Mathematics 16:642:621 Mathematics Finance I
Schedule
The course is offered during the
Fall
semester.
 Class meeting dates:
Please visit the University's academic
calendar.
 Schedule and
Instructor: Please
visit the University's schedule
of classes for the instructor, time, and room.
 Instructor
and Teaching Assistant Office Hours: Please visit the
Mathematical Finance program's office
hour schedule.
Course Abstract
This course is an introduction to the mathematical theory of derivative security (or option) pricing. Fundamental concepts are briefly
introduced first using the discretetime binomial model: financial markets, derivative securities, arbitrage, hedging and replicating
portfolios, riskneutral probabilities, riskneutral pricing formula, and market completeness. Basic ideas of probability and stochastic
processes are reviewed for finite probability spaces and discretetime processes: conditional expectation, martingales, and Markov processes.
After this introduction to finance using discretetime models, the emphasis shifts to continuoustime models and the main part of the course.
Topics covered include a summary of probability measure theory and conditional expectation, Brownian motion and quadratic variation,
martingales, Ito integral, stochastic calculus, replicating portfolios and hedging, BlackScholesMerton formulae for a Europeanstyle call
option price, change of measure and Girsanov's Theorem, riskneutral pricing pricing theory, noarbitrage and existence of riskneutral
measure, market completeness and uniqueness of riskneutral measure, Markov property, FeymanKac theorem and the connection between
stochastic calculus and partial differential equations, and local volatility and stochastic volatility models.
Prerequisites and Corequisites
Ordinary differential equations (01:640:244 or 01:640:252), multivariable
calculus (01:640:251), linear algebra (01:640:250), and undegraduate
probability theory with calculus (01:640:477 or 01:960:381). An undergraduate course
on analysis (01:640:311312 or 01:640:411412) or engineering
mathematics (01:640:421) or partial differential equations
(01:640:423) is recommended but not required.
Please visit the
prerequisites
page for descriptions of Rutgers undergraduate course prerequisites. A solid
understanding of undergraduate probability at the level of the
textbook by Sheldon Ross,
A First
Course in Probability, is especially important. Given this
background, the course should be accessible to Mathematical Finance
master's degree students and graduate students in Computer Science,
Economics, Finance, Engineering, Mathematics, Physics, Operations
Research, and Statistics.
Required Textbooks
Stephen E.
Shreve,
Stochastic Calculus for Finance II: ContinuousTime Models, Springer Verlag, 2004, ISBN 0387401018. (Text errata available from author's web site.)
Supplementary Textbooks: Stephen
E. Shreve,
Stochastic Calculus for
Finance I: The Binomial Asset Pricing Model, Springer Verlag,
2004; John C. Hull,
Options, Futures, and other Derivatives,
7th Edition, Prentice Hall, 2008.
Sakai
All
course content – lecture notes, homework assignments and
solutions,
exam solutions, supplementary articles, and computer programs
– are posted
on
Sakai
and available to registered students.
Grading
Class attendance 5%, homework 10%, two midterm exams at 20% each, and final exam 45%. Exams are inclass.
Class Policies
Please see the MSMF
common class policies.
Weekly Lecturing Agenda and Readings
The lecture schedule below is a sample; actual content may vary depending on the instructor. Please see the Sakai Wiki for the the latest lecture schedule.
Week 
Topics 
Reading 
1 
Financial markets and derivative securities; Noarbitrage condition;
One bond, onestock model; Forward contracts
No arbitrage pricing;
No arbitrage price of an option for the binomial model

Hull, § 1, 2, & 5
Hull, § 11; ShreveI, § 1; Pliska, § 1.1, 1.2

2 
First Fundamental Theorem of Asset Pricing for a
one period, finite state model; Stateprice vector
Stateprice vectors and riskneutral measure;
Risk neutral pricing formula. Examples.

