Rationale: Great advances in mathematical finance over the last two decades have lead to a field that requires advanced Mathematics as well as a background in Finance. Most notable is perhaps the development of the theory of option pricing developed by F. Black, B. Scholes and R. Merton that earned Merton and Scholes the 1997 Nobel Prize in Economics. The mathematics involved in these derivative pricing models involves a lot of advanced mathematical topics such as Brownian motion, stochastic calculus, binomial trees, numerical optimization, PDE's.

The proposed program aims to take advantage of interdisciplinary interaction and establish synergy between the Mathematics and Finance departments and their students while affording students an attractive and coherent program of study.

Graduates will also be exceptionally well equipped to enter graduate programs such as the DePaul's Master of Science in Finance (MSF) program as well as financial mathematics master's programs at the U of C, Columbia, Carnegie-Mellon, Cornell, Toronto, Cambridge and many others. We have found only one similar program at the undergraduate level, however far-flung, at the University of Singapore.  It should also be mentioned that graduates of this program will be well-prepared for graduate study in Applied Mathematics as well as Financial Mathematics and some MBA programs as well. Those who go on to earn graduate degrees in Financial Mathematics will be equipped to enter careers paths as described in the paragraph above, but at a higher level.  Those that earn a doctorate would be likely considered at the level of research director.

Compared with many peer Math departments, our Mathematical Sciences Department has a disadvantage due of a lack of an engineering school and relatively small physical science programs. This, along with the decline in traditional mathematics majors, puts our undergraduate program in a somewhat tenuous state. This concentration might increase our pool of majors overall, our advanced course enrollment and range of offerings. It should also increase the attractiveness of our discipline both intellectually and practically.

This program  will also satisfy the requirements for an minor in Finance for non-Commerce students, with courses designed to complement the Math concentration.  Also, a graduate of the program will be well on her or his way well to fulfilling requirements for the Masters of Science in Finance MSF program, which would take as little as 4 more quarters to complete. In any case, the graduate of the proposed program would have a very big head start in graduate programs in Financial Mathematics /Financial Engineering.

Careers: Financial Mathematics is an employable field in the Chicago area and in other financial centers (click here for a typical job ad). Students who earn this concentration will be well-equipped to enter the job market.  Employers recruit math majors for positions which draw on skills represented by the coursework in this program. Employers emphasize mathematical, financial, numerical, computational and competencies to varying degrees.  For Math majors, the knowlegede of and ability to absorb financial modalities (such as trading strategies and institutional factors) is important.  We believe that the attached Finance minor will give the graduate a boost because of the added numeracies and literacies from the Finance part of  this program. Typical jobs involve the following components: creation, maintenance, computer implementation and interpretation of mathematical models for financial risk management, especially those related to Black-Scholes-Merton models.  Career paths can be diverge in different directions,  so that one or more of these components might be emphasized.  Often a job will be to assist a research director on projects involving financial models by working mainly on implementation, data preparation and communication with traders. There are many firms in the Chicago and New York areas  that do
work in this area.  Other companies, e.g. large utilities, might employ graduates for work in modeling price fluctuations (e.g. in crude oil) and the attendant risk
management problems.

 Location: The program will not affect current offerings and scheduling initially. If the demand were to grow significantly, sections could be scheduled in the loop campus and in evenings in order to complement Commerce courses.

|sample curriculum |sample syllabus for methods course |rationale |requirements |top |

Core: 4 courses required by Commerce, Accounting 101, 102; Economics 105, 106 (micro).

Finance 310, 311 (Corporate Finance I,II), Finance 320 (Money and Banking), Finance 330 (Investments)
335 (Portfolio management), 337 (Options) and 362 (Risk Management)

Math part: (14 courses)
Core: 160-2 (or 150-2), 260-2, 215.
MAT 351-3 (Probability And Statistics sequence)
MAT 338 Differential Equations,
MAT 355 Stochastic Processes,
MAT 385 Numerical Analysis
Mathematical Methods in Finance

Other suggested courses: MAT 370 (advanced linear algebra), MAT 384 (math modeling).
Fin 339 (futures)
A course in any programming language (CSC 215 or 211 suggested).

Rationale for Advanced Math courses:
The course selected are course in applied mathematics that are relevant to the financial models, especially the Nobel laureate Black-Scholes-Merton option pricing theory mentioned above. The set of courses develops knowledge and competency that culminate in the derivation of the Black-Scholes model, it's implementation, and related topics. More specifically:

MAT 351-3:  This is our basic sequence in probability and statistics.  This is an introduction to the mathematical theory that is fundamental to the notiions of chance and risk, as discussed by Pascal and Bernoulli and axiomatized by Kolmogorov.  This course is absolutely foundational to mathematical finance is an essential part of the edifice of mathematical culture.
MAT 338: Differential equations.  The basic structure of mathematical models (see MAT 384) are these equations which form a vast subject themselves. They are canonical in applied mathematics.  Stochastic differential equations involve a probabilistic components as in MAT 355 below and are a starting point in the theory of options covered in the methods course.
MAT 355: Stochastic Processes are random variables as studied in 351-3 that can be thought of as evolving through time.  Prices with a random component are modeled by these processes, which form a part of the theory stochastic differential equations just mentioned.  The material in this course is a continuation of MAT 351-3, and constitutes on of the basic building block of the general theory.of the methods course.
MAT 385: Numerical Analysis.  This course consists of the study of sophisiticated number-crunching algorithms which allow for the implementation
of  financial models. The more practical side of the program is addressed here and in a programming course.

