Fall Semester 2016

Stochastic Methods + Lab

Syllabus

Summary:

This course is a first hands-on introduction to stochastic modeling. Examples will mostly come from the area of Financial Mathematics, so that this course plays a central role in the education of students interested in Quantitative Finance and Mathematical Economics.

Topics include binomial tree models, discrete Brownian paths, stochastic ODEs, Monte-Carlo methods, finite differences solutions for the Black-Scholes equation, and an introduction to time series analysis, parameter estimation, and calibration. Students will program and explore all basic techniques in a numerical programming environment and apply these algorithms to real data whenever possible.

Contact Information:

Instructor:	Marcel Oliver
Email:	m.oliver@jacobs-university.de
Phone:	200-3212
Office hours:	We/Th 10:00 in Research I, 107
TA:	Kim Philipp Jablonski
Email:	k.jablonski@jacobs-university.de

Time and Place:

Tu 8:15-11:00 in East Hall 4

Th 11:15-13:30 in East Hall 4

Grading:

The class grade is based on individually graded exercises. These are either worked on during lab sessions or as take-home work to be turned in up to one week later. The final task sheet will a larger project worth 20% of the grade.

Submission of Work:

Theoretical questions and explanations as handwritten answers,
The output of computer programs (if more than a few single numbers) as a print-out when requested,
The source code must be submitted via git; detailed instructions will be posted on this page.
You are permitted to talk to each other and exchange code fragments. Nontrivial code contributions from the internet or from your peers are to be acknowledged. This will not affect the grade, but I will mark down work which is clearly copied without acknowledgment. The presentation of explanations and of handwritten answers must be yours, and you should be prepared to explain your code submissions unannounced.

Literature:

Yuh-Dauh Lyuu, "Financial Engineering & Computation: Principles, Mathematics, Algorithms," Cambridge University Press, 2002.
Some chapters of this book are available from the author's web page.
Sheldon M. Ross, "An Introduction to Mathematical Finance: Options and Other Topics," Cambridge University Press, 1999.
Ruey S. Tsay, "Analysis of Financial Time Series," second edition, Wiley, 2005.
Desmond J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review 43 (2001), 525-546.
Desmond J. Higham, Black-scholes for scientific computing students, Computing in Science and Engineering 6 (2004), 72-79.
Kjersti Aas and Xeni K. Dimakos, Statistical modelling of financial time series: An introduction, Norwegian Computing Center Note SAMBA/09/04.

Scientific Python Resources:

Introduction to Scientific Python as HTML for browsing and PDF for printing

Course Schedule (subject to change!)

01/09/2016:	Introduction to Scientific Python
06/09/2016:	Basic notions from finance (present value, forward value, annuities, amortization schedule, yields, internal rate of return); Basic computational tools (vectorized formulation of problems, timing issues, root finding methods). Lyuu, Sections 3.1-3.4
08/09/2016:	Introduction to git
13/09/2016:	Bonds, zero coupon bonds, level coupon bonds Lyuu, Section 3.5
15/09/2016:	Immunization, Macaulay duration, immunization (begin) Lyuu, Sections 4.1-4.2
20/09/2016:	Immunization (ctd.), convexity (brief); The term structure of interest rates Lyuu, Sections 4.2-5.2
22/09/2016:	No class
27/09/2016:	Options: introduction, single-period model, risk-neutral probabilites, binomial tree mathod (start)
29/09/2016:	Binomial tree method (implementation)
04/10/2016:	Binomial tree method: scaling and calibration
06/10/2016:	Binomial tree method: visualization, convergenc; call-put parity
11/10/2016:	Central limit theorem for binomial distributions; derivation of the Black-Scholes formula
13/10/2016:	Brownian motion
18/10/2016:	Stochastic integrals
20/10/2016:	Stochastic differential equations, Euler-Maruyama method
27/10/2016:	Weak and strong convergence
01/11/2016:	Ito Lemma
03/11/2016:	Derivation of the Black-Scholes equation; finite differences
10/11/2016:	Stability; explicit vs. implicit timestepping, tridiagonal systems
15/11/2016:
17/11/2016:
22/11/2016:
24/11/2016:
29/11/2016:
01/12/2016:
06/12/2016:

Last modified: 2016/09/04
This page: http://math.jacobs-university.de/oliver/teaching/jacobs/fall2016/acm221/index.html
Marcel Oliver (m.oliver@jacobs-university.de)