FINANCIAL MARKETS AND FINANCIAL DERIVATIVES

AN INTRODUCTION TO OPTIONS AND MARKETS Finance is one of the fastest developing areas in modern banking and corporate world. This provides a rapidly growing impetus for new mathematical models and modern mathematical methods; the area is an expanding source for mathematics. The demand from financial institutions for well-qualified mathematicians is substantial and there is a corresponding need for professional training of existing staff.

Financial Markets Financial markets are markets for financial instruments, in which buyers and sellers find each other and create or exchange financial assets. Stock markets Bond markets Currency markets or foreign exchange markets Commodity markets Futures and options markets

A company that needs to raise money, for example to build a new factory or develop a new product, can do so by selling shares (stocks, or equities) in itself to investors. The company is then owned by its shareholders. If the company makes a profit, part of this may be paid out the shareholders as a dividend of so much per share, and the proceeds are distributed to shareholders. Thus, shares have a value that reflects the views of investors about the likely future dividend payments and capital growth of the company and this value is quantified by the price at which they are bought and sold on stock exchanges.

As markets have become more sophisticated, more complex contracts than simple buy and sell trades have been introduced known as: financial derivatives, derivative securities, derivative products, contingent claims or just derivatives, They can give investors of all kinds a great range of opportunities to tailor their dealings to their investment needs.

Mathematical finance is not about predicting the price of a stock. What it is about is figuring out the price of options and derivatives. In calculus, a derivative gives you a measure of the rate of change of a dependent variable as an independent variable is changed. In finance, an option is an example of a derivative, any financial instrument whose value is derived from that of an underlying security.

We will deal with some of the financial theory and models that have been developed to analyse derivatives which is a combination of mathematical modelling and analysis. We need to become familiar with some of the necessary financial jargon.

FINANCIAL DERIVATIVES Definitions: Practitioners’ Definition: Derivative securities are financial contracts that ‘derive’ their value from cash market instruments such as stocks, bonds, currencies and commodities. Academic Definition: A financial contract is derivative security or contingent claim if its value at expiration date T is determined by the market price of the underlying cash instrument at time T.

TYPES OF DERIVATIVES Options Futures and forwards Swaps Options, forwards and futures are basic building blocks. Swaps can eventually be decomposed into sets of basic forwards and options.

OPTIONS Definition: An option is the right, but not the obligation, to buy or sell a security such as a stock for an agrees upon price at some time in the future. Options have been around for a long time. The earliest ones were used by manufacturers and food producers to hedge their risk. A farmer might agree to sell a bushel of wheat at a fixed price six months from now then, take a chance on the vagaries of market prices. Similarly, a steel refinery might want to lock in the price of iron ore at a fixed price.

The most familiar type of option is the option to buy a stock at a given price at a given time. Suppose I have an option that allows me to buy a share of Microsoft for $50 in three months time, but does not compel me to do so. If Microsoft happens to be selling at $45 in three months time, the option is worthless. It would not be sensible to buy a share for $50 when I could call my broker and buy it for $45. So I would choose not to exercise the option. On the other hand, if Microsoft is selling for $60 three months from now, the option would be quite valuable. I could exercise the option and buy a share for $50. I could then turn around and sell the share on the open market for $60 and make a profit of $10 per share. Therefore this stock option I possess has some value. There is some chance it is worthless and some chance that it will lead me to a profit. The basic question is: how much is the option worth today?

The huge impetus in financial derivatives was the paper of Black and Scholes in Although many researchers had studied this question, Black and Scholes gave a definitive answer, and a great deal of research has been done since. These are not just academic questions; today the market in financial derivatives is larger than the market in stock securities. In other words, more money is invested in options on stocks than in stocks themselves.

TYPES OF OPTIONS There are two basic types of options, European and American options. Strike price: aggreed upon price for buying or selling Expiry: deadline by which the option must be exercised (also known as exercise time, strike time and expiry date) Call option: an option to buy a security Put option: an option to sell a security Definition: A European call option on a security is the right to buy the security at a present strike price K. This right may be exercised at the expiration date T of the option.

A European put option is similar, but gives the owner the right to sell an asset at a specified price at expiration. In contrast to European options, American options can be exercised any time between the writing and expiration of the contract. To the mathematician, American options are more interesting since they can be interpreted as free boundary value problems. Other types of options are: Exotic (Asian, Bermudan), look-back, etc.

FORWARDS AND FUTURES CONTRACTS A forward contract is an obligation to buy or sell an underlying asset at a specified price, known as forward price, on a specified date in the future, the delivery date or maturity date. A futures contract is essence a forward contract, but with some technical modifications. Because neither forward nor futures contracts contain the element of choice (to exercise or not to exercise) that is inherent in an option, they are easier to value.

REFERENCES The Mathematics of Financial Derivatives, by P. Wilmott,S. Howison and J. Dewynne, Cambridge University Press, An Elementary Introduction to Mathematical Finance. Options and Other Topics. (Second Edition), by Sheldon M. Ross, Cambridge University Press, An Introduction to the Mathematics of Financial Derivatives, by Salih N. Neftci, Academic Press, 2000.