# Introduction to Mathematical Finance

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1 Introduction to Mathematical Finance R. J. Williams Mathematics Department, University of California, San Diego, La Jolla, CA USA DO NOT REPRODUCE WITHOUT PERMISSION OF THE AUTHOR Copyright c R. J. Williams, 2005 January 5, 2005

2 2 1. FINANCIAL MARKETS AND DERIVATIVES 1 Financial Markets and Derivatives 1.1 Financial Markets the development in the continuous case. Binomial tree models are the most common example in the discrete case. Our treatment of discrete and continuous models is not meant to be exhaustive, but rather to provide an introduction that will enable students to go on to read more advanced material. An arbitrage opportunity is an opportunity for a risk free profit. A financial market is said to be viable if there are no arbitrage opportunities. Typically, liquid financial markets move rapidly to eliminate arbitrage opportunities. Accordingly, in this course, attention will focus on financial market models that are viable. A financial market consists of tradable securities such as stocks, bonds, currencies, commodities, or even indexes. One reason for the existence of financial markets is that they facilitate the flow of capital. For example, if a company wants to finance the building of a new production facility, it might sell shares of stock to investors who buy the shares based on the anticipation of future rewards, such as dividends or a rise in the stock price. A variety of stochastic models is used in modeling the prices of securities. All such models face the usual trade off, namely, more complex models typically provide a better fit to data (although there is the danger of overfitting), whereas simpler models are generally more tractable and despite their simplicity can sometimes provide useful qualitative insights. Finding a good balance between a realistic and a tractable model is part of the art of stochastic modeling. Both discrete (in time and state), and continuous (in time and state) models will be considered here. The treatment of discrete time models is intended as a means of introducing notions such as hedging and pricing by arbitrage in a simple setting and to provide a bridge to 1.2 Derivatives A derivative or contingent claim is a security whose value depends on the value of some underlying security. Examples of derivatives are forward contracts, futures contracts, options, swaps, etc. (We recognize that the reader may not be familiar with all of these terms. Some of them are described in more detail below. For the other terms, or more detailed descriptions in general, we refer the reader to the book by J. Hull [14] or Musiela and Rutkowski [18].) The underlying security on which a derivative is based could be a security in a financial market, such as a stock, bond, currency or commodity, but it could also be a derivative itself, such as a futures contract. Secondary financial markets can be formed from derivatives. The derivatives market is huge a book written in 1996 (by Baxter and Rennie [1]) described it then as a \$15 trillion market, and in the 1999 printing of a book by S. Pliska [19] this market is cited as having \$20 trillion in principal outstanding. A forward contract is an agreement to buy or sell an asset at a certain future time for a certain price. Forward contracts are traded over 1

5 1.3. EXERCISES 7 a European call option for 100 shares of stock is purchased and that at expiration there are two possible scenarios for the stock price it has gone up or down by 40% since purchase of the option.

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