Download Fin 603 Week 11

Fin 603
Week 11
Options
Fall 2005
Professor Ross Miller
University at Albany
School of Business
Copyright 2005 by Ross M. Miller. All rights reserved
What is a Derivative Security?
 A security whose value is “derived” from
something else, using a security, a collection of
securities, or anything that has a price or
probability attached to it
 Derivatives are primarily used to “shape risk”
 The three main kinds of derivative securities:
• Futures contracts
• Swaps
• Options
Professor Ross Miller • Fall 2005
1
A Few More Words on Futures
 Stock index futures
• Pricing works the same as gold futures—price
goes up by the full carryin cost (determined by
the risk-free rate of return)
 Futures, in general
• Typical energy and agricultural futures do not
rise in line with the full carrying cost
• Seasonal and temporary factors make their
prices fall more in line with the expectations of
the market
Professor Ross Miller • Fall 2005
2
A Full Carrying Charge Market
 A full carrying charge market occurs when the
futures price reflects the cost of storing and
financing the commodity until the delivery month
 The futures price is equal to the current spot
price plus the carrying charge:
F  St  C
Professor Ross Miller • Fall 2005
E=mc2
3
Contango and Backwardation
 Basis is the difference between the future price
of a commodity and the current cash price
• Normally, the futures price exceeds the cash
price (contango market)
• The futures price may be less than the cash
price (backwardation or inverted market)
Professor Ross Miller • Fall 2005
E=mc2
4
Normal Backwardation
(John Maynard Keynes)
 Locking in a future price that is acceptable
eliminates price risk for the hedger
 The speculator must be rewarded for taking the
risk that the hedger was unwilling to bear
 Thus, at delivery, the cash price will likely be
somewhat higher than the price predicated by
the futures market
Professor Ross Miller • Fall 2005
E=mc2
5
Stock Index Futures in March 2005
Fifteen-minute delayed quotes from CME
Professor Ross Miller • Fall 2005
6
Heating Oil Futures in March 2005
Professor Ross Miller • Fall 2005
7
A Few More Words on Swaps
 Interest rates swaps are just a bundle of forward
rate agreements (FRAs)
• FRAs are priced off the yield curve
• Swap prices (in percentage yields) are
approximately the average of the forward rates
• The swap price represents the fixed rate that is
an even trade-off for the bundle of FRAs
 Many other types of instruments called “swaps”
exist, but most of them are just complicated
forward contracts
Professor Ross Miller • Fall 2005
8
Some Option Basics
 Options provide the opportunity to buy (call
option) or sell (put option) something at a given
exercise price on or before a given expiration
date
 Options that restrict exercise to the expiration
date are known as European options. Such
options are rare in the U.S., and so the standard
kind of option is known as an American option
Professor Ross Miller • Fall 2005
9
The Benefits of Holding Call Options
 Insurance: Protection against major price
declines
 Leverage: Increases returns relative to holding
stock in an up market
 The Catch: You are paying for both the
insurance and the implicit “rental” of the stock
Professor Ross Miller • Fall 2005
E=mc2
10
Options are Everywhere
 We have already seen the prepayment option in
mortgages
 The ability of a company to default on its debt can
be viewed as an option
 The ability of equity holders to walk away from
their company and cede control to debt holders is
an option
 Even limit orders and market maker’s quote
provide options
Professor Ross Miller • Fall 2005
E=mc2
11
Where Traded Options Come From
 Unlike stocks and bonds (and like futures and
swaps), there is no fixed number of put or call
options
•
•
•
•
The number in existence changes every day
This number is called the open interest
Similarly, futures contracts have open interest
Stocks, on the other hand, have shares
outstanding, which the issuer can only change
by buying or floating shares
Professor Ross Miller • Fall 2005
E=mc2
12
Opening and Closing Transactions
 The first trade someone makes in a particular
option is an opening transaction for that person
 When the individual subsequently closes that
position out with a second trade, this latter trade
is a closing transaction
Professor Ross Miller • Fall 2005
E=mc2
13
Opening and Closing Transactions (cont’d)
 When someone buys an option as an opening
transaction, the owner of an option will ultimately
do one of three things with it:
• Sell it to someone else
• Let it expire
• Exercise it
Professor Ross Miller • Fall 2005
E=mc2
14
Opening and Closing Transactions (cont’d)
 When someone sells an option as an opening
transaction, this is called writing the option
 No matter what the owner of an option does, the
writer of the option keeps the option premium
that he or she received when it was sold
Professor Ross Miller • Fall 2005
E=mc2
15
Role of the Options Clearing Corporation
(OCC)
 The Options Clearing Corporation (OCC)
