Sample Course Outline Fall 2012 Page 1 of 7

Financial Mathematics I CKMT 801

This is a sample course outline only. It should not be used to plan assignments or purchase textbooks.

A current version of the course outline will be provided by the instructor once the course begins.

Every effort will be made to manage the course as stated. However, adjustments may be necessary at the

discretion of the instructor. If so, students will be advised and alterations discussed in the class prior to

implementation.

It is the responsibility of students to ensure that they understand the University’s policies and procedures,

in particular those relating to course management and academic integrity

COURSE DESCRIPTION

This course focuses on a core topic in Financial Mathematics: derivative pricing.

It intends to provide the students with theoretical foundations and main tools used in the pricing of bonds,

European, American and Exotic Options, Futures and options on Futures. To this end the syllabus takes

into account concepts and results studied in MTH500, i.e. martingales, Ito’s lemma and Girsanov theorem

to develop methods for pricing financial derivatives. The notions of completeness, viability and neutral

risk measure are examined. Pricing problems are considered in discrete and continuous-time frameworks.

The link between pricing and Partial Differential Equations(PDE) is reviewed. Existing numerical

methods to derivative prices are examined. It includes PDE’s, trees, Monte Carlo and Fast Fourier

Transform techniques.

Through the course, theoretical and numerical examples will be solved. Sample questions from previous

FRM/PRM exams will be discussed.

COURSE OBJECTIVE/LEARNING OUTCOMES

To provide students with:

tools to price basic fixed income securities, most common European, American and Exotic

options.

understanding about fundamental concepts in pricing theory.

partial preparation for FRM/PRM certificates through discussion of past sample exam problems.

SAMPLE COURSE OUTLINE

CKMT 801

FINANCIAL MATHEMATICS I

Sample Course Outline Fall 2012 Page 2 of 7

Financial Mathematics I CKMT 801

CORE TOPICS:

Fixed income pricing: The first part addresses to the pricing of basic fixed income derivatives, their

sensitivities and term structure under constant interest rates. Practical issues such as bond pricing, yield to

maturity, duration and convexity calculations are considered. The topic is continued in Financial

Mathematics II.

Option Pricing under Discrete-time Models: The link between martingales and pricing is studied. The

concept of pricing, following a Risk Neutral Market approach, is developed in details within the

framework of a Binomial Cox-Ross-Rubinstein Model. In this case a valuation formula for European

options is obtained. Important concepts of complete and viable markets, self-financing and admissible

portfolios are introduced.

Option Pricing under Continuous-time Models: Classical Black-Scholes model and its famous pricing

formula are studied. The later is obtained for European vanilla options, under risk neutral and arbitrage

arguments. Extensions to a variety of situations, such as underlying stocks paying dividends, or with

time-dependent, but deterministic, drift and volatility parameters are considered. Sensitivities to

parameters in the contract (Greeks) are analyzed, as well as the problem of hedging derivative positions

and parameter calibration. The second part of the topic deals with the valuation of American and Exotic

options. A segment is dedicated to the analysis of futures and options on futures.

Numerical Methods for Option Pricing: Pricing methods outlined in previous topics are discussed here in

details. It includes common approaches used by practitioners and academics such as Partial Differential

Equations, trees, Monte Carlo and Fast Fourier Transform techniques. Different algorithms are reviewed

and their practical implementation is considered.

TEXTBOOK AND READING LISTS

This is a sample course outline only. It should not be used to purchase textbooks. A current version

of the course outline will be provided by the instructor once the course begins.

Required Text:

Bond Markets, Analysis, And Strategies, Frank J. Fabozzi. Pearson. Prentice Hall, 6th edition (2007).

The Mathematics of Financial Derivatives, Wilmott, P., Howison, S., Dewynne, J. Cambridge Press. 2nd

edition ( 1996).

Reading and Related Material:

Options, Futures and Other Financial Derivatives, Hull, J.C., Prentice Hall, 8th edition.

PDE and Martingale Methods in Option Pricing. Pascucci, A.. Bocconi & Springer Series, 2011.

Tools for Computational Finance. R. U. Seydel, Universitext, 3rd Edition Springer 2000.

