16:642:611. Selected Topics in Applied Mathematics I (Mathematical Finance)

Autumn 2003

TTh 1:10-2:30, Hill 525 (tentative)

 

Professor Paul Feehan

Hill Center for Mathematical Sciences

Hill 544

Secretary: 732-445-2390

Fax: (509) 753-9730

 Email: feehan at rci dot rutgers dot edu

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

 

 

Overview

 

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.

 

Principal texts

 

Stochastic Differential Equations, Øksendal

Brownian Motion and Stochastic Calculus, Karatzas and Shreve

Derivatives, Wilmott

Modeling Extremal Events, Embrechts, Klüppelberg, and Mikosch

 

Supplementary texts

 

Quantitative Modeling of Derivative Securities,Avellaneda and Lawrence

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

 

 

Background probability texts

 

Stochastic Processes, Doob

An Introduction to Probability Theory and Applications I, II, Feller

A First Course on Stochastic Processes, Karlin and Taylor

A Second Course on Stochastic Processes, Karlin and Taylor

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.

 

Last updated: September 22, 2003 © Paul Feehan (feehan at rci dot rutgers dot edu)