Autumn 2003
TTh
1:10-2:30,
Hill 525 (tentative)
Professor Paul Feehan
Hill 544
Secretary: 732-445-2390
Fax: (509) 753-9730
Course e-mailing list:
math_611-at-rams.rutgers.edu
(documentation at Rutgers Automated
Mass-mailing System)
Overview | Course
outline | Principal
texts | Supplementary
texts | Background
probability texts |
Suggested
prerequisites or co-requisites
| Internet
resources for mathematical finance
| Internet
resources for computing |
C
program and other software downloads
| Problem
sets | Lecture
web log | Math_611 mailing
list
archive
Mathematical finance is an emerging discipline wherein mathematical tools are used to model financial markets and solve problems in finance. Finance as a sub-field of economics concerns itself with the valuation of assets and financial instruments as well as the allocation of resources. History and experience have produced fundamental theories about the way economies function and the way we value assets. Mathematics comes into play because it allows theoreticians to model the relationship between variables and represent randomness in a manner that can lead to useful analysis. Mathematics, then, becomes a tool chest from which researchers can draw to solve problems, provide insights and make the intractable model tractable.
Our course will primarily consist of an introduction to some of the most important mathematical tools in use today by practitioners of mathematical finance, stochastic partial differential equations, which is used to price options, derivatives, and other complex financial securities, and extreme value theory, used in risk analysis. After an introduction, as needed, to the background theory of probability and stochastic processes, we shall focus on diffusion processes, such as a Brownian motion, and jump diffusion processes. In discussing applications, such as the Black-Scholes option pricing formula, we shall give an introduction to derivative securities.
We shall also discuss the application of stochastic processes to risk analysis. Empirical observation suggests that large market movements occur more often than models based on a convenient normal probability distribution would indicate – so, in reality, probability distributions accurately modeling the behavior of the market should have “fat tails”. Hence, we shall discuss extreme value theory and its application to risk analysis. Time permitting, techniques for measuring and managing the risk of trading and investment positions will be discussed, including the portfolio risk management technique of Value-at-Risk, stress testing, and credit risk modeling. Important applications include risk analysis for credit derivatives and collateral debt obligations.
Stochastic Differential Equations, Øksendal
Brownian Motion and Stochastic Calculus, Karatzas and Shreve
Derivatives, Wilmott
Modeling Extremal Events, Embrechts, Klüppelberg, and Mikosch
Quantitative Modeling of Derivative Securities,Avellaneda
and
Financial Calculus: An Introduction to Option Pricing, Baxter and Rennie
Options, Futures, and Other Derivatives, Hull
Derivative Securities, Jarrow and Turnbull
Methods of Mathematical Finance, Karatzas and Shreve
Statistical Analysis of Extreme Events, Reiss and Thomas
Dynamic Hedging, Taleb
Stochastic Processes, Doob
An Introduction to Probability Theory and Applications I, II, Feller
A First Course on Stochastic Processes,
Karlin and
A Second Course on Stochastic Processes,
Karlin and
Probability, Random Variables, and Stochastic Processes, Papoulis
Suggested
prerequisites or
co-requisites
Our target audience is second graduate students or higher in all areas of mathematics, as well as graduate students in statistics, economics, and business, so we shall do our best to cater for a diverse audience. For all graduate students, familiarity with basic probability theory at the level of an applied text such as Papoulis is important background knowledge. Students in mathematics and statistics should ideally have an enthusiasm for learning how mathematics is used in finance, while economics and business students should have an interest in analytical, quantitative methods.
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