contributes substantially to the smooth operation
of the options market
 The OCC positions itself between every buyer
and seller and acts as a guarantor of all option
trades
 The OCC sets minimum capital requirements
and provides for the efficient transfer of funds
among members as gains or losses occur
Professor Ross Miller • Fall 2005
16
Exchanges
 Major options exchanges in the U.S.:
•
•
•
•
•
Chicago Board Options Exchange (CBOE)
American Stock Exchange (AMEX)
Philadelphia Stock Exchange (Philly)
Pacific Stock Exchange (PSE)
International Securities Exchange (ISE)
 These exchanges are equal owners of the OCC
 Options are traded on exchanges in most
countries with organized securities markets
Professor Ross Miller • Fall 2005
17
Over-the-Counter (OTC)Options
 With an over-the-counter option:
• Institutions enter into “private” option
arrangements with brokerage firms or other
dealers
• The striking price, life of the option, and premium
are negotiated between the parties involved
 Over-the-counter or OTC options are subject to
counterparty risk and are generally not fungible
 OTC options tend to be cost significantly more
than their “theoretical value,” while traded
options trade close to that value
Professor Ross Miller • Fall 2005
18
Some Not-So-Exotic “Exotic” Options
 Asian or average price option
• Payoff based on the average price of the
underlying over the life (or some part of the life)
of the option
 Barrier option
• Option that is either activated (an “in” option) or
cancelled (an “out” option) when the underlying
prespecified price level is reached
 Binary option
• Option that pays out zero or a fixed amount
Professor Ross Miller • Fall 2005
19
ClickOptions (Blurring the Boundary Between
Options and Gambling)
Professor Ross Miller • Fall 2005
20
Intrinsic Value and Time Value
 Intrinsic value is the amount that an option is
immediately worth given the relation between the
option striking price and the current stock price
• For a call option, intrinsic value =
stock price – striking price
• For a put option, intrinsic value =
striking price – stock price
• Intrinsic value cannot be < zero
Professor Ross Miller • Fall 2005
E=mc2
21
Intrinsic Value and Time Value (cont’d)
 An option with no intrinsic value is out-of-themoney
 An option whose striking price is exactly equal to
the price of the underlying security is at-themoney
 Options that are “almost” at-the-money are nearthe-money
Professor Ross Miller • Fall 2005
E=mc2
22
Intrinsic Value and Time Value (cont’d)
 Time value is equal to the premium minus the
intrinsic value
 As an option moves closer to expiration, its time
value decreases (time value decay), making it a
wasting asset
Professor Ross Miller • Fall 2005
E=mc2
23
Profit and Loss Diagrams
 Vertical axis reflects profits or losses on the
expiration day resulting from a particular strategy
 Horizontal axis reflects the stock price on the
expiration day
 Bends in the diagram occurs at the striking
prices
 By convention, diagrams ignore the effect of
transactions costs, including commissions
Professor Ross Miller • Fall 2005
E=mc2
24
Going Long a Call Option
 Example: buy a Microsoft October 25 call for
$3.70
• Maximum loss is $3.70
• Profit potential is unlimited
• Breakeven at expiration is a stock price of
$28.70
Professor Ross Miller • Fall 2005
E=mc2
25
Going Long a Call Option (cont’d)
Breakeven = $28.70
0
10
15
20
25
30
Maximum
loss = $3.70
Professor Ross Miller • Fall 2005
E=mc2
26
A Problem with the Standard Payoff Diagram
 You pay for the option now
 You get the option payoff (if any) in the future
 In looking at the payoff diagram, you should use
the future value of the option
 This future value is difficult to compute, in
particular, you cannot use the risk-free rate to
discount the option
 See my first Financial Engineering News
“Capital Notions” column for details
Professor Ross Miller • Fall 2005
E=mc2
27
Writing or Going Short a Call Option
 Ignoring commissions, the options market is a
zero sum game
• Aggregate gains and losses will always net
to zero
• The most an option writer can make is the
option premium
 Writing a call without owning the underlying
shares is called writing a naked (uncovered) call
Professor Ross Miller • Fall 2005
E=mc2
28
A Key Observation
 If an option expires out of the money (below the
exercise price in the case of a call option), it
does not matter how far out of the money it
expires
 If one compares options on two otherwise
identical stock with different volatility, the option
on the more volatile stock is worth more
• The more volatility stock has a greater expected
upside
• It also has a greater expected downside,
but that does not matter
Professor Ross Miller • Fall 2005
E=mc2
29
Writing a Call Option (cont’d)
Breakeven = $28.70
Maximum
Profit = $3.70
0
Professor Ross Miller • Fall 2005
10
15
20
25
30
E=mc2
30
Going Long a Put Option
 Example: buy a Microsoft April 25 put for $1.10
• Maximum loss is $1.10
• Maximum profit is $23.90
• Breakeven at expiration is a stock price of
