Fall Semester 2012

Derivatives Lab

Syllabus

Summary:

This lab is an introduction to Mathematical Finance, in particular derivatives pricing, via computer experiments. It is designed for second year ACM students, but should also appeal to a wider audience.

Prerequisites are one year of Engineering and Science Mathematics or equivalent. The lab does rely on elementary concepts from Linear Algebra and Probability; students who have not taken second semester ESM (such as second year GEM students in the quantitative specialization) may pick up these concepts "on the fly" provided their general mathematical background is strong. If in doubt, please contact me directly.

Contact Information:

Instructor:	Marcel Oliver
Email:	m.oliver@jacobs-university.de
Phone:	200-3212
Office hours:	Mo, Tu 10:00 in Research I, 107

Time and Place:

Mo, Tu 14:15-17:00 in the CS Lecture Hall (Research I)

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,
The source code of a program must be submitted by email to me, using the following file naming convention:
Lastname_Firstname_XX_YY.py
where XX is name of the Session number and YY the task number. (If a single program produces output for several tasks, please use task number ranges.)
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 differentialequations, 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

Session Overview

Session 1:	Introduction to Scientific Python; present value, forward value; vectorized implementation and timing issues Lyuu, Section 3.1, Introduction to Scientific Python
Session 2:	Annuities, amortization schedule, yields, internal rate of return; root finding methods Lyuu, Sections 3.2-3.4, Brent's method on Wikipedia
Session 3:	Bonds, zero coupon bonds, level coupon bonds Lyuu, Section 3.5
Session 4:	Price volatility, Macaulay duration, immunization (begin) Lyuu, Sections 4.1-4.2
Session 5:	Immunization (ctd.), convexity (brief); The term structure of interest rates Lyuu, Sections 4.2-5.2
Session 6:	Forward rates, risky bonds Lyuu, Sections 5.3, 5.6
Session 7:	Introduction to options, payoff functions Lyuu, Chapter 7 (parts)
Session 8:	Arbitrage-free pricing, risk-neutral probability in the one-period model, binomial tree model
Session 9:	Calibrating the binomial tree in terms of the annualized drift and volatility Lyuu, Section 9.3.1
Session 10:	Implied volatility, Put-call parity; Continuum limit of the binomial tree, Black-Scholes formula Lyuu, Section 8.3, 9.3.2
Session 11:	Session 10 topics continued.
Session 12:	Brownian motion, geometric Brownian motion Higham (2001), Section 2
Session 13:	Stochastic integrals, stochastic differential equations Higham (2001), Section 3
Session 14:	Euler-Maruyama method Higham (2001), Sections 4 and 5
Session 15:	Ito formula Higham (2001), Section 8; Ito's lemma on Wikipedia
Session 16:	Derivation of the Black-Scholes equation; banded solvers Lyuu, Sections 4.1-4.2
Session 17:	Implicit and explicit finite difference schemes for the Black-Scholes equation Lyuu, Section 18.1
Session 18:	Implementation and testing of explicit and implicit schemes for the Black-Scholes equation; stability of time-integrators
Session 19:	The Greeks, numerical computation of sensitivities Lyuu, Section 10.1
Session 20:	Implementation of sensitivity solver
Session 21:	Parameter estimation for geometric Brownian motion Tsay, Section 6.3.4
Session 22:	Autocorrelation plots, QQ-plots; Elements of time series analysis: random walk and AR(1) models Aas and Dimakos, Sections 2 and 4.2
Session 23:	Elements of time series analysis: very brief introduction to MA(1) and ARMA(1,1), outlook on ARCH and GARCH models Tsay, Sections 2.5, 2.6.1

Last modified: 2012/12/04
This page: http://math.jacobs-university.de/oliver/teaching/jacobs/fall2012/acm221/index.html
Marcel Oliver (m.oliver@jacobs-university.de)