Simon Benninga. Financial Modelling 3-rd edition
Подождите немного. Документ загружается.
III
Option-Pricing Models
Chapters 16–24 deal with option pricing and applications. Chapter 16 is
an introduction to options. After defi ning the option terminology, Chapter
16 discusses option payoffs and basic option pricing propositions. In
Chapter 17 we discuss the binomial option pricing model and its imple-
mentation in Excel. After showing how these binomial models work, we
use Visual Basic for Applications (VBA; see Chapters 37–40) to build
binomial option-pricing functions for both European and American
options. One of the applications discussed is the pricing of employee
stock options.
Chapter 18 discusses the lognormality of stock prices. The assumption
of lognormality underlies the Black-Scholes pricing formulas; in Chapter
18 we use Excel to simulate lognormal price processes.
Chapter 19 discusses the Black-Scholes pricing formulas for European
calls and puts. These formulas can be implemented either by direct cal-
culation in the spreadsheet or by using VBA to build new spreadsheet
functions. An extension of the Black-Scholes model to the pricing of
dividend-paying stocks (the so-called Merton model) is implemented.
We show how to apply the option pricing models to the valuation of
structured securities. Chapter 20 discusses the computation of “Greeks”—
the derivatives of the option-pricing formula that show the sensitivities
of the option valuation to its various parameters.
In Chapter 21 we discuss an application of the Black-Scholes model—
portfolio insurance. We use Excel to simulate the performance of port-
folio insurance strategies; these simulations use the lognormal simulations
developed in Chapter 18.
Many options cannot be priced by the closed-form formulas discussed
in Chapter 19. Chapters 22 and 23 show how to implement Monte Carlo
techniques for the pricing of Asian and barrier options. VBA and Excel
lend themselves readily to Monte Carlo.
Real options are illustrated in Chapter 24. Real options are an applica-
tion of option-like concepts to the capital budgeting and valuation prob-
lems discussed in Chapters 1–4. We show an application of the real-option
technique to the valuation of a bio-med startup.
16
An Introduction to Options
16.1 Overview
In this chapter we give a brief introduction to options. The chapter can,
at best, serve as an introduction to the already informed. If you know
nothing whatsoever about options, read an introduction to the topic in
a basic fi nance text.
1
We start with the basic defi nitions and options ter-
minology, go on to discuss graphs of option payoffs and “profi t diagrams,”
and fi nally discuss some of the more important option arbitrage proposi-
tions (sometimes referred to as linear pricing restrictions). In subsequent
chapters we discuss two methods of pricing options: The binomial option-
pricing model (Chapter 17) and the Black-Scholes option-pricing model
(Chapter 19).
16.2 Basic Option Defi nitions and Terminology
An option on a stock is a security that gives the holder the right to buy
or to sell one share of the stock on or before a particular date for a pre-
determined price. Here is a brief glossary of terms and notation used in
the fi eld of options:
•
Call, C: An option that gives the holder the right to buy a share of stock
on or before a given date at a predetermined price.
•
Put, P: An option that gives the holder the right to sell a share of stock
on or before a given date at a predetermined price.
•
Exercise price, X: The price at which the holder can buy or sell the
underlying stock; sometimes also referred to as the strike price.
•
Expiration date, T: The date on or before which the holder can buy or
sell the underlying stock.
•
Stock price, S
t
: The price at which the underlying stock is selling at date
t. The current stock price is denoted S
0
.
•
Option price: The price at which the option is sold or bought.
1. Good chapters can be found in the following books: John Hull, Options, Futures, and
Other Derivatives (6th edition, 2006, Upper Saddle River, NJ: Prentice Hall); Bodie,
Kane, and Marcus, Investments (6th edition, 2004, New York: McGraw Hill Higher
Education).
422 Chapter 16
American versus European options In the jargon of options markets,
an American option is an option that can be exercised on or before the
expiration date T, whereas a European option is one that can be
exercised only on the expiration date T. This terminology is confusing
for two reasons:
1. The options sold on both European and American options exchanges
are almost invariably American options.
