduction to
Mathematical Finance (9 units). The theme of this
course is pricing derivative securities by replication. The
simplest case of this idea, static hedging, is used to discuss
net present value of a non-random cash flow, internal rate of
return, and put-call option parity. Pricing by replication is
then considered in a one-period random model. Risk-neutral
probability measures, the Fundamental Theorems of Asset
Pricing, and an introduction to expected utility
maximization and mean-variance analysis are presented in this
model. Finally, replication is studied in a multiperiod
binomial model. Within this model, the replicating strategies
for European and American options are determined.
21-292 Operations Research
(9 units). Operations research offers a scientific approach to
decision making, most commonly involving the allocation of
scarce resources. This course develops some of the fundamental
methods used. Linear programming: the simplex method and
its linear algebra foundations, duality, post-optimality and
sensitivity analysis; the transportation problem; the critical
path method; non-linear programming methods.
21-370 Discrete Time Finance
(9 units). This course introduces the Black-Scholes option
pricing formula, shows how the binomial model provides a
discretization of this formula, and uses this connection to fit
the binomial model to data. It then sets the stage for
Continuous-Time Finance by discussing in the binomial model
the mathematical technology of filtrations, martingales,
Markov processes and risk-neutral measures. Additional
topics are American options, expected utility maximization, the
Fundamental Theorems of Asset Pricing in a multi-period
setting, and term structure modeling, including the Heath-Jarrow-Morton
model. Students are expected to read and write proofs.
21-420 Continuous-Time
Finance (9 units). This course begins with Brownian
motion, stochastic integration, and Ito’s formula from
stochastic calculus. This theory is used to develop the
Black-Scholes option pricing formula and the Black-Scholes
partial differential equation. Additional topics may
include models of credit risk, simulation, and expected utility
maximization.
STATISTICS:
36-201 Statistical Reasoning
and Practice (9 units). This course will introduce
students to the basic concepts, logic, and issues involved in
statistical reasoning, as well as basic statistical methods used
to analyze data and evaluate studies. The major topics to be
covered include methods for exploratory data analysis, an
introduction to research methods, elementary probability, and
methods for statistical inference. The objectives of this
course are to help students develop a critical approach to the
evaluation of study designs, data and results, and to develop
skills in the application of basic statistical methods in
empirical research. An important feature of the course will be
the use of the computer to facilitate the understanding of
important statistical ideas and for the implementation of data
analysis.
36-225 Introduction to
Probability and Statistics I (9 units). Provides an
introduction to probability and mathematical statistics for
students in mathematics and statistics. The use of probability
theory is illustrated with examples drawn from engineering, the
sciences, and management. Topics include elementary probability
theory, conditional probability and independence, random
variables, distribution functions, joint and conditional
distributions, law of large numbers, and the central limit
theorem.
36-226 Introduction to
Probability and Statistics II (9 units). This course is
the second half of a year long course in probability and
mathematical statistics. Topics include maximum likelihood
estimation, confidence intervals, and hypothesis testing. If
time permits there will also be a discussion of linear
regression and the analysis of variance.
36-410 Introduction to
Probability Modeling (9 units). An introductory-level
course in stochastic processes. Topics typically include
Poisson processes, Markov chains, birth and death
processes, random walks, recurrent events, and renewal theory.
Examples are drawn from reliability theory, queuing theory,
inventory theory, and various applications in the social and
physical sciences.
COMPUTER PROGRAMMING:
15-100
Introductory/Intermediate Programming (10 units). An
introduction to the process of program design and analysis using
the Java programming language. Topics to be covered
include basic data types and their operators, I/O, control
structures (selection, loops), classes (including methods and
fields), arrays, and simple sorting and searching algorithms.
21-369
Numerical Methods (9 units). This course provides an
introduction to the use of computers to solve scientific
problems. Methods for the computational solution of linear
algebra systems, nonlinear equations, the interpolation and
approximation of functions, differentiation and integration, and
ordinary differential equations. Analysis of round-off and
discretization errors and programming techniques.
BUSINESS & ECONOMICS:
70-122 Introduction to
Accounting (9 units). This course provides the
knowledge and skills necessary for the student to understand
financial statements and financial records and make use of the
information for management and investment decisions. Topics
include an overview of financial statements and business
decisions; the balance sheet, the income statement, and the cash
flow statement; sales revenue, receivables, and cash; cost of
goods sold and inventory; long-lived assets and depreciation,
and amortization; current and long-term liabilities; owners’
equity; investments in other corporations; an introduction to
financial statement analysis; and international issues dealing
with financial statements.
70-365 International Trade
and International Law (9 units). The course discusses
the international legal system and laws that affect
international trade. It covers the Foreign Corrupt Practices
Act, treaties and concessions, shipping and customs,
appointment of foreign sales agents, resolution of trade
disputes, international mergers and joint ventures,
international competition law, UN sales convention,
international trade organizations (IMF, WTO,
World Bank, etc.), risk insurance, cultural factors,
international E-Commerce and intellectual property.
