Mathematics 16:642:621 Syllabus - Fall 2007

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The goal of this course is stochastic calculus and its applications to finance. However, the first part of the course will focus on finite state-space/discrete period models, for which only finite probability spaces are needed. The material will be presented in lecture notes and occasional class handouts. It is covered in part in the texts of John C. Hull, Options, Futures, and Other Derivatives, Prentice-Hall, 6th Edition, 2006, and of Steven E. Shreve, Stochastic Calculus for Finance I: The Binomial Asset Pricing Model, Springer Verlag, 2004. Copies of these books and our primary text, Shreve, Stochastic Calculus for Finance II: Continuous Time Models, Springer Verlag, 2004, and will be placed on reserve in the Mathematics Library in Hill Center. (These texts are highly recommended to students interested in the field.) The advantage of starting with the discrete models is that the sophisticated machinery later used for the stochastic calculus models has a simple and transparent form.

In the second part of the course, we will develop the theory of stochastic integration and stochastic differential equations and apply it to the statement and analysis of continuous-time models. This material is in Volume II of the text by Shreve. Our objective is to cover the material of the first six chapters.

This page lists the topics we shall cover in each lecture, with links and information on the related problem assignments and readings. Reading assignments should be completed prior to each class.

Lecture	Topics	Reading [Lectures after 10/16/2007 on Sakai]	Assignments [#7 & after on Sakai]
1	Financial markets and derivative securities; No-arbitrage condition; One bond, one-stock model; Forward contracts	Lectures 1 & 2 (pdf) Hull, § 1, 2, & 5 Lecture 1 supplement (© D. Ocone, 2005) (pdf)
2	No arbitrage pricing; No arbitrage price of an option for the binomial model	Hull, § 11; Shreve-I, § 1; Pliska, § 1.1, 1.2 Lecture 2 supplement (© D. Ocone, 2005) (pdf)	Homework 1 (pdf) Due 9/11/2007.
3	First Fundamental Theorem of Asset Pricing for a one period, finite state model; State-price vector	Lectures 3 & 4 (pdf) Lecture 3 supplement (© D. Ocone, 2005) (pdf)
4	State-price vectors and risk-neutral measure; Risk neutral pricing formula. Examples.	Shreve-I, § 1; Hull, § 11; Pliska, § 1.3, 1.4, 1.5 Lecture 4 supplement (© D. Ocone, 2005) (pdf) Optional: Duffie, § 1; Supplement (pdf) on Hyperplane Separation Theorem and proof of First Fundamental Theorem of Asset Pricing (© D. Ocone, 2005)	Homework 2 (pdf) Due 9/18/2007.
5	Binomial trees (continued); Probability theory and discrete-time stochastic processes	Lectures 5 & 6 (pdf) Shreve-I, § 2 & 3; Lecture 5 supplement (© D. Ocone, 2005 (pdf)
6	Binomial trees (continued); Risk-neutral measure and option pricing	Shreve-I, § 2 & 3; Lecture 6 supplement (© D. Ocone, 2005) (pdf)	Homework 3 (pdf) Due 9/25/2007.
7	Binomial trees (continued)	Lectures 7 & 8 (pdf) Shreve-I, § 2 & 3
8	Probability spaces	Shreve-I, § 2; Shreve-II, § 1.1, 1.2, 1.3	Homework 4 (pdf) Due 10/2/2007.
9	Expectation, information, and σ-algebras	Lectures 9 & 10 (pdf) Shreve-I, § 2.2; Shreve-II, § 1.3, 1.5
10	Conditional expectation	Shreve-I, § 2.3, 2.4, 2.5; Shreve-II, § 2.1, 2.2, 2.3 Rutgers Math 591 Notes, Chicago Stat 313 Notes, Lyons Notes, Harvey Mudd Math 157 Notes	Homework 5 (pdf) Due 10/9/2007.
11	Brownian motion: Random walks and the central limit theorem	Lectures 11 & 12 (pdf) Shreve-II, § 3.2
12	Brownian motion: Definition, martingale property, quadratic variation	Shreve-II, § 3.3	Homework 6 (pdf) Due 10/16/2007.
13	Brownian motion: Markov property	Lectures 13 & 14 (pdf) Shreve-II, § 3.3
14	The Itô integral: Introduction	Shreve-II, § 4.2, 4.3, & 4.4
15	The Itô formula	Lectures 15 & 16 (pdf) Shreve-II, § 4.4
16	The Black-Scholes-Merton PDE and its solution for European-style call and put option prices.	Shreve-II, § 4.5
17	The Black-Scholes-Merton formula, geometry of hedging, put-call parity.	Lectures 17 & 18 (pdf) Shreve-II, § 4.5
18	Multivariable stochastic calculus, Lévy's characterization of Brownian motion, Gaussian processes, Brownian bridge.	Shreve-II, § 4.6 & 4.7
19	Change of measure, Radon-Nikodym derivative, Girsanov's theorem for single Brownian motion	Lectures 19 & 20 (pdf) Shreve-II, § 1.6, 5.1, & 5.2.1
20	Discounted stock and portfolio processes as martingales	Shreve-II, § 5.2.2, 5.2.3, & 5.2.4
21	Pricing under risk-neutral measure, derivation of Black-Scholes-Merton formula	Lectures 21 & 22 (pdf) Shreve-II, § 5.2.4, & 5.2.5
22	Martingale representation theorem, Multi-dimensional market model	Shreve-II, § 5.3, 5.4.1 & 5.4.2
23	Existence of risk-neutral measure, no arbitrage, and First fundamental theorem of asset pricing	Lectures 23 & 24 (pdf) Shreve-II, § 5.4.3
24	Uniqueness of risk-neutral measure, completeness, and Second fundamental theorem of asset pricing	Shreve-II, § 5.4.4
25	Option pricing and PDEs	Lectures 25 & 26 (pdf) Shreve II, § 6.1, 6.2, & 6.3
26	Option pricing and PDEs (continued)	Shreve II, § 6.4 & 6.6
27	Risk-neutral, martingale measure pricing theory and explicit portfolio hedge ratios	Lectures 27 & 28 (pdf) Steele § 14.3, Shreve II chapters 5 & 6
28	Overview of Dupire local volatility, Heston stochastic volatility, and jump models	Shreve II, chapters 6 and 11	Course sequels: Mathematical Finance II, Computational Finance

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