Stochastic Calculus for Finance I (46-944) Spring 2008 Syllabus

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1 Stochastic Calculus for Finance I (46-944) Spring 2008 Syllabus Introduction. This is a first course in stochastic calculus for finance. It assumes students are familiar with the material in Introduction to Probability and Multi-Period Asset Pricing. This course revisits the ideas of no-arbitrage pricing and risk-neutral probability measures covered in Multi-Period Asset Pricing, but now in a continuous-time context. The applications in this course are primarily to equity derivatives. In particular, the Black-Scholes partial differential equation and formula are developed in detail. However, the stochastic calculus content of the course is the foundation for fixed income and even credit derivative models. Topics covered by the course are probability theory in general spaces, independence and conditioning, Brownian motion, Itô integrals and the Itô formula, the Black-Scholes formula, and change of measure The sequel to this course, Stochastic Calculus for Finance II, discusses risk-neutral pricing in more detail, the relationship between stochastic calculus and partial differential equations, change of numéraire and its application to option pricing in the presence of a random interest rate, and interest rate term-structure models. Instructor Steven Shreve Department of Mathematical Sciences Carnegie Mellon University Wean Hall 6216 Pittsburgh, PA Fax: Voice: Text. The lectures detailed below will be taken from the text S. Shreve, Stochastic Calculus for Finance II: Continuous Time Models, Springer-Verlag, New York, Errata at 1

2 Lecture Schedule. Pittsburgh classes meet Mondays and Wednesdays, 3:30-5:00, in Cooper Auditorium, except that because of Martin Luther King Day, there will be no class on Monday, January 21. However, in that week there will be a class at 3:30 on Friday, January 25. These afternoon lectures will not be recorded. New York classes meet Tuesdays, 5:30 8:30, in Cooper Auditorium, and these lectures will be recorded. The instructor will be in New York for the classes on Tuesday, January 22, and Tuesday, February 26. Week of January 14: General Probability Theory Section 1.1: Infinite Probability Spaces Section 1.2: Random Variables and Distributions Section 1.3: Expectations Section 1.4: Convergence of Integrals Section 1.5: Computation of Expectations Section 1.6: Change of Measure Week of January 21: Information and Conditioning Section 2.1: Information and σ-algebras Section 2.2: Independence Section 2.3: General Conditional Expectations Week of January 28: Brownian Motion and Itô Integrals Section 3.3: Brownian Motion Section 3.4: Quadratic Variation Section 3.5: Markov Property Section 4.2: Itô Integral for Simple Integrands Section 4.3: Itô Integral for General Integrands Week February 4: Mid-term Exam, 5:30-7:30 pm, Tuesday, February 5, covering first three lectures. No class this week Week of February 11: Stochastic Calculus and Black-Scholes Section 4.4: Itô-Doeblin Formula Section 4.5: Black-Scholes-Merton Equation Section 4.6: Multivariable Stochastic Calculus 2

3 Week February 18: Risk-Neutral Pricing Section 5.2: Risk-Neutral Measure Section 5.3: Martingale Representation Theorem Section 5.4: Fundamental Theorems of Asset Pricing Week of February 25: Review Final Exam: Monday, March 3, 5:30-8:30 p.m., Cooper Auditorium Evaluation. There will be a mid-term exam, Tuesday, Feb. 5, 5:30 7:30 p.m. There will be no class the week of February 4. The mid-term exam will cover the first three lectures and will count for 32% of the course grade, except as noted below. There will be a final exam Monday, March 3, 5:30-8:30 p.m.. This exam will cover the full course. It will count for 48% of the course grade, except as noted below. You may not bring anything to the exams except writing instruments. During the exam, you may not give nor receive assistance. Violation of this policy will be treated seriously according to the procedures in the MSCF handbook current-graduatestudents/student-handbook/index.aspx and could result in expulsion from the program. There will be weekly homework assignments. To assist with these, each Saturday there will be a session with the teaching assistant to review the lectures and discuss examples related to the homeworks. You are encouraged to work together on the homeworks, and the teaching assistant will be available to help. If you receive help on the homework, you must write on your homework paper the names of all the people with whom you worked. Furthermore, you must write up your own solutions without reference to the solutions written by others. Working together is permitted, even encouraged. It aids understanding. Merely copying the work of others without understanding is unethical. It is our goal to identify people who are unethical and to prevent them from entering the finance industry. 3

4 A good homework score is useless if you do not perform well on the exams. The homeworks constitute 20% of the course grade, provided you score a weighted average of at least 60% on the exams. Your weighted exam average will be computed by assigning 40% weight to the midterm exam and 60% weight to the final exam. If you fail to obtain a weighted exam average of at least 60%, your homework grades will be disregarded and your course grade will be based on the exams alone. Approximately half the homework problems will be graded, and all the midterm exam problems and final exam problems will be graded. Solutions to all homework problems, both the graded and the ungraded ones, will be provided. Homeworks are due on the dates indicated below. These may be submitted at the site where you receive instruction or faxed or ed to the teaching assistant: David German Department of Mathematical Sciences Carnegie Mellon University Wean Hall 7213 Pittsburgh, PA Fax: Voice: If homeworks are mailed on or before the due date, we will regard them as submitted on time. Students in both Pittsburgh and New York may submit homeworks late, but credit will be deducted for late submission. No homeworks will be accepted on or after the day solutions are posted. The due dates for homeworks and dates the homeworks and solutions will be posted are given below. Homework Schedule Mon., Jan. 14: Homework 1 posted Mon., Jan. 21: Homework 2 posted Fri., Jan. 25: Homework 1 due Mon., Jan. 28: Homework 3 and Solutions to Homework 1 posted Wed., Jan. 30: Homework 2 due 4

5 Mon., Feb. 4: Homework 4 and Solutions to Homework 2 posted Tue., Feb. 5: Homework 3 due Mon., Feb. 11: Homework 5 and Solutions to Homework 3 posted Mon., Feb. 18 Homework 6 and Solutions to Homework 4 posted Mon., Feb. 25 Solutions to Homework 5 posted Wed., Feb. 13: Homework 4 due Wed., Feb. 20: Homework 5 due Wed., Feb. 27: Homework 6 due Fri., Feb. 29: Solutions to Homework 6 posted 5

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