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Options: the basics Readings for Options • Forget about the textbook. Use online free sources and the lecture notes as your main reference • Most of the lecture materials are drawn from Chicago Broad of Options Exchange learning center Online tutorials Click on “Options Basics” and read: [1] Options Overview [2] Introduction to Options Strategies [3] Expiration, Exercise and Assignment [4] Options Pricing 1 • If you want to know more about options, such as the specific regulations for trading options, you can go to: [1] The Option Industry Council Website [2] The Options Clearing Corporations Website [3] Check out the various websites for US exchanges. Content • What is option? • Terminology • No arbitrage • Pricing Options – The Binomial Option Pricing Model – The Black-Scholes Model Why do we study options? Economics: It is all about our happiness. Happiness = U(CCurrent,Cfuture) We are happier with more Cfuture, but because we are risk-averse, we are worried about the fluctuation of Cfuture. As you will see later, options provide a special payoff structure. In words: Our happiness is derived not only from current consumption but also from future consumptions which inherently involves uncertainty. This ultimately constitutes our risk concern over the future payoffs of assets that we own. Because of the special payoff structure of options, holding options enables us to adjust our risk exposure, and ultimately change our happiness level. Options? Terminology Arbitrage Binomial Black-Scholes Who trade options? A quote from Chicago Board Options Exchange (CBOE): “The single greatest population of CBOE users are not huge financial institutions, but public investors, just like you. Over 65% of the Exchange's business comes from them. However, other participants in the financial marketplace also use options to enhance their performance, including: 1. Mutual Funds 2. Pension Plans 3. Hedge Funds 4. Endowments 5. Corporate Treasurers” Options? Terminology Arbitrage Binomial Black-Scholes How big is option trading? Figure 6.1: Total number of option contracts traded in a year in ALL US Exchanges 1973 – 2003 (Source: CBOE) 1400 No. of Contracts in Millions 1200 1000 800 600 400 200 0 1973 1975 1977 1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 Year Options? Terminology Arbitrage Binomial Black-Scholes How big is option trading? Figure 6.2: CBOE average daily trading volume 1973 – 2003 (Source: CBOE) 1.6 No. of Contracts in Millions 1.4 1.2 1 0.8 0.6 0.4 0.2 0 1973 1975 1977 1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 Year Options? Terminology Arbitrage Binomial Black-Scholes How big is option trading? Figure 6.3: CBOE year-end options open-interest dollar amount (in thousands)1973 – 2003 (Source: CBOE) 350000000 Year-End Open Interest (in 300000000 thousand dollars $) 250000000 200000000 150000000 100000000 50000000 0 73 76 79 82 85 88 91 94 97 00 03 19 19 19 19 19 19 19 19 19 20 20 Year Options? Terminology Arbitrage Binomial Black-Scholes Where do we trade options? Trading of standardized options contracts on a national exchange started in 1973 when the Chicago Board Options Exchange (CBOE), the world's first listed options exchange, began listing call options. Options are also traded in US in several smaller exchanges: • New York - the American Stock Exchange (AMEX) - the International Securities Exchange (ISE) • Philadelphia - the Philadelphia Stock Exchange (PHLX) • San Francisco - the Pacific Stock Exchange (PCX) Options? Terminology Arbitrage Binomial Black-Scholes Where do we trade options? Trading of non-standardized options contracts occurs on the Over-the- counter (OTC) market. And it is a bigger market than the exchange-traded market for option trading. • The OTC market is a secondary market that trades securities (stocks or options or other financial assets) which are not traded on an exchange due to various reasons (e.g., an inability to meet listing requirements, special terms in contracts incompatible with exchanges’ standardized terms). • For such securities, broker/dealers negotiate directly with one another over computer networks and by phone, and their activities are monitored by the National Association of Securities Dealers. (Conversations are usually taped) • One advantage of options traded in OTC is that they can be tailored to meet particular needs of a corporate treasurer or fund manager. Options? Terminology Arbitrage Binomial Black-Scholes Standardized VS Non-Standardized Standardized Options • The terms of the option contract is standardized. • Terms include: 1. The exercise price (also called the strike price) 2. The maturity date (also called the expiration date) • For stock options, this is the third Saturday of the month in which the contract expires. 