70-391 Finance (9
units). The course examines the role of the financial manager
in the overall management and control of a firm. Stress is
placed on the use of analytical models for improving the
decision-making process. Both the short-term management of
working capital and the long-term planning of capital structure
and investment strategy are covered.
73-100 Principles of
Economics (9 units). Literally, an introduction to
economic principles, the goal of this course is to give students
an understanding as to what constitutes good “economic
thinking”. This thought process is grounded in the construction
and use of economics models. Drawing on issues in both
microeconomics and macroeconomics, fundamental principles are
shown to transcend particular examples and allow the field to be
seen as a coherent, unified whole.
73-200 Macroeconomics
(9 units). Through macroeconomic models built upon
microeconomic foundations, insights are developed into economic
growth processes and business cycles. Topics include
aggregation and measurement, national income, business cycle
measurement, economic welfare theorems and social
inefficiencies, the effect of government fiscal policy upon
employment and productivity, and the relationship between
investment, interest rates and economic growth.
73-226 Quantitative Economic
Analysis (9 units). Using and extending upon students’
introductory knowledge of probability and economic models, this
course introduces students to the tools of economic analysis.
Taking the perspective of active economic participants (rather
than outside observers), students gain experience with a
diversity of analytical techniques--such as regression analysis
and simulation--in the context of real world data decision
problems.
73-251 Economic Theory
(9 units). This course prepares students for advanced
coursework in economics by providing a mathematically intensive
overview of economic theory. Students take advantage of their
knowledge of multi-dimensional calculus and constrained
optimization techniques in order to understand the development
and logical consistency of the most commonly employed economic
models. Topics include consumer preferences and utility
function representations, consumer choice under a budget
constraint, substitution and income effects, compensated and
uncompensated demands, expected utility theory, risk and
insurance, technology and production functions, cost
minimization, profit maximizing firms, perfect competition,
single-firm markets, game theoretic analysis of markets with few
firms, introduction to general equilibrium models and the
welfare laws.
73-347 Game Theory for
Economists (9 units). An introduction to the theory of
non-cooperative games with an emphasis on economic
applications. After an initial examination of two-person,
zero-sum games, the notion of a Nash equilibrium in an
n-person, non-cooperative game is considered. Existence of and
refinements to the equilibrium concept are discussed in the
context of both normal and extensive form games. Economic
applications may include various topics, including Cournot
and Bertrand oligopoly models, general competitive exchange
equilibrium, and free rider problems.
73-365 Industrial
Organization (9 units). This course is concerned with
the economic analysis of industrial markets that are not
perfectly competitive. The effects of imperfect competition on
firms’ decisions (pricing, location, advertising, research and
development, among others) are reviewed. Implications of these
effects in terms of public policy are also discussed from a
variety of perspectives. Finally, applications to actual
markets are considered.
73-372 International Money
and Finance (9 units). Course concerns itself with the
determination of real, monetary, and financial aggregates and
the policies that influence them in an international context.
Topics include: monetary policy and its effects on employment
and inflation, the role of the banking system in the
transmission of monetary policy, credit markets, banks as
financial intermediaries, and the effect of domestic policies on
international trade and financial markets.
BROADENING COUSEWORK:
99-102 Computing Skills
Workshop (3 units). The course is comprised of mostly
Carnegie Mellon specific information and helps students
understand what resources are available to them and what
responsibilities they have as a user in our computing community.
33-106 Physics I for
Engineering Students (12 units). This is a first
semester, calculus-based introductory physics course. Basic
principles of mechanics and thermodynamics are developed.
Topics include vectors, displacement, velocity, acceleration,
force, equilibrium, mass, Newton’s laws, gravitation,
work, energy, momentum, impulse, temperature, heat, equations of
state, thermodynamic processes, heat engines, refrigerators,
first and second laws of thermodynamics, and the kinetic theory
of gases.
76-101 Interpretation and
Argument (9 units). This course will give students a
comprehensive grounding in communication processes. The class
focuses on the way in which interpretive arguments in the
processes of communication and social and personal development.
In the class, students will develop these skills by reading and
understanding the important issues and arguments regarding those
issues advanced by a variety of texts, both fiction and
non-fiction. They will then be asked to respond to these
positions by developing positions of their own, in their writing
and in their speaking. The course thus serves as an
introduction to the discourse and arguments of the academic
community, as well as serving as an introduction to some of the
broader issues that the academic community addresses.
79-104
Introduction to World History (9 units). Course
challenges students to think analytically about the major
historical processes that shaped and continue to shape cultures
and civilizations. The course is based on a series of case
studies that focus on shifting power relations between and
within civilizations. Three major themes connect the several
topics discussed throughout the semester: issues of authority
and inequality within civilizations; encounters and conflicts
between civilizations; and patterns of continuity and change
across space and time. The course demonstrates how historians
explain what has happened in the past and in various
civilizations and cultures; presents the kinds of evidence that
historians use to reconstruct the past; and examines the
interpretations historians make based on this evidence.
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