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Options: the basics

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Content What is option? Terminology No arbitrage Pricing Options –The Binomial Option Pricing Model –The Black-Scholes Model

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Why do we study options? Options?TerminologyArbitrageBinomialBlack-Scholes The underlying concern: Happiness = U(C now,C future ) We are happier with more C future, but we are worried about the fluctuation of C future. 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.

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Who trade options? Options?TerminologyArbitrageBinomialBlack-Scholes 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”

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How big is option trading? Options?TerminologyArbitrageBinomialBlack-Scholes Figure 6.1: Total number of option contracts traded in a year in all exchange 1973 – 1999 (Source: CBOE)

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How big is option trading? Options?TerminologyArbitrageBinomialBlack-Scholes Figure 6.1: CBOE average daily trading volume 1973 – 1999 (Source: CBOE)

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How big is option trading? Options?TerminologyArbitrageBinomialBlack-Scholes Figure 6.2: CBOE year-end options open-interest dollar amount (in thousands)1973 – 1999 (Source: CBOE)

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Where do we trade options? Options?TerminologyArbitrageBinomialBlack-Scholes 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 also trade now on several smaller exchanges, including New York - the American Stock Exchange (AMEX)AMEX - the International Securities Exchange (ISE)ISE Philadelphia - the Philadelphia Stock Exchange (PHLX)PHLX San Francisco - the Pacific Stock Exchange (PCX)PCX

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Where do we trade options? Options?TerminologyArbitrageBinomialBlack-Scholes Trading of non-standardized (tailor-made) options contracts occurs on the Over-the-counter (OTC) market. And it is in fact bigger 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). 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. One advantage of options traded in OTC is that they can be tailored to meet particular needs of a corporate treasurer or fund manager.

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Standardized VS Non-Standardized Options?TerminologyArbitrageBinomialBlack-Scholes 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, or the third Thursday of the month if the third Friday is a holiday. 3.Number of shares committed on the underlying stocks In US, usually 1 contract – 100 shares of stock 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.

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Option Quotation (standardized) Options?TerminologyArbitrageBinomialBlack-Scholes You will see how standardized options trading would be with a quick look of option quotation system. Option quotes follow a pattern that enables you to easily construct and interpret symbols once that formula is understood. The basic parts of an option symbol are: Root symbol + Month code + Strike price code A root symbol is not the same as the ticker symbol. Please refer to the option chain for that ticker to find the corresponding root. In conjunction with the option root symbol, you can utilize the tables below to assist you in creating or deciphering options symbols.

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Option Quotation (standardized) Options?TerminologyArbitrageBinomialBlack-Scholes Expiration Month codes: Strike Price codes: Exercise: quoting price for a call option on Microsoft (MSQ) at $27.5 expiring December 2006. (MSQ LY)

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What is an option contract? Options?TerminologyArbitrageBinomialBlack-Scholes 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

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What is an option contract? Options?TerminologyArbitrageBinomialBlack-Scholes For example, as of Nov 1, 2005 at around 5pm, Intel was selling at $22.65 per share. A call option that allows the holder to buy a share of Intel at the third Saturday of November 2005 for a price of $20 has a market price of $2.80 ONE stock option contract is a contract to buy or sell 100 shares. Thus, you need $280 to buy ONE such call option contract in the market.

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Call Option’s payoff Options?TerminologyArbitrageBinomialBlack-Scholes Assuming you hold ONE contract of that Calls, i.e., the call contract allows you to buy 100 shares of Intel on the third Saturday of December at an exercise price of $20/share. What is your payoff if Intel at that date is: (a) Selling @ $25 You will be very happy. To cash in, you do two things simultaneously: [1] exercise your right, and buy 100 Intel at $20. Total amount you use is $2,000. [2] sell 100 shares of Intel at the market price (i.e., $25/share). Total amount you get is $2,500. Your payoff is $2,500 - $2,000 = $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.

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Call Option’s payoff Options?TerminologyArbitrageBinomialBlack-Scholes Exercise Price = $20/share. If at maturity, market Price = $25 > $20 (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 < $20 (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 Intel’s stock price to skyrocket. If Intel is selling at $100, you will be really happier. 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.

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Put Option’s payoff Options?TerminologyArbitrageBinomialBlack-Scholes Assuming you hold ONE contract of that Puts, i.e., the put contract allows you to sell 100 shares of Intel on the third Saturday of December at an exercise price of $22.50/share. (current price of this option = 0.65) What is your payoff if Intel at that date is: (a) Selling @ $25 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 Intel at $19, total purchase = $1,900 [2] exercise your right, and sell 100 Intel at $22.5. Total amount you get is $2,250. Your payoff is $2,250 - $1,900 = $350.

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Put Option’s payoff Options?TerminologyArbitrageBinomialBlack-Scholes Exercise Price = $22.50/share. If at maturity, market Price = $25 > $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 Intel to go broke. If Intel 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.

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Bunch of Jargons Options?TerminologyArbitrageBinomialBlack-Scholes Option is a derivative – 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

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Bunch of Jargons Options?TerminologyArbitrageBinomialBlack-Scholes 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.

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Bunch of Jargons Options?TerminologyArbitrageBinomialBlack-Scholes 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 ECMC49F 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.

