MATH 6911 Numerical Methods in Finance

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1 MATH 6911 Numerical Methods in Finance Hongmei Zhu Department of Mathematics & Statistics York University Math6911 S08, 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 Math6911 S08, HM Zhu 1

2 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 4 Be able to answer 3 questions How does an option work? How can it be used? How is it priced? 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. 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 European 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 on 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 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 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 20 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? 21 7

8 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 a strike of $22.5 is $1 What are the alternative strategies? Investor s strategy S T =$15 S T =$27 Buy 100 shares Buy 2,000 call options 22 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? 23 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 24 8

9 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 lose is 1 billion dollars. Barings existing 200 years was bankrupted in 1995 Movie Rogue Trader 25 Master Rogue Traders 2008: French banker Jérôme Kerviel handed Société Générale a $7.1-billion loss 2006: Brian Hunter of Calgary oversaw the loss of $6- billion on hedge fund bets at Amaranth Advisors 2002: John Rusnak, who frittered away $750-million through unauthorized currency trading for Allied Irish Bank 1998: Yasuo Hamanaka squandered $2.6-billion on fraudulent copper deals for Sumitomo Corp. of Japan 1995: Nick Leeson brought down Britain's Barings Bank by blowing $1.4-billion Introduction to Options 1.2 More terminology and properties of stock options Math6911 S08, HM Zhu 9

10 More Terminology Moneyness : In-the-money option would give its holder a positive cash flow if it were exercised immediately At-the-money option would give its holder a zero cash flow if it were exercised immediately Out-of-the-money option would give its holder a negative cash flow if it were exercised immediately 28 Option value Option value = intrinsic value + time value Intrinsic value of an option is defined as the maximum of zero and the value the option would have if it were exercised immediately. For a call option, it is max (S K, 0) For a put option, it is max (K S, 0) 29 Option value An option's time value captures the possibility, however remote, that the option may increase in value due to volatility in the underlying asset In-the-money American options >= its intrinsic value 30 10

11 Notations c : European call option price p : European put option price S 0 :Stock price today K : Strike price T : Life of option σ: Volatility of stock price C : American Call option price P : American Put option price S T : Stock price at time T D : Present value of dividends during option s life r : Risk-free rate for maturity T with continuous compounding Note: risk-free rate is the rate of the interest that can be earned without assuming any risks 31 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) The effect on the price of a stock option of increasing one variable while keeping all others fixed (Hull, page 168) Variable S 0 K T σ r D c p C P

12 1. Introduction to Options (Hull s book, Section 4.2) 1.3 Interest rates and present values Math6911 S08, HM Zhu Interest Rates & Present Value For most of our course, assume that the short-term interest rate is a known function of time, not necessarily constant. For valuing options, an important concept concerning interest rates is that of present value or discounting. Ask the question: How much would I pay now to receive a guaranteed amount E at the future time T? 35 Group Discussion Assume that the constant annual interest rate is 10% and you deposit $100 in the bank today. 1.Assume your savings is compounded annually, how much money would I have in the bank after one years? 2.Assume your savings is compounded semiannually, how much money would I have in the bank after one years? 3.Assume your savings is compounded every three months, how much money would I have in the bank after one years? 36 12

13 Discrete Compounding In general, suppose that an amount M 0 is invested for t years at an annual interest rate r. If the rate is compounded m times per year, the final value of the investment is M 0 1 mt r + m 37 Discrete Compounding Compounding Frequency Annually (m=1) Value of $100 at end of one year Semiannually (m=2) Quarterly (m=4) Weekly (m=52) Daily (m=365) Continuous Compounding With continuous compounding, it can be shown that an amount M 0 invested for t years at an annual rate r grows to mt r rt lim M 0 1+ = M 0e m m 01. For example, 100e = $ For practical purpose, continuous compounding can be thought of as being equivalent to daily compounding. 1 Note: lim 1+ = e k k k 39 13