Sample Course Outline Fall 2012 Page 3 of 7

Financial Mathematics I CKMT 801

COURSE STRUCTURE AND ORGANIZATION

This is a sample course outline only. A current version of the course outline will be provided by the

instructor once the course begins.

The course is divided in four chapters corresponding to the pricing of fixed income instruments, options

pricing under discrete-time models, pricing under continuous-time models and numerical methods for

derivative pricing.

A mixing of theoretical aspects of pricing with practical implementations and calculation is

recommended. The instructor should rigorously derive pricing formulas and illustrate their use through

practical applications.

Class Topic Details

1

Introduction to Financial

Markets

Bond Pricing

An introduction to financial markets.

The basic theory of interest.

Fixed Income Markets. Bond Pricing, Fabozzi, Chap. 1-3

2

Bond pricing (cont.)

Duration and Convexity.

Term structure.

Computing yield to maturity, duration and convexity of a

bond. Term Structure, Fabozzi, Chap. 4-5

3 Derivative Pricing in

discrete-time models I

Derivative Markets. Option pricing and hedging. Portfolios

and strategies. Complete, Viable and Risk Neutral markets.

Wilmott, Chap. 1

4 Derivative Pricing in

discrete-time models II

The Cox- Ross-Rubinstein Binomial Model. Valuation

formula Tree methods. Convergence to Black-Scholes in

continuous-time trading. Wilmott, Chap. 2,10.

Assignment 1 Fixed Income Instruments Pricing

5 Derivative Pricing in

continuous-time models

The Black-Scholes model. Neutral risk markets and the

Girsanov theorem. Wilmott, Chap.3,5

6 Black-Scholes formula Black-Scholes formula. Sensitivities. Hedging. Pricing and

PDE’s. Wilmott, Chap. 3,5

7 Extensions to the Black-

Scholes model

Dividends. Exchange options. The Garman-Kohlhagen

model. Time dependent parameters. n Wilmott, Chap. 6

8 Pricing American Options

Assignment 2

American options. Binomial trees for American options.

Wilmott, Chap. 7,10

Sample Course Outline Fall 2012 Page 4 of 7

Financial Mathematics I CKMT 801

METHOD AND SCHEDULE OF STUDENT EVALUATION

This is a sample course outline only. It should not be used to plan assignments. A current version of the

course outline will be provided by the instructor once the course begins.

3 assignments worth 20% each

Due by 6:30pm in classes 4, 8, 11

60%

(20% each)

Final examination (non-lab) 40%

Total 100%

Assignments (graded):

Assignment 1: Fixed Income Instruments Pricing.

Assignment 2: Pricing European options under Binomial and Black-Scholes.

Assignment 3: Pricing American and Exotic Options

Assignments are due at the beginning of the evening set out in the schedule above. There will be a 10%

penalty for late handling.No lab assignment submission will be accepted for grading once graded labs

have been returned to students, normally at the next class after being handed in.

Pricing European option under Binomial and Black-

Scholes

9 Exotic options and options on

futures

Asian and path-dependent options. Pricing of Futures

derivatives and options on futures. Wilmott, Chap. 11-15

10 Monte Carlo methods to

price exotic options.

Computing expected values. Importance Sampling. Pricing

Asian and Path-dependent Options. Seyde, Chap. 2, 3

11

Finite-difference methods

and Fourier Transform

Methods

Applications of finite-difference methods to option pricing.

Fourier and Fast Fourier Transforms. Implementation in

European Put and Call options

Assignment 3 Wilmott, Chap. 4,8,9

Pricing American and Exotic Options

12 Review Review and problem solving session

13 Exam Exam: Multiple Choice/Short Answer

Sample Course Outline Fall 2012 Page 5 of 7

Financial Mathematics I CKMT 801

COURSE EXPECTATIONS/ASSESMENT TOOL:

Here is a list of theoretical and practical aspects students are expected to know by the course end:

Fixed Income Instruments.

Price a bond knowing its cash flow.

Compute the yield of a bond.

Compute the duration of a bond and its modified duration.

Compute the convexity of a bond.

Estimate the change in the price of a bond after a change in the yield

Future and present value and different ways of compounding.

