# Department of Statistics

## STAT 523

523—Financial Mathematics II. (3){=MATH 515} (3) (Prereq: MATH 514 or STAT 522 with a grade of C or better) Convex sets. Separating Hyperplane Theorem. Fundamental Theorem of Asset Pricing. Risk and expected return. Minimum variance portfolios. Capital Asset Pricing Model. Martingales and options pricing. Optimization models and dynamic programming.

Usually Offered: Spring even years.

Purpose: Complex mathematical techniques are now widely used in finance. This course is the second part of a two course sequence that provides an elementary introduction to these techniques and to the fundamental concepts of financial mathematics. We examine some mathematical models that are used in finance, e.g. to model risk and return of financial assets and to model the random nature of stock prices.

In the first part of the course we continue our investigation of the `No Arbitrage Principle'. A very important mathematical result known as the Separating Hyperplane Theorem will be proved in order to establish the `Fundamental Theorem of Asset Pricing' and the closely related `Arbitrage Theorem'. We also consider some other interesting mathematical consequences of the Separating Hyperplane Theorem such as the existence of a steady state vector for Markov matrices.

In the second part of the course we look at some mathematical problems arising in portfolio selection. Taking into account the investor's tolerance for risk leads to the problem of maximizing expected `utility'. Using the technique of `mean-variance analysis' we derive the main results of the famous Capital Asset Pricing Model which relates the riskiness of an individual stock to the overall market through a numerical coefficient known as `beta'. If time permits we will consider some discrete optimization problems such as the `knapsack problem' which arise in portfolio selection.

In the third part of the course we shall return to the problem of options pricing. Using the multiperiod binomial model we shall find an algorithm for the pricing of the American put option. The existence of an `equivalent martingale measure' for which the discounted stock process is a `fair game' leads us naturally to the general theory of martingales. We shall examine some of the basic concepts and results of this theory such as the notion of a `stopping time' and Doob's Optional Sampling Theorem.

Current Textbook: An Elementary Introduction to Mathematical Finance (second ed.) by Sheldon M. Ross. Cambridge University Press, 2003.

 Topics Covered This is a continuation of the syllabus for Financial Mathematics I (STAT 522). The emphasis given to individual topics will depend on the interests of the class and other factors. Convexity: Convex sets and convex hulls. The Separating Hyperplane Theorem. The Fundamental Theorem of Asset Pricing. Further applications of the Separating Hyperplane Theorem (e.g. to Markov matrices) Risk Management: Covariance and correlation. Utility functions. Jensen's Inequality. Maximization of expected utility. Risk and expected return of a portfolio. Covariance matrices. Minimum variance portfolios. Mean-variance analysis. Attainable portfolios: the Markovitz bullet and the efficient frontier. The Capital Asset Pricing Model. The market portfolio and the capital market line. Systematic and diversifiable risk. The beta coefficient and the security market line. Martingales and Option Pricing: Pricing of American options for the multiperiod binomial model. Discounted processes and martingales. Supermartingales. Stopping times. Doob's Optional Sampling Theorem. Estimation of the volatility parameter. Optimization Models: Examples of dynamic programming: the knapsack problem. Concave return functions. Investment allocation models.

The above textbook and course outline should correspond to the most recent offering of the course by the Statistics Department. Please check with the instructor for the course regulations, expectations, and operating procedures.

Contact Faculty: Stephen Dilworth (Mathematics)