Transcript Options - University of Toronto

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.
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 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.
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.
2.
3.
4.
5.
Options?
Mutual Funds
Pension Plans
Hedge Funds
Endowments
Corporate Treasurers”
Terminology
Arbitrage
Binomial
Black-Scholes
How big is option trading?
1400
1200
1000
800
600
400
200
2003
2001
1999
1997
1995
1993
1991
1989
1987
1985
1983
1981
1979
1977
1975
0
1973
No. of Contracts in Millions
Figure 6.1: Total number of option contracts traded in a year in ALL US
Exchanges 1973 – 2003 (Source: CBOE)
Year
Options?
Terminology
Arbitrage
Binomial
Black-Scholes
How big is option trading?
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
2003
2001
1999
1997
1995
1993
1991
1989
1987
1985
1983
1981
1979
1977
1975
0
1973
No. of Contracts in Millions
Figure 6.2: CBOE average daily trading volume 1973 – 2003 (Source: CBOE)
Year
Options?
Terminology
Arbitrage
Binomial
Black-Scholes
How big is option trading?
Year-End Open Interest (in
thousand dollars $)
Figure 6.3: CBOE year-end options open-interest dollar amount (in
thousands)1973 – 2003 (Source: CBOE)
350000000
300000000
250000000
200000000
150000000
100000000
50000000
0
3
6
9
2
5
8
1
4
7
0
3
7
7
7
8
8
8
9
9
9
0
0
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, including:
•
New York - the American Stock Exchange (AMEX)
- the International Securities Exchange (ISE)
Options?
•
Philadelphia - the Philadelphia Stock Exchange (PHLX)
•
San Francisco - the Pacific Stock Exchange (PCX)
Terminology
Arbitrage
Binomial
Black-Scholes
Where do we trade options?
Trading of non-standardized options contracts occurs on the Over-thecounter (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.
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 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).
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
•
•
e.g.,
Options?
Long – buy
Short – sell
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
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 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
Options?
•
You will be very sad. You would not exercise the rights. The contract
is thus expired without exercising.
•
Your payoff is $0!!!!
Terminology
Arbitrage
Binomial
Black-Scholes
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.
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 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
•
•
Options?
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.
Terminology
Arbitrage
Binomial
Black-Scholes
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-themoney” 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.
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
•
Options?
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.
Terminology
Arbitrage
Binomial
Black-Scholes
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.
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?
Strike price = X
Stock price at present = S0
Stock price at expiration = ST
Price of a call option = C
Payoff to Call Holder at expiration
(ST - X) if ST >X
0
if ST < X
Profit to Call Holder at expiration
Payoff – Purchase Price (time adjusted)
$
Payoff
Price of a put option = P
Risk-free interest rate = Rf
Profit
Expiration time = T
Present time = 0
Time to maturity = T – 0 = T
Options?
Terminology
Purchase price
adjusted by time
Arbitrage
ST
x
Binomial
Black-Scholes
Payoff of Short Call
If you sell (short) a call option, what is your payoff at expiration?
Strike price = X
Stock price at present = S0
Stock price at expiration = ST
Price of a call option = C
Payoff to Call seller at expiration
-(ST - X) if ST >X
0 if ST < X
Profit to Call seller at expiration
Payoff + Selling Price (time adjusted)
$
Price of a put option = P
Risk-free interest rate = Rf
Expiration time = T
Selling price
adjusted by time
Present time = 0
ST
x
Time to maturity = T – 0 = T
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?
Strike price = X
Stock price at present = S0
Stock price at expiration = ST
Price of a call option = C
Payoff to Put Holder at expiration
0 if ST >X
(X – ST) if ST < X
Profit to Put Holder at expiration
Payoff - Purchasing Price (time adjusted)
$
Price of a put option = P
Risk-free interest rate = Rf
Expiration time = T
Payoff
Present time = 0
x
Time to maturity = T – 0 = T
ST
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?
Strike price = X
Stock price at present = S0
Stock price at expiration = ST
Price of a call option = C
Payoff to Put seller at expiration
0
if ST >X
-(X – ST) if ST < X
Profit to Put seller at expiration
Payoff + Selling Price (time adjusted)
$
Price of a put option = P
Risk-free interest rate = Rf
Profit
Expiration time = T
Selling price
adjusted by time
Present time = 0
Options?
Terminology
ST
x
Time to maturity = T – 0 = T
Arbitrage
Payoff
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
x
ST
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 - ST) if ST < X
+
Stock price (ST) at time T =
X
X
if ST >X
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.
S0 + P – C = X/(1+Rf)T
Thus,
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