Syllabus

AUBG, Spring 2011

ECO 404b: Financial Calculus

Prerequisites MAT 103 (Calculus I), Elective for ECO and MAT

classes meet: TF 16:00-17:15, room 002 BAC

tutorials: TBA; tutor: TBA

Alexander GANCHEV

, http://home.aubg.bg/faculty/aganchev/ ,

office 304 (new bulding), phone: 480, office hours: TBA

Course description:

The course introduces the mathematics behind the correct pricing of financial derivatives (we will limit ourselves to European options) based on the no arbitrage principle – the Black-Scholes formula. We begin with the conceptually simpler discrete time approach (binary trees) and time permitting we extend to continuous time (Brownian motion, stochastic differential equations -- the Black-Scholes equation, stochastic integration).

Expected Outcomes:

At the end of the course the student should be able:

  • To understand the principle of no arbitrage and how it can be applied to produce a mathematical model for the pricing of European options
  • Understand risk-neutral option pricing and portfolio replication using assets and bonds
  • To understand concepts and use tools from probability theory such as conditional expectation values, stochastic processes, martingales, Markov processes, Brownian motion, the Black-Scholes stochastic differential equation, stochastic integration
  • Calculate prices of simple financial instruments, do simple calculations with martingales and Brownian motion

Besides the above we want to learn to think creatively, be able to attack a problem you have not seen before, develop tools for that, develop a mathematical model for a given “real life” situation, develop mathematical modeling skills.

Prerequisites: This course is an introduction to a nontrivial mathematical model in economics and as such requires a certain dose of mathematical maturity. Besides Calculus I it will be very helpful if you have taken a calculus sequence including some acquaintance with differential equations, some course(s) introducing probability theory, and some elements of linear algebra.

Textbooks:

required:

M. Joshi, The Concepts and Practices of Mathematical Finance (CUP 2007)

additional:

R.F. Bass, The Basics of Financial Mathematics, (lectures available at http://www.math.uconn.edu/~bass/finlmath.pdf)

M. Baxter, A. Rennie, Financial Calculus, an introduction to derivative pricing,

(Cambridge Univ Press, 1996)

A. Etheridge, A course in Financial Mathematics, (Cambridge Univ Press, 2002)

T. Bjoerk, Arbitrage Theory in Continuous Time, (Oxford Univ Press, 1998)

J.C. Hull, Options, Futures and Other Derivatives, (Prentice Hall 2000)

R. and E. Korn, Opton Pricing and Oirtfolio Optimization, (AMS, 2001)

N. Chriss, Black-Scholes and Beyond (McGraw-Hill 1997)

Assessment: Your grade will be formed by

Homework/projects 15+ points

participation/popup-quizes 10+

oral exams 15+

midterm exams 30+

final exam 40+

______

total 110+

Points/Grade Map:

D-  > 45, D > 50, D+ > 55, C- > 60, C > 65, C+ > 70,

B- > 75, B > 80, B+ > 85, A- > 90, A > 95

Exam policies: During quizzes, exams and the final you should work strictly by yourself – you should not communicate in any way with your classmates – violation of this will be considered cheating with all the ensuing consequences (see the AUBG documentation for the consequences of cheating). Cheating is not only talking to the person next to you (talking about anything: math, the problems, the weather, last nights party …) but also intentionally making your work available to others during the exam.

Attendance: Students are expected to attend classes regularly and should comply with the university attendance policies. I expect you to come to class prepared (having read the assigned text if there is such) and to show active participation during the lecture.

Office hours: If the “official” office hours above are not convenient for you please contact me to arrange some other time. Don’t be afraid to come and ask. There are no stupid questions.

Expanded Description: (sections from the textbook(s) we will cover)

The course will cover the material contained in the first seven chapters of the book of N. Chriss (the disrete-time/binomial model) and, time and background of the students permitting, we will make excursions into the continuous-time models as for example covered by the first five chapters of A. Etheridge, A course in Financial Mathematics, or the first three chapters of the book of M. Baxter, A. Rennie, Financial Calculus, an introduction to derivative pricing. As a text with more detailed explanation of the mathematics we can use the lecture notes of R.F. Bass, The Basics of Financial Mathematics. One can also look at the first six chapters of the book of T. Bjoerk, Arbitrage Theory in Continuous Time.

Online Components of the Course: The Course Schedule will be updated as we go along on and placed on my web space. You will be able to find additional material there that I will add as we proceed.

Disclaimer: This syllabus is subject to modification. The instructor will communicate with students on any changes.

Tentative schedule for the topics to be covered:

  • stocks and their derivatives (futures and options), hedging, no arbitrage principle and other simplifying assumptions leading to a manageable mathematical model
  • binomial tree model for an European option, replicating portfolio,
  • random walk as example of a stochastic process, basic notions form probability theory
  • risk-free probability, value of replicating portfolio as expectation value,
  • discrete parameter martingales, binomial representation theorem, overture to continuous time
  • Brownian motion
  • Ito calculus
  • change of measure (Girsanov theorem), martingale representation theorem
  • Black-Scholes model,
  • Black-Scholes differential equation, Feynman-Kac representation
  • solutions, the Greeks
  • further directions for study in Mathematical Economics and Finance