3. Number of shares committed on the underlying stocks • In US, usually 1 option contract – 100 shares of stock 4. Type: American VS European • Stocks options in exchanges are American Non-standardized options also involves these terms, but they can be anything. For example, 1 contract underlies 95 shares instead of 100 shares of stock. Terms being more flexible for non-standardized options and are traded in OTC market are the two distinct features. Options? Terminology Arbitrage Binomial Black-Scholes A screenshot from CBOE • As of Jun 26, 2006 at around 10am ET, GM was selling at $26.97 per share. • An american call option that allows the holder to buy a share of GM on or before the third Saturday of Aug 2006 for a price of $25 has a market price of $3.20. • ONE stock option contract is a contract to buy or sell 100 shares. Thus, you need $320 to buy ONE such call option contract in the market. Options? Terminology Arbitrage Binomial Black-Scholes What is an option contract? • There are 2 basic types of options: CALLs & PUTs • A CALL option gives the holder the right, but not the obligation • To buy an asset • By a certain date • For a certain price • A PUT option gives the holder the right, but not the obligation • To sell an asset • By a certain date • For a certain price • an asset – underlying asset • Certain date – Maturity date/Expiration date • Certain price – strike price/exercise price Options? Terminology Arbitrage Binomial Black-Scholes Bunch of Jargons Option is a DERIVATIVES – since the value of an option depends on the price of its underlying asset, its value is derived. In the Money - exercise of the option would be profitable Call: market price > exercise price Put: exercise price > market price Out of the Money - exercise of the option would not be profitable Call: market price < exercise price Put: exercise price < market price At the Money - exercise price and asset price are equal • Long – buy • Short – sell e.g., Long a put on company x – buy a put contract of company x. Short a call on company y – sell a call contract of company y Options? Terminology Arbitrage Binomial Black-Scholes Bunch of Jargons American VS European Options An American option – allows its holder to exercise the right to purchase (if a call) or sell (if a put) the underlying asset on or before the expiration date. A European option – allows its holder to exercise the option only on the expiration date. Options? Terminology Arbitrage Binomial Black-Scholes Call Option’s payoff • Assume you hold ONE contract of that Calls, i.e., the call contract allows you to buy 100 shares of GM on the third Saturday of August at an exercise price of $25/share. Assume it is European instead of American. What is your payoff on the expiration date if GM at that date is: (a) Selling @ $30 • You will be very happy. To cash in, you do two things simultaneously: • [1] exercise your right, and buy 100 GM at $25. Total amount you use is $2,500. • [2] sell 100 shares of GM at the market price (i.e., $30/share). Total amount you get is $3,000. • Your payoff is $3,000 - $2,500 = $500. (b) Selling @ $19 • You will be very sad. You would not exercise the rights. The contract is thus expired without exercising. • Your payoff is $0!!!! Options? Terminology Arbitrage Binomial Black-Scholes Call Option’s payoff (European) • Exercise Price = $25/share. • If at maturity, market Price = $30 > $25 (You exercise and get profit, the option you hold is said to be “in-the-money” because exercising it would produce profit) • If at maturity, market price = $19 < $25 (You do not exercise, the option you hold is said to be “out-of-the-money” because exercising would be unprofitable) • In general, if you hold a call option contract, you want GM’s stock price to skyrocket. If GM is selling at $100, you