Lecturer: Diane Wilcox  Course notes & information
my UCT webpage
	Dept. of Mathematics and Applied Mathematics

MAM502W - MDS module 

MAM502W - FIRM module (2006)

MAM502W - TA module (2005)

MAM502W - MDS module

LECTURE NOTES: PDE methods for Pricing Derivative Securities [ Download version 1.4 pdf ] Edited notes will be posted through 2007 Please email me at if you detect any typos (diane.wilcox@uct.ac.za)

REMARKS: These notes have been developed for a 24 lecture module in the UCT MSc Mathematics of Finance programme. No prior exposure to PDE is required. Students enter the programme with a four year degree in mathematics, physics, statistics, computer science or engineering and are assumed to have at least 2 years of university-level mathematics training.

HOMEWORK SETS:

Homework 1

Other references for content covered: 

      Part I

[1] E.Zauderer,  Partial Differential Equations of Applied Mathematics

[2] L.C. Evans, Partial Differential Equations

[3] K.E. Gustafson, Introduction to Partial Differential Equations and Hilbert Space Methods

     Part II

[4] Y.-K. Kwok, Mathematical models of financial derivatives

[5] J. Hull, Options, futures and other derivatives

[6] Wilmott, Howison and Dewynne, Option pricing: mathematical models and computation

[7] P.Wilmott, Derivatives

To be incorporated in future revisions:

• Chapters 2 and 4 to be edited to optimise content for applications for finance

• Chapter 6 is currently just a place holder

• Notes on Feynman-Kac and numerical implementations will be added

• Pictures must be added

• Bibliography to be added

• The longer term plan is to add more to Part II, on applications to derivatives, and possibly incorporate notes on equivalent martingale measure approach as well

MAM502W - TA module (2005)

Theory of Arbitrage (Discrete Time)

Course Outline

• THE SINGLE-PERIOD SECURITIES MARKET MODEL: Model description; Separating Hyperplane Theorem; No-Arbitrage Theorem; Probabilistic framework; complete and incomplete markets.

• INTRODUCTION TO PROBABILITY THEORY: Events, sigma-algebras, measures and random variables. Borel-Cantelli Lemma. The Lebesgue integral for use: Radon-Nikodym Theorem, L^p spaces, Convergence Theorems, Conditional Expectation. Convergence of random variables; Uniform integrability;

• THEORY OF MARTINGALES: Random walk and other examples; Doob’s inequalities; Martingale Convergence Theorem; martingale transforms;

• MULTIPERIOD SECURITIES MARKET MODEL: Model description; risk-neutral pricing, no-arbitrage and equivalent martingale measures; complete markets; Fundamental Theorem of Asset Pricing.

• THE BINOMIAL MODEL: Pricing and hedging european derivatives; calibration of binomial trees; from Cox-Ross-Rubinstein to Black-Scholes.

• STOPPING TIMES AND AMERICAN OPTIONS: Pricing and hedging american contingent claims; Snell envelope; dynamic programming approach.

Notes: Course notes by D. Wilcox (2005)

Further References:

[1] S.R.Pliska, Introduction to Mathematical Finance: Discrete Time Models

[2] N.H.Bingham and R.Kiesel, Risk-Neutral Valuation : Pricing and Hedging of Financial Derivatives

[3] J.M.Steele, Stochastic Calculus and Financial Applications

[4] D.Lamberton and B.Lapeyre, Introduction to Stochastic Calculus Applied to Finance

[5] G.Grimmett and D.Stirzaker, Probability and Random Process

[6] R.Bartle, Measure and Integration