$23.90
Professor Ross Miller • Fall 2005
E=mc2
31
Buying a Put Option (cont’d)
$23.90
Breakeven = $23.90
0
10
15
20
25
30
$1.10
Professor Ross Miller • Fall 2005
E=mc2
32
Some Nov 2005 Google Call Options
From 11/15/2005 with GOOG at 392.00
Strike
Symbol
Last
Chg
Bid
Ask
370.00 GGDKN.X 22.20 – 5.10 22.10 22.40
Vol Open Int
561
5,923
380.00 GOPKP.X 12.90 – 4.70 12.70 13.00 2,369 10,638
390.00 GOPKR.X
5.40 – 3.70
5.40
5.60 4,075 17,460
400.00 GOPKT.X
1.70 – 2.10
1.60
1.75 5,959 19,564
Professor Ross Miller • Fall 2005
33
Some Nov 2005 Google Put Options
From 11/15/2005 with GOOG at 392.00
Strike
Symbol
Last
Bid
Ask
Vol Open Int
370.00 GGDWN.X 0.20 – 0.10
0.20
0.30
315
380.00 GOPWP.X
0.80 + 0.15
0.80
0.90 1,482 17,939
390.00 GOPWR.X 3.50 + 1.35
3.40
3.50 4,235 17,409
400.00 GOPWT.X
9.60
9.90 1,276
Professor Ross Miller • Fall 2005
Chg
9.60 + 3.10
9,150
5,382
34
Writing (or Going Short) a Put Option
 The put option writer has the obligation to buy if
the put is exercised by the holder
Professor Ross Miller • Fall 2005
E=mc2
35
Writing a Put Option (cont’d)
Breakeven = $23.90
$1.10
0
10
15
20
25
30
$23.90
Professor Ross Miller • Fall 2005
E=mc2
36
A Note on Margin Requirements
 A margin requirement is analogous to posting
collateral and can be satisfied by a deposit of
cash or other securities into your brokerage
account
 The margin system is to reduce the likelihood
that option writers will be unable to fulfill their
obligations
Professor Ross Miller • Fall 2005
E=mc2
37
Protective Puts
 A protective put is a descriptive term given to a
long stock position combined with a long put
position
 Investors who use protective puts may anticipate
a decline in the value of an investment but
cannot or may not want to sell
Professor Ross Miller • Fall 2005
E=mc2
38
Microsoft Example
Assume you purchased Microsoft for $28.51
Profit or loss ($)
0
28.51
Stock price at
option expiration
28.51
Professor Ross Miller • Fall 2005
E=mc2
39
Microsoft Example (cont’d)
Assume you purchased a Microsoft APR 25 put for
$1.10
23.90
23.90
0
25
Stock price at
option expiration
1.10
Professor Ross Miller • Fall 2005
E=mc2
40
“Add” the Two Diagrams Together
Protective put
25
0
29.61
Stock price at
option expiration
4.61
Professor Ross Miller • Fall 2005
E=mc2
41
The Logic Behind Protective Puts
 A protective put is like an insurance policy
• You can choose how much protection you want
 The put premium is what you pay to make large
losses impossible
• The striking price puts a lower limit on your
maximum possible loss, like the deductible in car
insurance
• The more protection you want, the higher the
premium you are going to pay
Professor Ross Miller • Fall 2005
E=mc2
42
The “Greenspan Put”
 It was generally believed that Alan Greenspan
would act (by easing monetary policy)
 This protection that he was thought to provide to
the stock market became known as the
Greenspan put
Professor Ross Miller • Fall 2005
E=mc2
43
Option Combinations
 Options are usually used in combinations to
either place specific types of “bets” on a security
(or related securities) or to create a specific
hedge
 Straddles (the simultaneous purchase of a
matching put and call) is a common way to
betting on increasing volatility
 Options are so flexibility that any payoff diagram
that you can draw can be approximated with a
combination of options
Professor Ross Miller • Fall 2005
E=mc2
44
Synthetic Options and Synthetic Equity
 The term synthetic option describes a collection
of financial instruments that are equivalent to an
option position
 A protective put is an example of a synthetic call
long stock + long put  long call
 More importantly, this also means (in an
approximate sense) that one can create
synthetic equity as follows:
long stock  long call - long put
Professor Ross Miller • Fall 2005
E=mc2
45
One Technicality
 The synthetic stock in the previous slide is not
stock “now”
 Instead, it is a forward contract on stock that is
delivered at the expiration date of the two
options
 The price of this forward contract can be
computed using the standard FV formula with
the risk-free rate as the full carrying cost
Professor Ross Miller • Fall 2005
E=mc2
46
Put-Call Parity
 Basic Idea: The price of the synthetic stock
should (by an arbitrage argument) be the
same as the difference in prices between the
call the and the put
 Two important assumptions
• All options are European (no early exercise)
• The stock pays no dividends (or them are known
and can be PV’ed away)
Professor Ross Miller • Fall 2005
E=mc2
47
The Put-Call Parity Formula
(rearranged slightly from the BKM versions)
X
C  P  S0 
T
(1  rf )
Professor Ross Miller • Fall 2005
E=mc2
48
Putting a Value on Options Prior to Expiration
 Two related approaches
• Look at all the future possibilities and create an
“average” payoff for the option that is the basis
for its value
• Decompose the option into pieces with known
values and find the value of the options by
adding up the values of the pieces
 Key conceptual issues
• What is the distribution of future values?
