Office: Melcher 240B
Phone: 713-743-4466
Fax: 713-743-4789
E-mail cpirrong@uh.edu
website: www.bauer.uh.edu/spirrong
This course studies the fundamentals of innovations in
quantitative finance. I view this course as consisting of three basic
interlocking conceptual "rings": valuation theory, numerical methods,
and statistical/econometric methods. This course will examine each in
detail. We will explore the major valuation techniques--PDE techniques
and martingale methods in a variety of different contexts including equity option,
currency option, fixed income derivative, exotic derivative, and stochastic
volatility models. This analysis will be built on an extended
introduction to basic stochastic calculus. We will also implement PDE and
martingale models using numerical methods--PDE solvers and
My overheads can be downloaded from my web page. I will distribute additional readings from time to time during the semester. The book by Daniel Duffy, Finite Difference Methods for Financial Engineering is optional.
The grading for the course is based two major valuation projects and several homework assignments. The valuation projects are team/group endeavors. Each valuation project will involve the use of numerical techniques to value a derivative contract. Grading will be based on an instructor-assigned grade and a team evaluation of each team member. Homework assignments are individual work. The projects will account for 75 percent of the grade. The homeworks account for 25 percent of the grade.
This course will be demanding. I require and expect that students have a strong background in multivariate derivative calculus and integral calculus. Moreover, programming experience (in any common language such as C or variations thereon, Visual Basic, Fortran, or a higher level language such as Matlab, Mathematica, or Gauss) is very useful but not required. The material in the class is highly mathematical and you will be expected to understand the mathematical concepts and use them.
This outline provides an overview of the material that we will cover in this course and the order in which it will be covered. The timing is tentative. Best efforts will be devoted to keeping on schedule, but things may get pushed back somewhat to ensure that we don't have to rush through crucial material. I will let you know if we will deviate from the schedule: absent any announcements in class or on the web, you may assume that the schedule holds.
January 19. Derivatives Basics Review. To make sure that everyone is familiar with the basics of the kinds of derivatives contracts we will be valuing, we’ll spend our first week reviewing forwards, futures, swaps, and options.
January 26. Math Review and an Intro Into Stochastic Calculus. This week we will review some math basics and begin to explore stochastic calculus.
February 2. Stochastic Calculus and Ito's Lemma; Matlab Introduction. Stochastic calculus is the bedrock of quantitative finance. It is a very deep subject. Fortunately, the key concepts can be understood and used even if you don't know stochastic calculus in all its complexity. This week we will cover the basics, most notably Ito's lemma; by the end of this course, you should be able to use Ito's lemma in your sleep. In addition, we will have a MATLAB tutorial. MATLAB is a high-level mathematical programming language with some excellent matrix capabilities and a variety of useful toolboxes Homework Assignment 1 Issued: Ito's Lemma Exercises.
February 9. PDE Methods: Derivative Valuation in the Style of Black-Scholes-Merton. There are two basic (and related) approaches to value derivatives. The original approach was derived by Black, Scholes, and Merton in the early-1970s. This approach utilizes arbitrage arguments to derive partial differential equations that can be solved (subject to boundary conditions) to determine the price of a derivative. This section will examine these arguments in detail, focusing on simple equity derivatives (such as puts and calls).
February 16. PDE Methods When the Underlying is Not Traded. The traditional B-S-M approach assumes that the asset/contract underlying a derivative contract is traded (e.g., the stock underlying a stock option contract is traded). However, many derivatives are based on underlyings that are not traded. Electricity and weather are prominent examples. Also, if volatility is itself random, options on stocks depend on a non-traded quantity. This week will explore how to extend the basic PDE framework to encompass non-traded underlyings.
February 23-March 7. Implementing
PDE Models. Deriving PDEs is the easy
part--solving them is the hard part! These two classes will examine in detail
how to solve PDEs using finite difference methods. We
will focus on the Crank-Nicolson method, but will discuss implicit and explicit
schemes as well. We will also discuss successive overrelaxation
methods for pricing American options. Project 1 Assigned (3/9/10): Option
Valuation Using Finite Difference Methods.
March 15. Spring Break!
March 22. Finite Difference Solutions to Models with More Than
March 29. Martingale Methods: The Girsanov
Theorem, Numeraires, and the Kolmogorov
and Feynman Kac Theorems. The main competitor to
PDE methods for valuing derivatives is the Martingale approach. This approach
was developed in the 1980s and 1990s. It exploits the fact that the lack of
arbitrage implies (under certain technical conditions) the existence of an
"equivalent" probability measure. An option value is determined by
calculating expected payoffs under this equivalent measure and then discounting
at a risk free rate. We will first discuss the concept of a measure, and then
analyze the linkage between the lack of arbitrage and the existence of such a
measure. The most useful "trick" in applying Martingale methods
is the Girsanov Theorem (also known as the Cameron-Girsanov-Martin Theorem.) We'll examine this theorem and
see how to exploit it in pricing derivatives. Another nice trick that
frequently makes pricing easier is to change "numeraires."
The numeraire implicitly assumed in the B-S-M model
is a money market account, but in some circumstances other numeraires
make it easier to value particular derivatives. This week we'll see how to
exploit this "trick" by examining the pricing of "Quantos." PDE methods and Martingale methods
both have die-hard advocates. This week we will show that these methods are
effectively equivalent. The mathematical link between them is given by two
important theorems--the Kolmogorov backward equation
and the Feynman Kac theorem. We will analyze these
theorems in some detail.
March 29. Project 1 Due!
April 5. Martingale Methods: Numerical Implementation using Explicit
Integration, Numerical Integration Monte Carlo. Martingale methods imply
that derivatives can be valued by calculating the appropriate integral. There
are many ways to calculate integrals. We will discuss three alternative
methods, and determine the circumstances that make each appropriate. We will
spend the most time on
April 12-19. More
April 26. Stochastic Volatility and Jumps.
The B-S-M formula assumes that volatility is a constant (or a function of
time). We will discuss the evidence that is inconsistent with this. Moreover,
we will discuss alternative models that can potentially address the failings of
the B-S-M model--the stochastic volatility and jump models. We will examine the
derivation of these models, and appraise their strengths and weaknesses. We
will also discuss implementation.
May 3. Project 2 Due at 5 pm.
I joined the