2. The simplest option-pricing formulas (including the famous Black-
Scholes option-pricing formula discussed in Chapter 19) are for
European options. As we show in section 16.6, in many cases we can
price American options as if they were European options.
We use C
t
to denote the price of a European call on date t, and P
t
to
denote the European put price. If it is clear that the option price refers
to today’s price, we often drop the subscript, writing C or P instead of
C
0
or P
0
. When we need fuller notation, we write C
t
(S
t
, X, T) for the price
of a call on date t when the price of the underlying stock is S
t
, the exercise
price is X, and the expiration date is T. If we wish to specify that our
option-pricing formula relates to an American option, we use the super-
script A: C
t
A
, C
t
A
(S
t
, X, T) or P
t
A
(S
t
, X, T). When written without
superscripts, the symbols refer to European options.
At the Money, in the Money, out of the Money If the exercise price X
of a call or a put is equal to the current price of the stock S
0
, then the
option is at the money. If a positive cash fl ow could be made by immedi-
ately exercising an American option (that is, S
0
− X > 0 for a call and
X − S
0
> 0 for a put), then the option is in the money.
2
16.2.1 Writing Options versus Purchasing Options: Cash Flows
The purchaser of a call option acquires the right to buy a share of stock
for a given price on or before date T and pays for this right at the time
of purchase. The writer or seller of this call option is the seller of this
right: The writer collects the option price today in return for obligating
herself to deliver one share of stock in the future for the exercise price,
2. It is of course not logical that you can ever make an immediate profi t by buying an
American option and immediately exercising it. Thus, for American calls, C
0
> S
0
− X,
and for American puts, P
0
> X − S
0
. Thus in the money and out of the money refer only
to the relation between S
0
and X without taking into account the option price.
423 An Introduction to Options
if the purchaser of the call demands. In terms of cash fl ows, the purchaser
of an option always has an initial negative cash fl ow (the price of the
option) and a future cash fl ow that is at worst zero (if it is not worthwhile
exercising the option) and otherwise positive (if the option is exercised).
The cash-fl ow position of the writer of the option is reversed: An initial
positive cash fl ow is followed by a terminal cash fl ow that is at best
zero.
Time 0 Time T
Purchase call option,
cash flow < 0
Terminal call payoff,
Max[S
T
- X,0] ≥ 0
Cash flows
of call buyer
Between times 0 and T:
Cash flow = 0 for European option
Cash flow ≥ 0 for American option
Write (i.e., issue) call option,
cash flow > 0
Pay terminal call payoff
= - Max[S
T
- X, 0] ≤ 0
Cash flows
of call writer
Between times 0 and T:
Cash flow = 0 for European option
Cash flow ≤ 0 for American option
Call Option Payoff Patterns
A similar payoff pattern holds for the cash fl ows of the purchaser and
writer of a put option on a stock:
Time 0 Time T
Purchase put option,
cash flow < 0
Terminal put payoff,
Max[X - S
T
,0] ≥ 0
Cash flows
of put buyer
Between times 0 and T:
Cash flow = 0 for European option
Cash flow ≥ 0 for American option
Write (i.e., issue) call option,
cash flow > 0
Pay terminal call payoff
= - Max[X - S
T
, 0] ≤ 0
Cash flows
of put writer
Between times 0 and T:
Cash flow = 0 for European option
Cash flow ≤ 0 for American option
Put Option Payoff Patterns
424 Chapter 16
16.3 Some Examples
The following spreadsheet shows the most actively traded options on 3
October 2006:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
AB C DEF GH I J
Rank Symbol Name
Option
expiration
Option
exercise
price
Put or
call?
