Options: the basics. Readings for Options The textbook involves intensive mathematics. I therefore suggest you use online free sources and the lecture

Options: the basics

Readings for Options

• The textbook involves intensive mathematics. I therefore suggest you use online free sources and the lecture notes as your main reference

• Main lecture materials are drawn from (compulsory): 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.

http://www.cboe.com/LearnCenter/Tutorials.aspx

http://www.optionscentral.com/

http://www.optionsclearing.com/

Content

• What is option?

• Terminology

• No arbitrage

• Pricing Options– The Binomial Option Pricing Model– The Black-Scholes Model

Why do we study options?

Options? Terminology Arbitrage Binomial Black-Scholes

Economics: It is all about maximizing 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 vary our level of happiness.

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 Funds2. Pension Plans3. Hedge Funds4. Endowments5. Corporate Treasurers”

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)

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Figure 6.2: CBOE average daily trading volume 1973 – 2003 (Source: CBOE)

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Figure 6.3: CBOE year-end options open-interest dollar amount (in thousands)1973 – 2003 (Source: CBOE)

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350000000

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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, including:

• 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)

http://www.amex.com/

http://www.iseoptions.com/

http://www.phlx.com/

http://www.pacificex.com/

Where do we trade options?


Trading of non-standardized options contracts occurs on the Over-the-counter (OTC) market. And it is an even bigger market than the exchange-traded market for option trading.

• The OTC market is a secondary market that trades securities (stocks, options, and 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, brokers/dealers negotiate directly with one another over computer networks or by phone. Their activities are monitored by the National Association of Securities Dealers. (Conversations over the phone 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.

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.

A screenshot from CBOE


• As of Oct 30, 2006 at around 12pm ET, Verizon Commuications (VZ) was selling at $37.28 per share.

• An american call option that allows the holder to buy 100 shares of VZ on or before the third Saturday of Nov 2006 for a price of $37.50 has a market price of $240 (=$2.40x100).

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

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 profitableCall: market price > exercise pricePut: exercise price > market price

Out of the Money - exercise of the option would not be profitableCall: market price < exercise pricePut: 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/write a call contract of company

y

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.

Call Option’s payoff


• Assume you hold ONE contract of that Calls, i.e., the call contract allows you to buy 100 shares of VZ on the third Saturday of November at an exercise price of $37.50/share. Assume it is European instead of American.

What is your payoff on the expiration date if VZ at that date is:

(a) Selling @ $40

• You will be very happy. To cash in, you do two things simultaneously:

• [1] exercise your right, and buy 100 VZ at $37.50. Total amount you use is $3,750.

• [2] sell 100 shares of VZ at the market price (i.e., $40/share). Total amount you get is $4,000.

• Your payoff is $4,000 - $3,750 = $250.

(b) Selling @ $30

• You will be very sad. You would not exercise the rights. The contract is thus expired without exercising.

• Your payoff is $0!!!!

Call Option’s payoff (European)


• Exercise Price = $37.50/share.

• If at maturity, market Price = $40 > $37.50 (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 = $20 < $27.50(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 VZ’s stock price to skyrocket. The higher is the share price of VZ, the happier you are. Because the value of the call option is higher if the underlying asset’s price is higher than the exercise price.

• In contrast, the value of a call option is zero if the underlying asset’s price is lower than the exercise price. Whether it is $30, $29 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.

Put Option’s payoff


• Assume you hold ONE contract of that Puts, i.e., the put contract allows you to sell 100 shares of VZ on the third Saturday of November at an exercise price of $35/share. (current price of this option = $0.05) Assume it is European instead of American.

What is your payoff on expiration date if Intel at that date is:

(a) Selling @ $40

• You will be very sad. You would not exercise the rights. The contract is thus expired without exercising.

• Your payoff is $0!!!

(b) Selling @ $30

• You will be very happy. To cash in, you do two things simultaneously:

• [1] you buy 100 shares of VZ at $30, total purchase = $3,000

• [2] exercise your right, and sell 100 VZ at $35. Total amount you get is $3, 500.

• Your payoff is $3,500 - $3,000 = $500.

Put Option’s payoff (European)


• Exercise Price = $35/share.

• If at maturity, market Price = $40 > $35 (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 = $30 < $35(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 VZ to go broke. If VZ is selling at a penny, you will be very rich.

• That means, the value of a put option is higher if the underlying asset’s price is lower than the exercise price.

• In contrast, the value of a put option is zero if the underlying asset’s price is higher than the exercise price. Whether it is $40, $39 or $1000, it does not matter, the put option will still worth zero.

• Again, try to derive the payoff of the seller.

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

• ECMC49 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 are interviewed by 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, futures/forwards and warrants.

Stock options VS stocks


Let’s say you hold a option contract for VZ. How does that differ from directly holding Verizon Communications’ stocks?

Similarities:

• VZ’s options are securities, so does VZ’s stocks.

• Trading VZ’s options is just like trading stocks, with buyers making bids and sellers making offers.

• Can easily trade them, say in an exchange.

Differences:

• VZ’s options are derivatives, but VZ’s stocks aren’t

• VZ’s options will expire, while stocks do not.

• There is no fixed number of options. But there is fixed number of shares of VZ’s stocks available at any point in time.

• Holding VZ’s common stocks entitles voting rights, but holding VZ’s option does not

• VZ has control over its number of stocks. But it has no control over its number of options.

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

Payoff of Long Call


Strike price = X






Expiration time = T

Present time = 0


If you buy (long) a call option, what is your payoff at expiration?

Payoff to Call Holder at expiration

(ST - X) if ST >X

0 if ST < XProfit to Call Holder at expiration

Payoff – Purchase Price (time adjusted)

STx

Payoff

Profit

$

Purchase price adjusted by time

Payoff of Short Call


Strike price = X






Expiration time = T

Present time = 0


If you sell (short) a call option, what is your payoff at expiration?

Payoff to Call seller at expiration

-(ST - X) if ST >X

0 if ST < XProfit to Call seller at expiration

Payoff + Selling Price (time adjusted)

STx

Payoff

Profit

$

Selling price adjusted by time

Payoff of Long Put


Strike price = X






Expiration time = T

Present time = 0


If you buy (long) a put option, what is your payoff at expiration?

Payoff to Put Holder at expiration

0 if ST >X

(X – ST) if ST < XProfit to Put Holder at expiration

Payoff - Purchasing Price (time adjusted)

STx

Payoff

Profit

$

Purchasing price adjusted by time

Payoff of Short Put


Strike price = X






Expiration time = T

Present time = 0


If you sell (short) a put option, what is your payoff at expiration?

Payoff to Put seller at expiration

0 if ST >X

-(X – ST) if ST < XProfit to Put seller at expiration

Payoff + Selling Price (time adjusted)

STx

Payoff

Profit

$

Selling price adjusted by time

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?


-(ST - X) if ST >X

0 if ST < X+


0 if ST >X

(X – ST) if ST < X

STx

Payoff

$

STx

Payoff

$

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?


-(ST - X) if ST >X

0 if ST < X+


0 if ST >X

(X – ST) if ST < X

STx

$

Payoff

= -(ST – X)= (X - ST)

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?

if ST >X

if ST < X+ Stock price (ST) at time T =

STx

$

Payoff (short call & long put)

-(ST – X) (X - ST)

X if ST >XX if ST < X

Payoff (long the stock)

Total Payoff (risk-free)x

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

STx

$




The payoff at maturity/expiration is always X !!!

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.

STx

$




Documents

Options: the basics. Readings for Options The textbook involves intensive mathematics. I therefore suggest you use online free sources and the lecture