Instructor: Adam Sikora 115 Mathematics Building
Email: asikora at buffalo dot edu
Office hours: Tu,Th 2-3pm, and by appointment.

Class meets: Tu,Th 12:30-1:50, 250 Math Bldg.

Course Description: The course will focus on the mathematical models and solution methods for a number of important concepts in mathematical finance: mean-variance portfolio theory, the capital asset pricing model, derivative securities, asset dynamics, and the Black-Scholes equation for options pricing. The course does not assume extensive mathematical background as the necessary mathematical theory is introduced in the context of the financial models. The mathematical aspects of the course include probability and statistics, optimization, models for asset dynamics (discrete and continuous), random variables, stochastic differential equations, and solution methods for the Black-Scholes partial differential equation for option pricing.

Prerequisites: Calculus, elementary differential equations. Familiarity with (or willingless to learn to use) a spreadsheet program on a computer.

Text: Investment Science by D. G. Luenberger, (Oxford Press, 1998) Chapters 1,2,6-8,11-13.

Topics (tentative):

• Introduction: cash flows, investments and markets, comparison, arbitrage, dynamics, risk, pricing, hedging, random cash flow streams, derivative securities.
• Mean-variance Portfolio Theory: asset return, random variables, random returns, portfolio mean and variance, Markowitz minimum variance portfolio, two-fund theorem, inclusion of a risk-free asset, one fund theorem.
• Capital Asset Pricing Model (CAPM): market equilibrium, market lines, performance evaluation, CAPM as a pricing formula.
• Models and Data: data and parameter estimation.
• Models of Asset Dynamics: binomial lattice model, introduction to stochastic differential equations (random walks, Ito's lemma, connection to lattice models).
• Basic Options Theory: basic options, put-call parity.
• Black-Sholes Equation for Option Pricing: derivation of the Black-Scholes equation, analytical solutions for special cases, numerical solutions (Monte Carlo, lattice and finite difference solutions).

Class notes are available here.

Homework: With each lecture there will be assigned homework. Some of the questions will be identified as an assignment to be turned in with a specified due date and then graded. For each homework question, write a clear exposition of your solution (including sentences to explain your work where appropriate). To make it easier for record keeping, include the following information: name, MTH 458 or MTH 558, assignment due date. Homework is due at the beginning of class on the day stated (late homework will not be accepted). The cumulative homework score will be graded on a curve to determine the homework grade for the course. Homework guidelines:

• Use 8.5" by 11" paper, staple sheets together.
• If you want to protect the privacy of your your homework grade, then keep the first page of your homework empty, except for your name (which must always appear on the first page). Your grade will appear on the second page of your homework.
• Label each homework problem clearly.
• Write clearly and legibly.

Homework assignments:
Due Sep 10: Ch. 2: #2,3,6, Ch. 6: #2.
Due Sep 17: Ch. 6: # 3,4,5.
Extra credit problem: Consider spinner on a disk of radius 1ft. Let X be the height of the head of the spinner. (The center is at height zero.) It is a random variable with values between -1ft and 1ft. Find the probability distribution of X. (Hint: Requires inverse trig functions.)
Due Sep 24: Roll n dice. Let X_i be the number of dice which show i spots.Then X_1+...+X_6=n.
(1) Find the standard deviation of each X_i. (Use the fact that it is a sum of n independent variables. Which ones?)
(2) Find cov(X_i,X_j) for all i and j. Hint: All these covariances are equal. You can assume that fact in your solution.
Due Oct 1: (a) Ch. 6 #7. (For (c) use the method descibed in class, c.f. eq (6.10).)
(b) Write system of euqations for the 2nd Markovitz problem: given n assets and correlations between them and given sigma find weights of the portfolio which maximizes mean return with a given sigma. You need to use Lagrange multipliers here.
(c) Extra credit problem: Ch. 6 #8.
Due Oct 13: Ch. 7 #1,2,4,6.
Due Oct 20: here.
Due Oct 29: Ch 8 #1 (assume alphas equal 0), #2, Least squares problem .
Additional problem for Math 558 students (and extra credit problem for Math 458): Ch 8 #4.
Due Nov 5: Ch. 8 #6abc, Ch 11. #1.
Due Nov 12: Ch. 11.#2,3,4,5. In #5, you can use an alternative approach: use var(X)=E(X^2)-E(X)^2.
Due Nov 19: Ch 11. #6,7,8,10. For #10, t=1 refers to 1 year. Simulate many years, but hand in simulation for 2 years only. You can use a computer program of your choice. In Excel, for example, you can use ``=NORMINV(RAND(),0,1)" to generate random numbers with mean=0, sd=1, and normal distribution.

Tests: There will be three tests. Tentative dates: Oct 6, Nov 3, Dec 10

You are permitted one 3"x5" notecard with notes/formulas for each exam. Material covered in the exam is anything in the book or presented in lecture. The exam will consist of a mix of questions: some easy, some hard, and may also contain essay-answer questions. Each exam will be graded on a curve.
There will be no Final Exam during the Examination Period.

MTH 558: In accordance with Graduate School policy regarding dual-listed 400/500 level courses, students taking the course for graduate credit will be assigned additional or more difficult homework questions and will have additional or more difficult questions on exams. Grades for graduate students will be assigned based on graduate school standards.

Academic Honesty: You are expected to adhere to the letter and spirit of academic honesty. For homework assignments, you can discuss assignments with other students, but the details of the written solution as turned in are originally yours.
You must have your student ID for all exams.

Grades: There are 500 points available:
Exams: 100 points each.
Homework (including possible in-class quizzes): 200 points.
There will be a curve for each exam and for the homework.
The average Mth 458 final grade distribution in the recent years was as follows:
A 39%, A- 9%, B+ 3%, B 19.4%, B- 8%, C+ 2%, C 11%, C- 5%, D+ 0%, D 6%, F 15%.
Most likely the grade distribution in this class will be similar.

Incompletes will be given only in the most extreme emergencies, e.g. surgery the last week of the course.
Please note that you must be passing the course to receive an incomplete.