Mathematical Methods in Finance

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The purpose of this course is to help students develop advanced skills for formulating and analyzing mathematical models in the economics and finance. Rigorous mathematical analysis of theoretical models can lead to a better understanding of economic problems.

The course is dedicated to advanced undergraduate students of economics.

1. Non-linear programming:

constrained optimization; equality constrains and the Lagrange problem; the constraint qualification; Lagrange multipliers; Kuhn-Tucker multipliers.

2. Differential equations:

Constant coefficient linear differential equation (ODE) systems, fundamental matrix; qualitative solution: phase portrait diagrams; nonlinear systems; fixed points; linearization of dynamic system in the plane.

3. Difference equations:

review of difference equations; linear difference equations; non-linear difference equations and phase diagram; first order difference equations systems.

4. Optimal control:

maximum principle; transversality conditions; transversality conditions in infinite horizon problem; second variations and sufficient conditions.

5. Dynamic programming:

dynamic programming problems; the principle of optimality; the value function; Bellman equation; Euler equations.

6. Stochastic differential equations and partial differential equations:

probability spaces; random variables and stochastic processes; Brownian motion; construction of the Ito integral; the Ito integral; properties of the Ito integral; 1-dimensional Ito processes; 1-dimensional Ito formula; the martingale representation theorem; stochastic differential equations - examples and some solution methods.

Mandatory literature:

K. Sydsater, P. Hammond, A. Seierstad, A. Strom, Futher mathematics for economic analysis, Prentice Hall, 2005

Supplementary literature:

1. A. Chiang, Elements of dynamic optimization, McGraw-Hill 1992

2. A. Chiang, Fundamental methods of mathematical economics, McGraw-Hill 1967

3. Z. Brzeźniak, T. Zastawiak, Basic stochastic processes., Springer 2003

A student should be able to:

- solve constrained optimization problems,

- solve simple differential and difference equations,

- analyze nonlinear differential and difference equations and systems of equations,

- solve and analyze optimal control problems,

- calculate Ito integrals,

- use the above techniques in economic modeling and finance.

To complete the course, the student has to complete the assignments, pass the midterm exam and pass the final exam. The passing threshold is 50%.

During the midterm and final exam student will have to solve by hand five problems.

Additionally, there will be home assignments. A student will be asked to solve some problems from economics and finance in each homework.

There will be no oral exams.

The final grade will be determined as follows:

Midterm Exam - 30%, Final Exam - 50%, Assignments - 20%.