QF343 Introduction to Stochastic
Course Catalog Description
Introduction
This course is designed for undergraduate students in Quantitative Finance. In order for students to be able to price derivatives, they must first understand the behavior of the underlying. To this end, the students will need to understand how many of the most common stock models are created and utilized. At the core of this is understanding Brownian motion and other adapted stochastic processes from a conceptual level, which requires the knowledge of quadratic variation, martingales, and filtrations. Using these building blocks, many different structures of stock and interest rate processes can be developed: geometric Brownian motion, stochastic volatility models, CoxIngersollRoss, and Vasicek to name a few. After this course, the students will be able to price simple derivatives based on some of these models. In addition, through the use of stochastic integration, students will have a better understanding of portfolio valuation. By the end of this course, the student will be able to derive in its entirety the BlackScholesMerton European option pricing formula for Geometric Brownian Motion. The students are supposed to have a strong background in applied mathematics (analysis and calculus) and probability at an undergraduate level. Any student who does not already have this previous knowledge will have much greater difficulty learning the material.
Campus  Fall  Spring  Summer 

On Campus  X  X  
Web Campus 
Instructors
Professor  Office  

Thomas Lonon

Tlonon@stevens.edu  Altorfer 303 
More Information
Course Outcomes
By the end of this course, the students will be able to:
 Price derivative securities using the Binomial Asset Pricing Model
 Classify stochastic processes as martingales, Markov, or both/neither
 Simplify stochastic (Ito) integrals
 Determine the differentials of functions of stochastic processes
 Derive BlackScholesMerton’s European option pricing model
Course Resources
Textbook
Stochastic Calculus for Finance vol I and II, by Steven E. Shreve, Springer Finance, 2004, ISBN13: 9780387249681 (vol I) and 9780387401010 (vol II).
Additional References
Optional Readings:
Introduction to Probability Models, 10th edition, by Sheldon M. Ross, Academic Press, 2009, ISBN10: 0123756863, ISBN13: 9780123756862.
Probability and Random Processes, by Geoffrey Grimmett and David Stirzaker, Oxford University Press 2001. Stochastic Differential Equations, by Bernt Oksendal, 6th edition, 2010, ISBN10: 3540047581, ISBN13: 9783540047582. Introduction to the Mathematics of Financial Derivatives, by Salih N Neftci, 2nd ed, Associated Press, 2000, ISBN 0125153929.
Grading
Grading Policies
Assignments: There will be weekly homework assignments throughout the semester, with problems inspired from your textbook and on necessary skills.
The assignments and their weights are as shown below:
 Homework  30%
 Midterm  30%
 Final Exams  40% TOTAL  100%
Please note that your grade will be determined solely on the work you present over the course of the semester. No consideration such as your desire for a better grade for academic standing or job offers will be considered.
Extra Credit: On some of the homework assignments and possibly on the exams, there will be the occasional extra credit problem. This is the only source of extra credit for the course. There are no "extra assignments" that students can do to raise their average outside of the ones assigned. There are no exceptions, don't even bother coming to me and asking about extra work at the end of the semester, as I will only direct your attention to this part of the syllabus.
Lecture Outline
Topic  Reading  

Week 1  One Period Binomial Asset Pricing Models:
Introduction of the simplest asset pricing model 
Ch. 1 in vol. I 
Week 2  Multiperiod Model and Finite Probability Spaces:
Expand the model to include multiple time steps and define what all is involved in working in a finite probability space. 
Ch. 1 and 2 in vol. I 
Week 3  Expectation in Binomial Asset Pricing Models:
How to utilize expectation in the context of these asset pricing models 
Ch. 1 and 2 in vol. I 
Week 4  Martingales and Markov:
Definition and use of martingales and Markov processes in the context of a finite probability space. 
Ch. 2 in vol. I 
Week 5  Stopping Times and Random Walks:
Evolving our model to represent random walks and what is meant by a stopping time (and how/why we use them) 
Ch. 4 & 5 in vol. I 
Week 6  Scaled Symmetric Random Walks and Brownian Motion  Ch. 3 in vol. II 
Week 7  Quadratic Variation and Markov Property:
What is meant by quadratic variation, why it wasn’t seen before and what we mean by Markov property now that we are in a continuous probability space 
Ch. 3 in vol. II 
Week 8  Midterm  
Week 9  Reflection Prop. and Cont. Passage Times:
How we can use the reflection property to develop first passage times for Brownian motions in the context of a continuous probability space. 
Ch. 3 in vol. II 
Week 10  Stochastic Calculus Integrands:
What is meant by a stochastic integral and how we can represent/simplify one? 
Ch. 4 in vol. II 
Week 11  Ito’s Formula:
Use Ito to take differentials and functions of stochastic processes, provide examples of stochastic interest rate models  Ch. 4 in vol. II 
Week 12  BlackScholes, Levy:
Show the development of the BlackScholesMerton model and provide an alternative formulation of Brownian motion. 
Ch. 4 in vol. II 
Week 13  Change of Measure:
Use the change of measure to transition from the actual to the riskneutral probability space in order to price derivatives. 
Ch. 5 in vol. II 
Week 14  Review & Catchup 