ShreveI, § 1; Hull, § 11; Pliska, § 1.3, 1.4, 1.5
Optional: Duffie, § 1;

3 
Binomial trees (continued);
Probability theory and discretetime stochastic processes
Binomial trees (continued);
Riskneutral measure and option pricing
 ShreveI, § 2 & 3;

4 
Binomial trees (continued)
Probability spaces

ShreveI, § 2 & 3
ShreveI, § 2; ShreveII, § 1.1, 1.2, 1.3

5 
Expectation, information, and σalgebras
Conditional expectation 
ShreveI, § 2.2; ShreveII, § 1.3, 1.5
ShreveI, § 2.3, 2.4, 2.5; ShreveII, § 2.1, 2.2, 2.3
Rutgers Math 591 Notes,
Chicago Stat 313 Notes,
Lyons Notes,
Harvey Mudd Math 157 Notes (pdf)

6 
Brownian motion: Random walks and the central limit theorem
Brownian motion: Definition, martingale property, quadratic variation

ShreveII, § 3.2
ShreveII, § 3.3

7 
Brownian motion: Markov property
The Itô integral: Introduction

ShreveII, § 3.3
ShreveII, § 4.2, 4.3, & 4.4

8 
The Itô formula
The BlackScholesMerton PDE and its solution for
Europeanstyle call and put option prices.

ShreveII, § 4.4
ShreveII, § 4.5

9 
The BlackScholesMerton formula, geometry of hedging,
putcall parity
Multivariable stochastic calculus, Lévy's characterization
of Brownian motion, Gaussian processes, Brownian bridge.

ShreveII, § 4.5
ShreveII, § 4.6 & 4.7

10 
Change of measure, RadonNikodym derivative,
Girsanov's theorem for single Brownian motion
Discounted stock and portfolio processes as martingales

ShreveII, § 1.6, 5.1, & 5.2.1
ShreveII, § 5.2.2, 5.2.3, & 5.2.4

11 
Pricing under riskneutral measure,
derivation of BlackScholesMerton formula
Martingale representation theorem,
Multidimensional market model

ShreveII, § 5.2.4, & 5.2.5
ShreveII, § 5.3, 5.4.1 & 5.4.2

12 
Existence of riskneutral measure, no arbitrage, and
First fundamental theorem of asset pricing
Uniqueness of riskneutral measure, completeness, and
Second fundamental theorem of asset pricing

ShreveII, § 5.4.3
ShreveII, § 5.4.4

13 
Option pricing and PDEs

Shreve II, § 6.1, 6.2, 6.3, 6.4, & 6.6

14 
Riskneutral, martingale measure pricing theory
and explicit portfolio hedge ratios
Overview of Dupire local volatility,
Heston stochastic volatility, and jump models
Course sequels Introduction

Steele § 14.3, Shreve II chapters 5 & 6
Shreve II, chapters 6 and 11
Mathematical
Finance II,
Computational
Finance

Library Reserves
All textbooks referenced on this page should be on reserve in the Hill
Center Mathematical Sciences
Library (1st floor). Please contact the instructor if reserve copies
are insufficient or unavailable. Please visit the
Mathematical
Finance Reference Text List blog for additional textbook
suggestions.
Additional Textbooks
Class lectures will draw on material from the following texts and
current research articles. Please see the
Rutgers
Mathematical Finance Reference Texts blog for additional
textbooks.
K. Back,
A Course in Derivative Securities: Introduction to Theory and Computation, Springer, 2005
M. Baxter and A. Rennie,
Financial Calculus: An Introduction to
Option Pricing, Cambridge, 1996
T. Björk,
Arbitrage Theory in Continuous Time, Oxford, 2004
J. C. Hull,
Options, Futures, and other
Derivatives, 6th Edition, Prentice Hall, 2006
P. Hunt and J. Kennedy,
Financial Derivatives in Theory and Practice, Wiley, 2004
M. Jackson and M. Staunton,
Advanced Modelling in Finance using Excel and VBA, Wiley, 2001
R. Jarrow and S. Turnbull,
Derivative Securities, 2nd edition,
SouthWestern College,1999
I. Karatzas and S. E. Shreve,
Brownian Motion and Stochastic
Calculus, Springer, 1997
D. G. Luenberger,
Investment science, Oxford, 1997
S. R. Pliska,
Introduction to Mathematical Finance: Discrete Time Models, Blackwell, 1997
S. E. Shreve,
Stochastic calculus and Finance I: Binomial
Model, Springer, 2004
J. M. Steele,
Stochastic calculus and financial applications,
Springer, 2000
P. Wilmott,
Paul Wilmott on Quantitative Finance, 2nd edition, 3 volume set, Wiley, 2006
Software
Depending on the application, Excel/VBA or MATLAB may be used in the course. Please visit the
Quantitative
Finance Software blog for a guide to platforms, installation
guides, and sample code.