Elements of a bond. Different types of bonds.

Interpretation of duration and convexity.

Qualitative analysis of how the change of different factors affect the price of a bond.

Proof of certain properties related to bonds.

Pricing under discrete and continuous-time trading

Price European calls and puts under the binomial model(tree methods).

Price European calls and puts under the Black-Scholes models.

Computing the sensitivities of an option to its parameters ( Greeks).

Estimate the changes in the option price when the parameters changes.

Compute the hedging portfolio of European Options.

Price Exchange options.

Price American Options by binomial trees.

Price options with dividends.

Estimate the volatility from historical data and from option prices.

Price Futures derivatives and options on Futures.

Price Exotic options by Monte Carlo simulations.

Elements of an option: payoff, maturity, strike price, underline asset. European and American

options. Positions in the contract.

Portfolios: portfolio value, replicating, arbitrage, admissible, self-financing portfolios.

Risk Neutral probability and risk neutral market.

Complete and viable markets.

Relationship between pricing and martingale.

Describe the binomial model.

Parity call-put formula and arbitrage.

Risk neutral probability in the binomial model.

Black-Scholes continuous time model.

Knowledge of definition and payoffs of different Exotic Options.

General knowledge of the Monte Carlo Method: What is Monte Carlo method, algorithm to

simulate stochastic differential equations, algorithm to compute expected values.

Use of Ito's formula and Girsanov theorem in pricing.

Sample Course Outline Fall 2012 Page 6 of 7

Financial Mathematics I CKMT 801

Interpretation of the sensitivities.

Implementing finite-difference methods.

Implementing Inverse Fourier Transform Methods.

MISSED TERM WORK OR EXAMINATIONS

Students are expected to complete all assignments, tests, and exams within the time frames and by the

dates indicated in this outline. Exemption or deferral of an assignment, term test, or final examination is

only permitted for a medical or personal emergency or due to religious observance. The instructor must

be notified by e-mail prior to the due date or test/exam date, and the appropriate documentation must be

submitted. For absence on medical grounds, an official student medical certificate, downloaded from the

Ryerson website at http://www.ryerson.ca/senate/forms/medical.pdf or picked up from The Chang

School at Heaslip House, 297 Victoria St., Main Floor, must be provided. For absence due to religious

observance, visit http://www.ryerson.ca/senate/forms/relobservforminstr.pdf to obtain and submit the

required form.

PLAGIARISM

The Ryerson Student Code of Academic Conduct defines plagiarism and the sanctions against students

who plagiarize. All Chang School students are strongly encouraged to go to the academic integrity

website at www.ryerson.ca/academicintegrity and complete the tutorial on plagiarism.

ACADEMIC INTEGRITY

Ryerson University and The Chang School are committed to the principles of academic integrity as

outlined in the Student code of Academic conduct. Students are strongly encouraged to review the student

guide to academic integrity, including penalties for misconduct, on the academic integrity website at

www.ryerson.ca/academic integrity and the Student code of Academic conduct at

www.ryerson.ca/senate/policies.

RYERSON STUDENT EMAIL

All students in full and part-time graduate and undergraduate degree programs and all continuing

education students are required to activate and maintain their Ryerson online identity at

www.ryerson.ca/accounts in order to regularly access Ryerson’s E-mail (Rmail), RAMSS, my.ryerson.ca

portal and learning system, and other systems by which they will receive official University

communications.

Sample Course Outline Fall 2012 Page 7 of 7

Financial Mathematics I CKMT 801

COURSE REPEATS:

Senate GPA policy prevents students from taking a course more than three times. For complete GPA

policy see policy no. 46 at www.ryerson.ca/senate/policies.

RYERSON ACADEMIC POLICIES

For more information on Ryerson’s academic policies, visit the Senate website at www.ryerson.ca/senate.

Course Management Policy No. 145

Student Code of Academic conduct No. 60

Student code of non-Academic Conduct No. 61

Examination Policy No. 135

Policy on Grading, Promotion, and Academic Standing Policy No. 46

Undergraduate Academic consideration and Appeals Policy No. 134

Accommodation of Student Religious Observance Obligations Policy no. 150