Office: Melcher 240B
Phone: 713-743-4466
Fax: 713-743-4789
E-mail cpirrong@uh.edu
website: www.bauer.uh.edu/spirrong
This course is a thorough introduction to topics in the pricing of derivatives. After an introduction to stochastic calculus, we will plunge into the traditional derivation of the Black-Scholes model, and then discuss option pricing when the underlying is not traded (our first foray into incomplete markets.) We will then focus on the more "modern" method of pricing derivatives--martingale methods--beginning with an examination of the Girsanov Theorem, and then proceeding into a discussion of the linkage between absence of arbitrage and the existence of an equivalent martingale measure, and the linkage between market completeness and the uniqueness of the EMM. An examination of the role of numeraires and the utility of different numeraires follows. After this exploration of derivative pricing under classical assumptions, we will examine pricing the complications associated with pricing in incomplete markets, where incompleteness is due to jumps or stochastic volatility. We will then examine interest rate models, including traditional spot rate models and more current market models. We will close (time permitting) with a discussion of numerical methods.
Overheads can be downloaded from my web page. These will be the main source of readings, and will be supplemented throughout the semester. The textbook is Shreve, S. (2004). Stochastic Calculus Models for Finance. Volume 2. There are numerous additional books that you can obtain as well that may be of use to you. A bibliography is attached.
Grading is 30 percent homework, 70 percent final/final project.
This course will be demanding. I require and expect that students have a strong background in multivariate derivative calculus and integral calculus. 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 17. Stochastic Calculus and Ito's Lemma. 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.
January 24. Derivation of the Black-Scholes Model. Here we will reproduce the derivation of the Black-Scholes model using the original hedging-based approach pioneered by Black, Scholes, and Merton. We will also investigate how these hedging methods permit the determination of option prices when the underlying is not traded.
January 31-February 28. 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. 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.
Note: NO CLASS ON 7 FEBRUARY.
March 7-March 28. Pricing in Incomplete Markets: 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 review alternative models including jump-diffusion models, stochastic volatility models, and variance-gamma models. We will also discuss implementation. [Note: Spring Break is 13-17 March, so no class on 16 March.]
April 4-April 18. Interest Rate Models. Interest rate products make up the largest share of the derivatives market, and pose some unique challenges. We will examine the traditional approach for modelling these products based on spot rate models. We will then examine more contemporary approaches, most notably the market models such as the LIBOR model and the Swap Rate Model. In addition to being of interest in their own right, these models will also give additional insight on the importance of numeraire choice.
Apri1 25-May 2. Numerical Methods & Review. Most derivative models don't give us nice closed form solutions, only partial differential or integral equations that must be solved numerically. We will examine the three basic numerical approaches--finite difference methods, Monte Carlo methods, and explicit integration.
Take home final due May 9.
Bibliography. Here is a list of some useful books on the subjects
covered in this course. They differ in their completeness and difficulty. Joshi
and Baxter-Rennie give good, accessible overviews along with the practically
necessary math. Many of the other books are much more formal, and go into a
level of detail that is not really necessary for most applications. Most of
these books present the modern martingale approach. Wilmott's book gives an
encyclopedic introduction to PDE methods and the associated numerical
techniques.
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