• How do we discount them?
Professor Ross Miller • Fall 2005
E=mc2
49
What is Special About Option Values
 While stock values depend on beta (and not
volatility), option values depend on volatility (and
not beta)
Professor Ross Miller • Fall 2005
E=mc2
50
Closed Formulas vs. Numerical Methods
 Only under special conditions will we be able to
use a formula to compute the value of an option
 For European call options, no dividends, and a
normal distributions of returns, the Black-Scholes
formula gives the value of the option
 For other options, some version of the binomial
method for numerically computing the value of
an option is used.
Professor Ross Miller • Fall 2005
E=mc2
51
A Quick Binomial Example
 The binomial option model assumes simplifies
return to merely up and down rather than a full
normal distribution
• Divided the time until expiration into small
enough pieces ensures that the binomial model
approximates the normal model
 Consider an option with an exercise price of 50
• Stock A has a 50% chance of expiring at 62 and
a 50% chance of expiring at 40
• Stock B has a 50% chance of expiring at 72 and
a 50% chance of expiring at 30
Professor Ross Miller • Fall 2005
E=mc2
52
A Quick Binomial Example (continued)
 Value[Option on Stock A] = 0.5(62-50)+0.5(0) = 6
 Value[Option on Stock B] = 0.5(72-50)+0.5(0) = 11
 Because Stock B is more volatile, it is more
valuable everything else being equal
 Dividing the time until expiration into smaller and
smaller pieces is equivalent to assuming the the
change in stock price is normally distributed (this a
consequence of the famous Central Limit Theory in
statistics)
Professor Ross Miller • Fall 2005
E=mc2
53
The Black-Scholes Formula
 The value of a call option in a single messy
formula
 Underlying the messy formula is the idea that
call options can be replicated with two pieces:
• A fractional share of the underlying stock
• A loan at the risk-free rate to finance the cost of
the fractional share
 The reason that the formula is a mess:
• The fractional share (delta) and the cost of the
loan both change over time
Professor Ross Miller • Fall 2005
E=mc2
54
Delta
 Delta is the value of N(d1) in the messy formula
• It is between 0 and 1
• Delta is greater than the higher the price of the
stock relative to the exercise price of the option
• For at-the-money options, delta is a bit over 0.5
 Delta is also the probability that the option will be
in-the-money at expiration
Professor Ross Miller • Fall 2005
E=mc2
55
One Last Kind of Options:
Employee Stock Options
 Follow the same mechanics as traded options
except that they are usually issued with 10 years
until expiration and may not vest fully for several
years after issuance
 Whether Black-Scholes or another method
should be used to value them (and if so, how) is
a topic that is currently being hotly debated
 The main reason that Black-Scholes may not
work is that many employees never vest their
stock options and those that do may exercise
them early
Professor Ross Miller • Fall 2005
E=mc2
56
The Current State of Employee Stock Options
 Firms now have to account for the cost of
employee stock options when they are issued
rather than when they are exercised
 Issuance of employee stock options has,
predictably, declined
 This is not so horrible because they are at best a
horribly imperfect way of motivating employees
and can serve to demotivate them
Professor Ross Miller • Fall 2005
E=mc2
57
For Next Time
 Student presentations will be the main order of
business for the final two class meetings
 Click here to visit the spreadsheet where Fin 525
and Fin 603 Google submission are tracked
Professor Ross Miller • Fall 2005
E=mc2
58