Links to course descriptions:
        Click here for Math course descriptions
        Click here for Finance course descriptions

Thus the program has 24 required courses. This leaves ample room for Liberal Studies requirements. It is understood that Economics 105 and 106 both fall into the "Self and Society" domain. Students who test out of Calculus via AP have even greater flexibility. One advantage of this major is that the student can easily switch into another mathematics concentration, or enroll as a Commerce student in midstream, and will be well prepared to enter graduate programs in either Math, Finance or hybrid programs. The finance requirements are just two courses short of those Finance courses required for a Finance major (students would still be missing some Commerce core requirements).

The Liberal Studies requirement, (which is exactly the same as for a standard math major) requirement is fulfilled. Learning goals can be addressed just as they are in the two resident departments. In addition this program is obviously interdisciplinary and requires the development of the multiple literacies of the financial, economic, computational, and mathematical communities.

|sample curriculum |sample syllabus for methods course |rationale |requirements |top |

 New Course: Mathematical Methods in Finance

This new course will be required of all students in the concentration. It will draw heavily on the previous coursework, especially concerning differential equations, stochastic processes and options. The main topics will be stochastic calculus and partial differential equations leading up to a derivation of the Black-Scholes option pricing model and its variants, along with discrete methods. This course also should be of interest to students of applied mathematics and physics. There will be ample opportunity to analyze real option data in conjunction with theory. Possible additional topics would include discrete methods. If there is sufficient interest, and additional quarter course can be added.

Comparison with Existing Programs: The proposed concentration is merely a major in mathematics that is tailored to financial mathematical modeling. Concurrently, a student would earn a Finance Minor while in LA&S. The relatively hefty minor Commerce requirements are by design and are unavoidable.

Utilization of Existing Resources: The program will draw on existing resources.
All of the required courses are listed and are taught at least annually. The new course might be taught by several Math Faculty members, e.g. J. Goldman, C. Georgakis, G. Wang, W. Chin, Y. Kashina.

|sample curriculum |sample syllabus for methods course |rationale |requirements |top |

 Sample  Curriculum

 Freshman Year
Autumn           Winter               Spring
MAT 160       MAT 161           MAT 162
Econ 105        Econ 106            Self/Soc.
 History 1        Compos. 1            Compos. 2
Disc. Chi.        Focal Point          Physics 170

Sophomore Year
Autumn              Winter                 Spring
MAT 260         MAT 261             MAT 262
ACC 101         ACC 102              MAT 215
Multicult            History 2               Arts/Lit 1
Arts/Lit 2          CSC 215 or 211    Elective

Jr. Year
Autumn         Winter             Spring
MAT 350         MAT 351         MAT 352
Fin 310             Fin 311            Fin 320
Arts and Lit 3    Philosophy 1      Philosophy 2
Religion 1          Religion 2          Experiential Learning

Sr. Year
Autumn         Winter                 Spring
MAT 338     MAT 355            MAT 385
Self /Society   Math Methods   Capstone
Fin 335          Fin 337               Fin 362
Fin 330           Elective                Elective

Remarks:
I've include Physics as a lab science in my example, since many financial models are analogous to physical systems as their analogs (e.g. Brownian motion, the heat equation). CSC 215 (Programming in C++) seems to be the most popular choice for financial applications (its MAT 140 prerequisite will be waived by CSC), though Java (CSC 211) is gaining in popularity.   Econ 105-6 both count toward the Self/Society requirement.

|sample curriculum |sample syllabus for methods course |rationale |requirements |top |

Textbook:
The mathematics of financial derivatives: A student introduction, by Paul Wilmott, Sam Howison, and Jeff Dewynne, Cambridge University Press, 1997.

Supplementary Texts:
1.  Options, futures, and other derivatives, by John Hull, Prentice Hall, 4th Edition, 1999;
2.  Derivative securities, by Robert Jarrow and Stuart Turnbull, South-Western College Publishing,
1996;
 3. Quantitative Models of Derivative Securities: From Theory to Practice, by Marco Avellaneda and
 Peter Laurence, Chapman & Hall, 1999.

Detailed schedule:

Week 1: Introduction to Financial Instruments: European options; put and call options;
forwards and futures. Arbitrage Pricing Theory. Put-Call parity. Pricing Futures.

Week 2:  Evolution of the stock prices; drift and volatility; standard deviation for the price of the stock.
Stochastic calculus: Ito's formula.

Week 3: Lognormal distribution of the stock prices. Derivative securities pricing on
one-period models; risk-neutral probability.

Week 4: Binomial trees simulation of the evolution of the stock price. Relationship between the binomial
tree model and the lognormal distribution. Binomial trees methods for european option pricing.

Week 5: American Puts. Derivation of the Black-Scholes formula from binomial trees pricing. Risk neutral
valuation.

Week 6: Dependence of the price of the call and put options on the underlying parameters -- heuristics
and proofs. Portfolio replication of the plain vanilla options.

Week 7: The diffusion equation. Fourier transform. Solution of the heat equation. Derivation of the Black-Scholes formula using portfolio replication.

Week 8: Estimation of the volatility of the stock price: historical volatility and implied volatility. Volatility
smile. Delta of an option. Delta hedging. Other partial derivatives: the "Greeks" G, q, vega, etc.

Week 9: Black-Scholes formula for dividend-paying assets. Continuously paid and discretely paid dividends. Pricing asset-or-nothing and cash-or-nothing options.

Week 10:  Options on Futures. Index Options.

Remark:  The are other topics which might be included in such a course but I believe that the focus on options
brings together the a good range of mathematical topics in the curriculum and makes for a coherent course.

|sample curriculum |sample syllabus for methods course |rationale |requirements |top |