# MATH20070 Optimization in Finance

This course is an introduction to mathematical methods of optimization in financial models.

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Curricular information is subject to change

Learning Outcomes:

On successful completion of this module, the student will(i) be familiar with basic mathematical techniques of constrained and unconstrained extrema in financial models;(ii) be aware of the relationship between constraint parameters and shadow prices;(iii) be able to set up and solve linear optimization problems by the simplex method and duality;(iv) be able to apply convexity methods to certain extremum problems.

Indicative Module Content:

Contents
1 Unconstrained Optimization
1.1 Introduction
1.2 Background and Notation. . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 The Second Derivative Test . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 The Hessian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.2 Second Derivative Test . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Lagrange Multipliers
2.1 Lagrange Multipliers with one constraint . . . . . . . . . . . . . . . . . . 17
2.2 Lagrangian method with n variables and one constraint . . . . . . . . . 20
2.3 The Lagrangian Method: n variables, m constraints . . . . . . . . . . . 23
2.4 Economic interpretation of Lagrange Multipliers . . . . . . . . . . . . . 26

3 Linear Programming: The Graphical Method
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Linear Programming (General Problem) . . . . . . . . . . . . . . . . . . . 28
3.3 The Graphical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 No Solution to Linear Programming Problem . . . . . . . . . . . . . . . . 34

4 Linear Programming: The Simplex Method
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Standard Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3 Gaussian reduction from linear algebra (Recall) . . . . . . . . . . . . . . 41
4.4 The Simplex Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.5 Two Choices for Optimal row . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 The Simplex Method with Mixed Constraints
5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.2 Transforming the Objective Function . . . . . . . . . . . . . . . . . . . . 52
5.3 The Simplex Method with Mixed Constraints . . . . . . . . . . . . . . . 54
5.4 Solving a Minimisation Problem . . . . . . . . . . . . . . . . . . . . . . . 57
5.5 When the Simplex Method breaks down . . . . . . . . . . . . . . . . . . 61
5.5.1 When the Feasible set is Empty . . . . . . . . . . . . . . . . . . . 62
5.5.2 When the Feasible set is Unbounded . . . . . . . . . . . . . . . . 63

6 Linear Programming: The Dual Problem
6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.2 The Dual Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.3 Interpretation of the Dual Problem . . . . . . . . . . . . . . . . . . . . . 70

7 Nonlinear Programming
7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7.2 Nonlinear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7.2.1 Extreme-Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . 77
7.2.2 The general case: Kuhn-Tucker theory . . . . . . . . . . . . . . . 81

8 Convex and Concave Functions
28.1 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
8.2 Concave and Convex Functions . . . . . . . . . . . . . . . . . . . . . . . 92
8.3 Positive and negative (semi)definite matrices . . . . . . . . . . . . . . . . 95
8.4 The n-variable case: convex and concave . . . . . . . . . . . . . . . . . . 99
8.5 Quasi-concave and quasi-convex functions . . . . . . . . . . . . . . . . 102
8.5.1 Upper Level Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8.5.2 Quasi-concave and quasi-convex functions . . . . . . . . . . . . 104
8.5.3 A determinant criterion for quasi-concavity . . . . . . . . . . . . 106

Student Effort Hours:
Student Effort Type Hours
Lectures

30

Tutorial

12

Specified Learning Activities

32

Autonomous Student Learning

32

Total

106

Approaches to Teaching and Learning:
Lectures, tutorials, enquiry and problem-based learning..

Requirements, Exclusions and Recommendations

Not applicable to this module.

Module Requisites and Incompatibles
Not applicable to this module.

Assessment Strategy
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade
Assignment(Including Essay): In class Quiz n/a Alternative linear conversion grade scale 40% No

10

Exam (In-person): Midterm Examination n/a Alternative linear conversion grade scale 40% No

20

Exam (In-person): End of Semester Examination n/a Alternative linear conversion grade scale 40% No

70

Carry forward of passed components
No

Resit In Terminal Exam
Autumn Yes - 2 Hour