will be really happy. • That means, the value of a call option is higher if the underlying asset’s price is higher than the exercise price. • That also means, the value of a call option is zero if the underlying asset’s price is lower than the exercise price. Whether it is $18, $19 or $2.50, it does not matter, the call option will still worth zero. • Try to derive the payoff for the seller of call using this example. Options? Terminology Arbitrage Binomial Black-Scholes Put Option’s payoff • Assume you hold ONE contract of that Puts, i.e., the put contract allows you to sell 100 shares of GM on the third Saturday of August at an exercise price of $22.50/share. (current price of this option = $0.75) Assume it is European instead of American. What is your payoff on expiration date if Intel at that date is: (a) Selling @ $30 • You will be very sad. You would not exercise the rights. The contract is thus expired without exercising. • Your payoff is $0!!! (b) Selling @ $19 • You will be very happy. To cash in, you do two things simultaneously: • [1] you buy 100 shares of GM at $19, total purchase = $1,900 • [2] exercise your right, and sell 100 GM at $22.5. Total amount you get is $2,250. • Your payoff is $2,250 - $1,900 = $350. Options? Terminology Arbitrage Binomial Black-Scholes Put Option’s payoff (European) • Exercise Price = $22.50/share. • If at maturity, market Price = $30 > $22.50 (You do not exercise, and the option you hold is said to be “out-of-the- money” because exercising would be unproductive) • If at maturity, market price = $19 < $22.50 (You exercise, and the option you hold is said to be “in-the-money” because exercising would be profitable) • In general, if you hold a put option contract, you want GM to go broke. If GM is selling at a penny, you will be even happier. • That means, the value of a put option is higher if the underlying asset’s price is lower than the exercise price. • That also means, the value of a put option is zero if the underlying asset’s price is higher than the exercise price. Whether it is $23, $24 or $1000, it does not matter, the put option will still worth zero. • Again, try to derive the payoff of the seller. Options? Terminology Arbitrage Binomial Black-Scholes Some more Jargons If the underlying asset of an option is: (a) A stock – then the option is a stock option (b) An index – the option is an index option (c) A future contract – the option is a futures option (d) Foreign currency – the option is a foreign currency option (e) Interest rate – the option is an interest rate option • ECMC49S will only focus on stock option. But you should know that there are other options trading in the market. You should definitely know them when you do interview with a firm or an i-bank for financial position. You will fail your CFA exam if you don’t know them. You also need to know the differences between options and futures/forwards. Options? Terminology Arbitrage Binomial Black-Scholes Stock options VS stocks Let’s say you hold a option contract for IBM. How does that differ from holding IBM’s stock? Similarities: • IBM’s options are securities, so does IBM’s stocks. • Trading IBM’s options is just like trading stocks, with buyers making bids and sellers making offers. • Can easily trade them, say in an exchange. Differences: • IBM’s options are derivatives, but IBM’s stocks aren’t • IBM’s options will expire, while stocks do not. • There is not a fixed number of options. But there is fixed number of stock shares available at any point in time. • Holding stocks of IBM entitles voting rights, but holding IBM’s option does not • IBM has control over its number of stocks. But it has no control over its number of options. Options? Terminology Arbitrage Binomial Black-Scholes Notations Strike price = X Stock price at present = S0 Stock price at expiration = ST Price of a call option = C Price of a put option = P Risk-free interest rate = Rf Expiration time = T Present time = 0 Time to maturity = T – 0 = T Options? Terminology Arbitrage Binomial Black-Scholes Payoff of Long Call If you buy (long) a call option, what is your payoff at expiration? Payoff to Call Holder at expiration Strike price = X (ST - X) if ST >X Stock price at present = S0 0 if ST < X Profit to Call Holder at expiration Stock price at expiration = ST