Stock
closing
price
Option
closing
price
Volume
Open
Interest
1 HAL Halliburton 20-Apr-07 27.5 Put 26.75 3.025 60,945 7,305
2 HAL Halliburton 19-Jan-07 27.5 Put 26.75 2.450 59,767 61,121
3 HAL Halliburton 19-Jan-07 30.0 Put 26.75 4.100 59,131 68,782
4 QQQQ Nasdaq 100 Index 20-Oct-06 40.0 Put 40.31 0.475 57,073 262,601
5 HAL Halliburton 20-Apr-07 22.5 Put 26.75 1.025 53,901 5,530
6 IWM Russell 2000 Index 17-Nov-06 70.0 Put 71.22 1.650 49,387 203,975
7 QQQQ Nasdaq 100 Index 17-Nov-06 39.0 Put 40.31 0.575 45,666 93,169
8 IWM Russell 2000 Index 20-Oct-06 72.0 Put 71.22 1.650 43,432 154,148
9 SPY S&P Depository Receipts ("Spider") 15-Dec-06 120.0 Put 133.36 0.400 40,125 32,090
10 SPY S&P Depository Receipts ("Spider") 15-Dec-06 139.0 Call 133.36 0.850 40,116 21,036
11 SPY S&P Depository Receipts ("Spider") 15-Dec-06 126.0 Put 133.36 0.925 40,082 32,249
12 IWM Russell 2000 Index 20-Oct-06 71.0 Put 71.22 1.200 39,015 168,035
13 QQQQ Nasdaq 100 Index 20-Oct-06 40.0 Call 40.31 0.875 37,502 196,355
14 BMY Bristol-Myers Squibb 20-Oct-06 22.5 Call 24.82 2.325 31,989 10,934
15 SPY S&P Depository Receipts ("Spider") 20-Oct-06 125.0 Put 133.36 0.075 31,964 25,569
16 S Sprint Nextel 17-Nov-06 17.5 Call 16.97 0.525 31,350 9,078
17 IWM Russell 2000 Index 20-Oct-06 69.0 Put 71.22 0.600 30,777 111,620
18 SPY S&P Depository Receipts ("Spider") 20-Oct-06 137.0 Call 133.36 0.125 30,199 6,261
19 QQQQ Nasdaq 100 Index 20-Oct-06 41.0 Put 40.31 0.975 29,881 94,955
20 MMM MMM 20-Oct-06 85.0 Put 74.03 11.000 25,732 6,364
Open interest: Number of open positions at end of day. A position is closed out (i.e., not open) if the option has been sold by day end.
Volume: Number of options traded on 4 Oct. 2006. Each option is for 100 shares, but price quotes are per share.
MOST ACTIVE OPTIONS, 3 OCTOBER 2006
Stock
Here are some comments on this list:
Out of the 20 most active options, 14 are on exchange-traded funds
(ETFs) and 6 are on stocks. For purposes of analysis, we can regard the
ETFs as if they are stocks.
•
QQQQ is an index of the 100 largest nonfi nancial companies traded
on Nasdaq.
•
SPY tracks the Standard & Poor’s 500 index.
•
IMW tracks the Russell 2000 index.
Out of the 20 most active options, 5 are calls and 15 are puts. On 3
October 2006, U.S. markets closed at or near their seven-year highs, so
the preponderance of puts is perhaps indicative of a tendency to bet on
market declines.
425 An Introduction to Options
Of the 20 most active options, 10 are for the closest option expiration
date—20 October 2006. The most active options are often the closest-
term options.
In many cases the volume of trading exceeds the open interest at the end
of the day. This fact indicates that option positions opened during the
day were closed out by the day’s end. An option position is closed out if
the holder has sold her options.
We examine the January 2007 options of Halliburton (HAL) some-
what more closely.
