INTRODUCTION TO FINANCIAL MATHEMATICS

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INTRODUCTION TO FINANCIAL MATHEMATICS

• Introduction and Preliminaries

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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 professianl
  training of existing staff.

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• Financial Markets
• Stock markets
• Bond markets
• Currency markets or foreign exchange markets
• Commodity markets
• Futures and options markets

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• 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 if the company is taken over or otherwise
  wound up, 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.

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• 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.

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• 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.

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• In this course, we 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.

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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.

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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.

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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.

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• The most familiar type of option is the option to
  buy a stock at a given price at a given time.
  Suppose we 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. I would be silly
  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?

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• The huge impetus in financial derivatives was the
  paper of Black and Scholes in 1973. 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.

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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.

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• 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.

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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.

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REFERENCES

• The Mathematics of Financial Derivatives, by P.
  Wilmott,S. Howison and J. Dewynne, Cambridge
  University Press, 1995.
• An Elementary Introduction to Mathematical
  Finance. Options and Other Topics. (Second
  Edition), by Sheldon M. Ross, Cambridge
  University Press, 2003.
• An Introduction to the Mathematics of Financial
  Derivatives, by Salih N. Neftci, Academic Press,
  2000.