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Stock options VS stocks Options?TerminologyArbitrageBinomialBlack-Scholes Let’s say you hold a option contract for GE. How does that differ from holding GE’s stock? Similarities: GE’s options are securities, so does GE’s stocks. Trading GE’s options is just like trading stocks, with buyers making bids and sellers making offers. Can easily trade them, say in an exchange. Differences: GE’s options are derivatives, but GE’s stocks aren’t GE’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 GE entitles voting rights, but holding GE’s option does not GE has control over its number of stocks. But it has no control over its number of options.

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Notations Options?TerminologyArbitrageBinomialBlack-Scholes Strike price = X Stock price at present = S 0 Stock price at expiration = S T Price of a call option = C Price of a put option = P Risk-free interest rate = R f Expiration time = T Present time = 0 Time to maturity = T – 0 = T

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Payoff of Long Call Options?TerminologyArbitrageBinomialBlack-Scholes Strike price = X Stock price at present = S 0 Stock price at expiration = S T Price of a call option = C Price of a put option = P Risk-free interest rate = R f Expiration time = T Present time = 0 Time to maturity = T – 0 = T If you buy (long) a call option, what is your payoff at expiration? Payoff to Call Holder at expiration (S T - X) if S T >X 0if S T < X Profit to Call Holder at expiration Payoff – Purchase Price STST x Payoff Profit $ Purchase price

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Payoff of Short Call Options?TerminologyArbitrageBinomialBlack-Scholes Strike price = X Stock price at present = S 0 Stock price at expiration = S T Price of a call option = C Price of a put option = P Risk-free interest rate = R f Expiration time = T Present time = 0 Time to maturity = T – 0 = T If you sell (short) a call option, what is your payoff at expiration? Payoff to Call seller at expiration -(S T - X)if S T >X 0if S T < X Profit to Call seller at expiration Payoff + Selling Price STST x Payoff Profit $ Selling price

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Payoff of Long Put Options?TerminologyArbitrageBinomialBlack-Scholes Strike price = X Stock price at present = S 0 Stock price at expiration = S T Price of a call option = C Price of a put option = P Risk-free interest rate = R f Expiration time = T Present time = 0 Time to maturity = T – 0 = T If you buy (long) a put option, what is your payoff at expiration? Payoff to Put Holder at expiration 0if S T >X (X – S T )if S T < X Profit to Put Holder at expiration Payoff - Purchasing Price STST x Payoff Profit $ Purchasing price

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Payoff of Short Put Options?TerminologyArbitrageBinomialBlack-Scholes Strike price = X Stock price at present = S 0 Stock price at expiration = S T Price of a call option = C Price of a put option = P Risk-free interest rate = R f Expiration time = T Present time = 0 Time to maturity = T – 0 = T If you sell (short) a put option, what is your payoff at expiration? Payoff to Put seller at expiration 0 if S T >X -(X – S T ) if S T < X Profit to Put seller at expiration Payoff + Selling Price STST x Payoff Profit $ Selling price

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Payoff of Long Put & Short Call Options?TerminologyArbitrageBinomialBlack-Scholes 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 -(S T - X)if S T >X 0if S T < X + Payoff to Put Holder at expiration 0if S T >X (X – S T )if S T < X STST x Payoff $ STST x $

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Payoff of Long Put & Short Call Options?TerminologyArbitrageBinomialBlack-Scholes 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 -(S T - X)if S T >X 0if S T < X + Payoff to Put Holder at expiration 0if S T >X (X – S T )if S T < X STST x $ Payoff = -(S T – X) = (X - S T )

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Long Put & Short Call & Long stock Options?TerminologyArbitrageBinomialBlack-Scholes If you buy (long) a put option and sell (short) a call, as well as holding 1 stock. Assuming the options’ exercise prices are the same, what is your payoff at expiration? if S T >X if S T < X + Stock price (S T ) at time T = STST x $ Payoff (short call & long put) -(S T – X) (X - S T ) X if S T >X Xif S T < X Payoff (long the stock) Total Payoff (risk-free) x

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Put-Call Parity Options?TerminologyArbitrageBinomialBlack-Scholes What we just do introduces a very important concept for pricing options. Holding a portfolio with (a) 1 stock (which costs S 0 ) (b) selling one call (which earns C) (c) buying one put (which costs P) Total value of constructing portfolio = S 0 + P - C STST x $ Payoff (short call & long put) Payoff (long the stock) Total Payoff (risk-free) x The payoff at maturity/expiration is always X !!!

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Put-Call Parity Options?TerminologyArbitrageBinomialBlack-Scholes Total value of constructing portfolio = S 0 + 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, S 0 + P – C = X/(1+R f ) 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. STST x $ Payoff (short call & long put) Payoff (long the stock) Total Payoff (risk-free) x

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Put-Call Parity S 0 + P – C = X/(1+R f ) T Let’s do an exercise. What is the risk-free interest rate? From the data below, S 0 = 22.65, 05 DEC 22.50 Call sells at 0.90, C = 0.90, X = 22.50 05 DEC 22.50 Put sells at 0.65, P = 0.65, X = 22.50 Time to maturity is roughly 6 weeks. Thus, T = 6/52 We have 22.65 + 0.65 – 0.90 = 22.50/(1+R f ) 6/52 Options?TerminologyArbitrageBinomialBlack-Scholes

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Readings for next class on pricing options Forget about the textbook. Read online free sources and the lecture notes. Many of the course materials are drawn from CBOE learning center Online tutorials CBOE 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 There are some other online sources you may find useful. For example, optionscentral.com.optionscentral.com Check out the websites for US exchanges.

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