# Math 194 Introduction to the Mathematics of Finance Winter 2001

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1 Math 194 Introduction to the Mathematics of Finance Winter 2001 Professor R. J. Williams Mathematics Department, University of California, San Diego, La Jolla, CA USA DO NOT REPRODUCE WITHOUT PERMISSION OF THE AUTHOR

2 2

3 Contents 1 Financial Markets and Derivatives FinancialMarkets Derivatives Exercises

4 Chapter 1 Financial Markets and Derivatives 1.1 Financial Markets A financial market consists of tradable securities such as stocks, bonds, currencies, commodities, or even indexes (see the excerpt from the Wall Street Journal). 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 are 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, 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. For reasons of tractability (to avoid measure theory), mostly discrete models will be considered here. 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. 1

6 1.3. EXERCISES 3 invested in stock, the profit would have been \$18 32 = 576 (on an investment of \$ =\$3,264 which is a 57600/3264% = 17.6% profit. Scenario 2: Suppose the price of CISCO stock on January 21 is \$67 per share. The holder of the option will not exercise it and makes a loss of \$33 per share (the cost of the option per share) and hence a net loss of \$3,300. This is a 100% loss on the \$3,300 initial investment. On the other hand, if the \$3,300 had been invested directly in stock, the loss would have been 35 32=\$1120 or a 34.3% loss on an investment of \$3,264 in stock. There are two main uses of derivatives, namely, speculation and hedging. For example, the buyer of a call option on a stock is leveraging his/her investment. The option may not cost much compared to the underlying stock and the owner of the call option can benefit from a rise in the stock price without having to buy the stock. Of course, if the stock price goes down substantially and the option is not exercised, the buyer of the call loses what he/she paid for the call option. In this case, the seller of the call option would benefit, without necessarily having to buy the stock. Of course, if the stock price goes up substantially and the option seller does not already own the stock, then he/she will have to pay the price of buying the stock at the increased price and turning it over to the owner of the option. Options may be used to leverage and speculate in the market. Depending on how the investment is used, the down side losses can be substantial. On the other hand, options can be used to hedge risk. For example, an owner of stock who wants to hedge against a dramatic decrease in the stock price could buy a put option to sell the stock at a certain price. 1.3 Exercises 1. Using the list of option prices given out in class, perform a similar calculation to that done in Example 1 of the text with AmOnline (America Online) in place of CISCO. For this, use the first AmOnline call option expiring in January for which a price is listed. You should assume that 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 10% since purchase of the option.

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