Payoff – Purchase Price (time adjusted) Price of a call option = C $ Price of a put option = P Payoff Risk-free interest rate = Rf Expiration time = T Profit Present time = 0 ST Time to maturity = T – 0 = T Purchase price x adjusted by time Options? Terminology Arbitrage Binomial Black-Scholes Payoff of Short Call If you sell (short) a call option, what is your payoff at expiration? Payoff to Call seller at expiration Strike price = X -(ST - X) if ST >X Stock price at present = S0 0 if ST < X Profit to Call seller at expiration Stock price at expiration = ST Payoff + Selling Price (time adjusted) Price of a call option = C $ Price of a put option = P Risk-free interest rate = Rf Expiration time = T Selling price adjusted by time Present time = 0 ST Time to maturity = T – 0 = T x Profit Payoff Options? Terminology Arbitrage Binomial Black-Scholes Payoff of Long Put If you buy (long) a put option, what is your payoff at expiration? Payoff to Put Holder at expiration Strike price = X 0 if ST >X Stock price at present = S0 (X – ST) if ST < X Profit to Put Holder at expiration Stock price at expiration = ST Payoff - Purchasing Price (time adjusted) Price of a call option = C $ Price of a put option = P Risk-free interest rate = Rf Expiration time = T Payoff Present time = 0 ST Time to maturity = T – 0 = T x Purchasing price adjusted by time Profit Options? Terminology Arbitrage Binomial Black-Scholes Payoff of Short Put If you sell (short) a put option, what is your payoff at expiration? Payoff to Put seller at expiration Strike price = X 0 if ST >X Stock price at present = S0 -(X – ST) if ST < X Profit to Put seller at expiration Stock price at expiration = ST Payoff + Selling Price (time adjusted) Price of a call option = C $ Price of a put option = P Risk-free interest rate = Rf Profit Expiration time = T Selling price adjusted by time Payoff Present time = 0 ST Time to maturity = T – 0 = T x Options? Terminology Arbitrage Binomial Black-Scholes Payoff of Long Put & Short Call If you buy (long) a put option and sell (short) a call, assuming their exercise prices are the same, what is your payoff at expiration? Payoff to Call seller at expiration Payoff to Put Holder at expiration -(ST - X) if ST >X 0 if ST >X + 0 if ST < X (X – ST) if ST < X $ $ Payoff ST ST x x Payoff Options? Terminology Arbitrage Binomial Black-Scholes Payoff of Long Put & Short Call If you buy (long) a put option and sell (short) a call, assuming their exercise prices are the same, what is your payoff at expiration? Payoff to Call seller at expiration Payoff to Put Holder at expiration -(ST - X) if ST >X 0 if ST >X = -(ST – X) + 0 if ST < X (X – ST) if ST < X = (X - ST) $ ST x Payoff Options? Terminology Arbitrage Binomial Black-Scholes Long Put & Short Call & Long stock If you buy (long) a put option and sell (short) a call, as well as holding 1 stock. If the two options have the same exercise price and the same expiration date, what is your payoff at expiration? -(ST – X) if ST >X X if ST >X + Stock price (ST) at time T = X if ST < X (X - ST) if ST < X Payoff (long the stock) $ Total Payoff (risk-free) x ST x Payoff (short call & long put) Options? Terminology Arbitrage Binomial Black-Scholes Put-Call Parity What we just do introduces a very important concept for pricing options. Holding a portfolio with (a) 1 stock (which costs S0) (b) selling one call (which earns C) (c) buying one put (which costs P) Total value of constructing portfolio = S0 + P - C Payoff (long the stock) The payoff at maturity/expiration $ is always X !!! Total Payoff (risk-free) x ST x Payoff (short call & long put) Options? Terminology Arbitrage Binomial Black-Scholes Put-Call Parity Total value of constructing portfolio = S0 + P – C Get back X at maturity for sure. Thus X discounted at the risk-free rate should equal to the portfolio value now. Thus, S0 + P – C = X/(1+Rf)T In words: “Current stock price plus price of a corresponding put option at exercise price X minus the price of a corresponding call option with exercise price X is equal to the present value of X at maturity discounted at risk-free rate. Payoff (long the stock) $ Total Payoff (risk-free) x ST x Payoff (short call & long put) Options? Terminology Arbitrage Binomial Black-Scholes

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