AK, AS, SC/MATH 4143 Scientific Computations for Finance Applications

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1 AK, AS, SC/MATH 4143 Scientific Computations for Finance Applications Hongmei Zhu Department of Mathematics & Statistics York University Math4143 W08, HM Zhu Objectives Master fundamentals of financial theory Develop basic models commonly used to analyze financial derivatives Be able to solve these models numerically and assess numerical solutions properly. Have a big picture on computational finance and be able to use skills in mathematical modeling, analysis, and numerical computation to analyze or characterize basic derivative products 2 1. Introduction to Options (Hull s book, Chapters 1, 8, and 9) 1.1 Options Math4143 W08, HM Zhu 1

2 Outline Options: European call and put, American call and put Payoff diagrams How and why options are traded 4 Derivatives In the last 25 years, derivatives have become increasingly important in finance Derivatives is a financial instrument whose value depends on the values of other, more basic, underlying variables The underlying variable are the prices of traded assets from oil, gold, to stocks Forward and future contracts are simple derivatives In this course, we focus on options 5 Options Have been around for many years, but were first traded on an exchange on April 26, 1973 An contract that gives holder the right to do something Its holder does not have to exercise the option; it s a right, not an obligation; therefore, he/she has to pay for the right Its writer does have a potential obligation; therefore, he/she needs compensation What is the fair price for an option? 6 2

3 Call and Put Options A Call Option is an option to buy a certain asset by a certain date for a certain price A Put Option is an option to sell a certain asset by a certain date for a certain price Note: The underlying asset of an option could be for example, stocks, foreign currency, stock indices, and future. 7 Specification of an option K: the price described in the contract, called exercise or strike price T: the date described in the contract, called expiration date or maturity Option price is called the premium. Types, such as European or American Put or call Dividends and stock splits Position limits and exercise limits 8 Option Positions There are two sides to each option: The party that has agreed to buy has what is termed a long position (purchaser) The party that has agreed to sell has what is termed a short position (writer) The writer of an option receives cash up front, but has potential liabilities later The writer s profit/loss is the reverse of the purchaser of the option 9 3

4 European options: simple options European Call Option is a contract which gives its holder the right to buy the underlying asset at the expiry date for a prescribed amount European Put Option is a contract which gives its holder the right to sell the underlying asset at the expiry date for a prescribed amount 10 European options: an example You wants to buy a house. After a few weeks of searching, you discovers one you really likes. Unfortunately, you won't have enough money for a substantial down payment for another six months. So, you approaches the owner of the house and negotiates an option to buy the house within 6 months for $200,000. The owner agrees to sell me the option for $2,000. Scenario 1: During this 6-month period, you discovers an oil field underneath the property. Scenario 2: You discovers a toxic waste dump on the property. - finance.yahoo.com 11 Long Call on one IBM share Profit from buying an IBM European call option: option price = $5, strike price = $100, option life = 2 months Profit ($) Terminal stock price ($) Exercised if S > K Call option holders like to see increase of S 12 4

5 Short Call on one IBM share Profit from writing an IBM European call option: option price = $5, strike price = $100, option life = 2 months Profit ($) Terminal stock price ($) Long Put on one Exxon share Profit from buying an Exxon European put option: option price = $7, strike price = $70, option life = 3 months Profit ($) Terminal stock price ($) Exercised if S < K Put option holders like to see decrease of S 14 Short Put on one Exxon share (Figure 1.5, page 9) Profit from writing an Exxon European put option: option price = $7, strike price = $70, option life = 3 months Profit ($) Terminal stock price ($)

6 Payoffs from Options Payoff: The cash realized by the holder of an option or other derivative at the end of its life K = Strike Price S T = S(T) = price of asset at maturity Payoff depends on K and S T. For instance, payoff from a long position in a European call option is max (S T K, 0) 16 Payoffs from Options What is the Option Position in Each Case? Payoff Payoff K S T K S T Payoff Payoff K S T K S T 17 American options European options can be exercised only at the expiration date American option can be exercised at any time up to the expiry date Most of the options are traded on exchanges are American options European options are easier to analyze Some properties of American options are often deduced from those of European Two major issues: determine a fair value and the best time to exercise the option 18 6

7 Other Options (Hull, Chap. 22, page 529) Exotic options or path-dependent options. It depends on the history of an asset price, not just on its value on exercise. Common ones are Barrier options (can either come into existence or become worthless if the underlying asset reaches some prescribed value before expiry) Asian options (the price depends on some form of average) Lookback options (the price depends on the asset price maximum or minimum during its life) 19 How are options traded? Options can be traded on official exchanges Most exchanges use market makers to facilitate options trading A market maker quotes both bid and ask prices when requested;the market maker does not know whether the individual requesting the quotes wants to buy or sell Options can be traded directly between large financial institutions (so called over-the-counter) 20 Reading Financial Press: Prices of options on Intel, May 29, 2003 CALLS PUTS Option Strike Price ($) June July Oct. June July Oct. Intel (20.83)

8 Purposes of Buying Options Hedging: to reduce the risk that one may face from potential future movements in market variables farmers, manufactures; without hedging, they may do better but may be worse Speculation: to bet on the future direction of market variables bet on increase or decrease of S Arbitrage: involves locking in a risk-free profit by simultaneously entering into transactions in two or more markets attractive, large trade, very short time span, very small chance Here, we assume no arbitrage opportunities 22 Hedging Example (p.10, Hull) An investor owns 1,000 Microsoft shares currently worth $28 per share. Concerns about the possible share decline in the next 2 months and therefore wants protection Could buy 10 two-month put options with K = $ Each option contract costs $100. What happens if the Microsoft share falls below or stays above $27.50 two months after? 23 Speculation Example (p. 12, Hull) An investor with $2,000 to invest feels that Amazon.com stock price will increase over the next 2 months. The current stock price is $20 and the price of a 2-month call option with K=$22.5 is $1 What are the alternative strategies? Investor s strategy S T =$15 S T =$27 Buy 100 shares Buy 20 call options 24 8

9 Arbitrage Example (p. 14, Hull) A stock price is quoted as 100 in London and $172 in New York Stock Exchange The current exchange rate is per pound What is the arbitrage opportunity? 25 No Arbitrage No arbitrage is the key principle in option price There is never an opportunity to make a risk-free profit that gives a greater return than that provided by the interest from a bank deposit 26 Business Snapshot The Barings Bank Disaster (p. 15, Hull) Derivatives are versatile. Nick Leeson, an employee of Barings Bank in the Singapore office in 1995, has a mandate to hedge or arbitrage became speculator He began to make losses which he was able to hide. Then he took bigger speculative positions in an attempt to recover the losses, but only succeeded in making the losses worse Finally, his total loss is 1 billion dollars. Barings existing 200 years was bankrupted in 1995 Movie Rogue Trader 27 9