14 Which offer to take? With continuous compounding, if someone were to make you the offer of a) $100 immediately ( time t=0), or b) $ 100e rt at time t, which one do you choose? We regard both offers a) and b) as being of equal value: a) M 0 immediately ( time t=0), or rt b) M t = M e at time t () 0 40 How much would I deposit now to receive a guaranteed amount $100 at the future time t? Similarly, a deal that is guaranteed to produce exactly $100 at time t is worth exactly $ 100e rt at time zero. Transferring from $100 to $ 100e rt discounting for interest is called ( ) () = Or if M T = E, the value at time t of the certain payoff E is rt ( t) M t Ee 41 Present Value or Discounting Compounding a sum of money at a continuously compounded rate r for t years involves multiplying it rt by e Discounting it at a continuously compounded rate r for t years involves multiplying it by e rt 42 14

15 1. Introduction to Options 1.4 More on Stock Options (Hull, Sec ,6, page 205) Math6911 S08, HM Zhu American vs European Options An American option is worth at least as much as the corresponding European option In addition, In particular, C c P p S 0 C c K P p; Ke -rt p 44 Lower Bound for European Call Prices (No Dividends; Hull, page ) Consider the following 2 portfolios: Portfolio A: 1 European call on a stock + Cash Ke -rt Portfolio B: 1 unit of stock At the expiry date, what the portfolio A worth? At the expiry date, what the protfolio B worth? 45 15

16 Lower Bound for European Call Prices (No Dividends; Hull, page ) c max{s 0 Ke rt,0} 46 Lower Bound for European Put Prices (No Dividends; Hull, page ) Consider the following 2 portfolios: Portfolio C: 1 European put on a stock + 1 unit of stock Portfolio D: Cash Ke -rt At the expiry date, what the portfolio C worth? At the expiry date, what the protfolio D worth? 47 Lower Bound for European Put Prices; (No Dividends; Hull, p ) p max{ke -rt S 0, 0} 48 16

17 Put-Call Parity; No Dividends (Hull, page 212, Section 9.4) Consider the following 2 portfolios: Portfolio A: European call on a stock + cash Ke -rt Portfolio C: European put on the stock + 1 stock At the expiry date, what the portfolios A and C worth? 49 Put-Call Parity; No Dividends (Hull, page 212, Section 9.4) Consider the following 2 portfolios: Portfolio A: European call on a stock + cash Ke -rt Portfolio C: European put on the stock + 1 stock Both are worth MAX(S T, K ) at the maturity of the options They must therefore be worth the same today This means that c + Ke -rt = p + S 0 50 Early Exercise Usually there is some chance that an American option will be exercised early An exception is an American call on a nondividend paying stock This should never be exercised early 51 17

18 American call on non-dividend paying stocks should never be exercised early For an American call option: S 0 = 100; T = 0.25; K = 60; D = 0 Should you exercise immediately? What if you exercise immediately and hold the stock for the next 3 months? you do not feel that the stock is worth holding for the next 3 months? 52 American call on non-dividend paying stocks should never be exercised early For such an American call, it satisfies C c S 0 Ke rt Given r>0, it follows C > S 0 K 53 Reasons For Not Exercising a Call Early (No Dividends ) No income is sacrificed We delay paying the strike price (time value of money) Holding the call provides insurance against stock price falling below strike price 54 18

19 It can be optimal to exercise European puts on non-dividend paying stock early when S is sufficiently low. Are there any advantages to exercising an American put when S 0 = 0; T = 0.25; r=10% K = 100; D = 0 P p Ke -rt S 0 In fact, a stronger condition holds P K S 0 55 References on Background Knowledge Appendix A for Introduction to MATLAB, Appendix B for review of probability theory, (Brandimarte, Numerical Methods in Finance ) Chapter 1, Appendices A & B for quick reviews of linear algebra, calculus, and probability theory, Mathematical Techniques in Finance: tools for incomplete markets by Aleš Černý 56 19