Fall Semester 2012

Derivatives Lab


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
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)

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:


Scientific Python Resources:

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)