Mathematics 16:643:622 Mathematics Finance II


The course is offered during the Spring 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 continues the development of the mathematical theory of derivative security pricing begun in Mathematics 16:642:621 and, in addition, focuses on applications to financial models. Topics covered include exotic options (such as barrier, lookback, and Asian options), stopping times, American-style (early exercise) options, McKean's formula for the perpetual American put, change of numéraire and risk-neutral measure, interest rate models, bonds and options on bonds, term-structure models (Heath-Jarrow-Morton model, forward LIBOR model, swap market model), Black's formulae for caps and swaptions. The course ends with an introduction to jump models, including compound Poisson, jump diffusion, and Lévy processes, stochastic calculus and change of measure for jump processes, and option pricing with jump processes.

Pre-requisites and Co-requisites

Math 16:643:621 (Mathematical Finance I).

Required Textbooks

Stephen E. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models, Springer Verlag, 2004, ISBN 0-387-40101-8. (Text errata available from author's web site.)


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.


Class attendance 5%, homework 15%, midterm exam 30%, quiz 10%, and final exam 40%. Exams and quizzes are in-class.

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.

WeekTopicsReading Assignments
1 Review of stochastic calculus;
Markov processes, martingales;
interest rate models and solution of SDEs
Vasicek, Hull-White and CIR models;
PDE for a zero-coupon bond price
Shreve II, §4.1-4.4,
Examples 4.4.10, 4.4.11, 6.2.2, & 6.2.3
Shreve II, § 6.2, & 6.5
2 Reflection principle; first passage time;
Maximum of Brownian motion, without and with drift
Barrier options
Shreve II, § 3.7, 7.1, 7.2

Shreve II, § 7.3
3 Barrier options (continued), stopping times,
maximum of Brownian motion
Stopped processes, Doob's Optional Sampling
Theorem, barrier options and PDEs
Shreve II § 3.6, 3.7, 7.3

Shreve II, § 8.2, 7.3, 7.4; Wilmott, pp 408,
409, 410, 411, 412, 413, 414, & 415 (pdf)
4 Lookback options and PDEs
Lookback options and closed-form formulae
via probability
Shreve II § 7.4.1-3
Shreve II § 7.4.4, Willmott (2006) § 26
Wilmott, pp. 445-452 (pdf)
5 Asian options and PDEs
American options; perpetual put
Shreve II § 7.5.1, 7.5.2, Willmott (2006) § 25
Shreve II § 8.1, 8.2, 8.3, Willmott (2006) § 9
6 American options; finite-maturity put
American options; finite-maturity call
Shreve II § 8.4, Jarrow & Turnbull, § 7
Shreve II § 8.5.1,
Karatzas & Shreve, § 2.4, 2.5, 2.6
7 Forwards and futures
Change of numéraire and risk-neutral measure
Shreve II § 5.6, 9.1; Hull § 2, 3, 5
Shreve II § 9.2
  Spring Break No Lectures, Class or Office Hours
8 Forward measures; stochastic interest rates and
Black-Scholes-Merton formula
Foreign exchange market model; domestic and
foreign risk-neutral measure
Shreve II § 9.4

Shreve II § 9.3.1, 9.3.2, & 9.3.3
9 Affine yield interest rate models
Affine yield interest rate models (continued),
Heath-Jarrow-Morton model

Shreve II § 10.1, 10.2
Shreve II § 10.2, 10.3
10 Review of affine-yield and HJM models;
Heath-Jarrow-Morton model implementation
Forward LIBOR model
Shreve II § 10.1-10.3.5

Shreve II § 10.4.1-10.4.4
11 Forward LIBOR model (continued);
Caps, caplets, and Black caplet formula
Forward LIBOR term structure model and calibration;
Swaps, swaptions, and swap market model
Shreve II § 10.4.4-5

Shreve II § 10.4.6, Björk § 25,
Brigo & Mercurio § 6.1-7
Notes on HJM and LIBOR market models (pdf)
12 Swaps, swaptions, swap market model, and
Black's formula for swaptions
Introduction to jump models, Poisson,
compound Poisson, and jump processes
Björk § 25 (pdf), Brigo & Mercurio § 6.7
Expository paper on swaps (pdf)
13 Stochastic calculus for jump processes
Change of measure for jump processes
Shreve II § 11.5
Shreve II § 11.6
14 Final  

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.

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.

T. Björk, Arbitrage Theory in Continuous Time, 2nd Edition, Oxford, 2009
R. Cont and P. Tankov, Financial Modeling with Jump Processes, Wiley, 2004
D. Brigo and F. Mercurio, Interest Rate Models - Theory and Practice, with Smile, Inflation, and Credit, 2nd Edition, Springer 2006
J-P. Fouque and G. Papanicolaou and K. R. Sircar, Derivatives in financial markets with stochastic volatility, Cambridge, 2000
J. Gatheral, The Volatility Surface: A Practitioner's Guide, Wiley, 2006
J. C. Hull, Options, Futures, and other Derivatives, 7th Edition, Prentice Hall, 2008
M. S. Joshi, The Concepts and Practice of Mathematical Finance, Cambridge, 2003
I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, 1997
A. Lipton, Mathematical methods for foreign exchange: a financial engineer's approach, World Scientific, 2001
P. Wilmott, Paul Wilmott on Quantitative Finance, 2nd Edition, 3 volume set, Wiley, 2006


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.