3
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
ABCDEFGH
Exercise
price
Closing
option
price
Volume
Open
interest
Closing
option
price
Volume
Open
interest
15.0 11.90 20 1,724 0.05 50 1,153
17.5 12.70 23 9,449 0.05 140 3,312
20.0 7.17 5 13,801 0.20 50 18,277
22.5 5.00 339 4,595 0.60 49 15,247
25.0 3.20 200 3,306 1.25 423 21,793
27.5 1.85 571 3,918 2.50 53,267 61,121
30.0 0.95 2,140 12,008 4.10 52,631 68,782
32.5 0.55 810 10,527 6.00 216 21,562
33.8 0.40 78 2,276 6.90 168 3,378
35.0 0.30 231 69,727 8.10 142 20,258
36.3 0.20 460 5,989 8.00 2 1,280
37.5 0.15 1,249 78,762 9.10 5 5,530
40.0 0.10 32 85,037 11.01 16 322
42.5 0.05 170 24,439 15.70 135 1,920
45.0 0.05 6 8,509 17.10 76 0
47.5 0.05 10 6,591 14.60 76 0
50.0 0.05 20 10,528 17.00 52 0
55.0 0.05 12 694 22.10 28 0
HALLIBURTON JAN 07 OPTIONS, PRICES ON 3 OCT 06
Closing price of stock: 26.75, down 1.29
Calls Puts
Note the imbalance in trading volume between the calls and the puts.
Note also that most of the put trading volume is concentrated on the
at-the-money puts.
3. Why were four Halliburton put options among the most active? On 3 October 2006
the stock itself traded at twice its average daily volume (29 million shares as opposed
to 14 million daily for the three months previously). Strong rumors were circulating
about a possible cover-up at the company related to bribes paid to government offi cials
in Nigeria.
426 Chapter 16
16.4 Option Payoff and Profi t Patterns
One of the attractions of options is that they allow their owners to
change the payoff patterns of the underlying assets. In this section we
consider
•
The basic payoff and profi t patterns of a call and a put option and a
share.
•
The payoff patterns of various combinations of options and shares.
16.4.1 Stock Profi t Patterns
16.4.1.1 Payoff Pattern from a Purchased Stock
Suppose you buy a share of General Pills stock in July at its then-current
market price of $40. If in September the price of the stock is $70, you
will have made a $30 profi t; if its price is $30, you will have a loss (or a
negative profi t) of $10.
4
Generalize this pattern by writing the price of
the stock in September as S
T
and its price in July by S
0
. Then we write
the profi t function from the stock as
Profi t from stock = S
T
− S
0
16.4.1.2 Payoff from the Short Sale of a Stock
Suppose we had sold one share of GP stock short in July, when its market
price was $40. If in September the market price of GP was $70, and if at
that point we undid the short sale (i.e., we purchase a share at the market
price in order to return the share to the lender of the original short),
then our profi t would be −$30:
Profi t from short sale of stock = S
0
− S
T
= − (Profi t from purchase of stock)
Notice that the profi t from the short sale is the negative of the profi t from
the purchase; this is always the case (also for options, considered in
section 16.4.2).
4. Our use of the word profi t in this section constitutes a slight abuse of the English lan-
guage and the standard fi nance concept of the word, since we are ignoring the interest
costs associated with buying the asset. In the case at hand, this abuse of language is
both traditional and harmless.
427 An Introduction to Options
16.4.1.3 Graphing Stock Profi t Patterns
The following Excel fi gure graphs the profi t patterns from both a pur-
chase and a short sale of the GP stock:
Stock Profit Patterns
–50
–40
–30
–20
–10
0
10
20
30
40
50
0 1020304050607080
Terminal stock price, S
T
($)
Profit ($)
Bought stock Shorted stock
16.4.2 Call Option Profi t Patterns
16.4.2.1 Payoff Pattern from a Purchased Call
We return to the General Pills (GP) options of the previous section.
Suppose that in July you bought one GP September 40 call for $4.
5
In
September you will exercise the call only if the market price of GP is
higher than $40. If we write the initial (July) call price as C
0
, we can write
the profi t function from the call in September as follows:
Call profit in September ,
,
if
=−−
=−−
=
−
max ( )
max ( )
0
0404
4
0
SXC
S
S
T
T
T
≤≤
−>
⎧
⎨
⎩
40
44 40SS
TT
if
5. Because the exercise price of this call is equal to the current market price of the stock,
it is called an at-the-money call. When the exercise price of the call is higher than the
current market price, it is called an out-of-the-money call, and when the exercise price
is lower than the current market price, the call is an in-the-money call.