Syllabus and Assignments 
Math 621 -- Fall 2005

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 class handouts. It is covered in part in the texts of Hull, Options, Futures, and Other Derivatives, Prentice-Hall 2003, and of Shreve, Stochastic Calculus I: Binomial Model, Springer Verlag, 2004. Copies of these books will be placed on reserve in the Mathematics Library in Hill Center. (I highly recommend these books to students interested in the field.) The advantage of starting with the discrete models is that the fancy 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 the course text by Shreve. Our object will be to cover the material of the first six chapters.

This page will record the topics we cover, with links and information on the related problem assignments and readings. Reading the material from the books on reserve books listed is strongly suggested, but not absolutely necessary. Reading the required text and handouts is required reading. The student should have read the material before coming to the class in which it is discussed!
Lecture Topics Reading Assignments
1  Financial Markets and derivatives;
Arbitrage and the no-arbitrage condition;
One bond/one-stock model; Forward contracts
 (Reserve) Hull, Chapters 1-3
Lecture 1 outline(pdf file)		 Problem Set 1
Solutions to Problem Set 1
2  No arbitrage pricing of forward contracts
No arbitrage price of a contingent claim for the binomial model.		 Hull, Chapter 3. Shreve, Chapter 1
Lecture 2 outline(pdf file)		  
3  Separation of convex sets
Fundamental Theorem of Asset Pricing
for a one period/finite state model;
State-price vector
Handout distributed on first day of class
PDF file of handout without figures
Lecture 3 outline(pdf file) 		 
4  State-price vectors and risk-neutral measure.
Multi-period binomial model (Binomial trees). 		Shreve, Volume 1, chapter 1(reserve)
Hull, Chapter 9, reserve. 		Problems 4-11
Hand in 4, 7, 8(Check the revised version) on Sept. 22
Hand in 11 on Sept. 27. Some solutions
5  Binomial trees continued.
 Start reading Chapter 2 of Shreve, Volume 2
Lecture notes for lectures 4 and 5 		 
6  Binomial trees, continued.
Solving and interpreting contingent
pricing equations 		Start reading Chapter 1 of Shreve, Volume 2
Lecture notes for lecture 6 		 
7  Probability spaces (introduction); Shreve, Volume 1
 
8  Probability spaces (introduction); Shreve, Volume 2, Chapter 1  
9  Expectation Shreve, Volume 2, Chapter 1  
10  Conditional expectation Shreve, Volume 2, Chapter 2 Shreve, Volume 2, Chapter 1: 1,3, 1.4, 1.5, 1.6, 1.9, 1.11
Hand in 1.5, 1.9, 1.11 on October 11.
Shreve, Volume 2, Chapter 2: 2.1, 2.2, 2.4, 2.6, 2.8, 2.9, 2.10
Hand in 2.6, 2.8, 2.9 October 18.
11  Brownian motion. Random walk and the central limit theorem Shreve, Vol. 2, section 3.2  
12  Brownian motion. Definition and fundamental properties Shreve, Vol. 2, section 3.3  
13  Brownian motion; martingale properties and quadratic variation Shreve, Vol. 2, section 3.3  
14  The Ito integral; introduction Shreve, Vol. 2, section 4.2  
15  The Ito integral Shreve, Vol. 2, section 4.3  
16  Ito' formula for Brownian motion and for
Ito processes. Application to solving the
Black-Scholes stock price eqn. 		Shreve, Vol. 2, section 4.4  
17  Derivation of the Black-Scholes pde. Shreve, Volume 2, section 4.5
Hull, Chapter 11		 Chapter 4: 4.1, 4.2, 4.3, 4.4
Hand in 4.4
Solutions
18  The Black-Scholes formula for the European call and put Shreve, Volume 2, section 4.5
Hull, Chapter 14		 Chapter 4: 4.6-4.12
Hand in 4.10, 4.11 on Nov. 10.
Solutions
19  Generalization of the Black-Scholes model and solution
to time varying coefficients and several dimensions 		Shreve, Vol. 2, section 4.6 Nov. 3: Take home due!
Take home solutions
20  Absolute continuity of measures;
Significance for price models;		 Shreve, Vol. 2, sections 1.6, 5.2  
21  Girsanov's theorem and construction of the risk-neutral probability measure. Shreve, Vol. 2, section 5.2  
22  Pricing with the risk-neutral probability measure;
martingale representation theorem and market completeness 		Shreve, Vol. 2, sections 5.2, 5.3  
23  Multi-stock price models:
Multi-dimensional stochastic calculus		 Shreve, Vol. 2, sections 4.5, 5.4 Problems , Shreve chapter 4: 13, 16, 18;
Chapter 5: 1, 2, 3, 4
Hand in (bold-faced) problems Dec 1.
24, Nov. 29 Applications of the theory; Forward and futures, coupons with interest Shreve, Vol. 2, sections 5.5, 5.6 Problems, Chapter 5: 7, 8, 11: Hand in (bold-faced) problems Dec. 6.
25, Dec. 1 Local volatility models
Stochastic diff. eqns and pde.		 Shreve, Vol. 2, chapter 6.
26, Dec 6. Stochastic diff. equations and
local volatility models		 Shreve, Vol. 2, chapter 6. Final take home given out and discussed
27, Dec. 8 Paper of Lyukov on volatility smile.  
28, Dec. 13 Stochastic volatility model of Heston