Mahlerʼs Guide to
Financial Economics
Joint Exam MFE/3F
prepared by
Howard C. Mahler, FCAS
Copyright 2012 by Howard C. Mahler.
Study Aid 2012-MFE/3F
Howard Mahler
hmahler@mac.com
www.howardmahler.com/Teaching
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 1
Mahlerʼs Guide to Financial Economics
Copyright 2012 by Howard C. Mahler.
Concepts in Derivatives Markets by Robert L. McDonald are demonstrated.
Information in bold or sections whose title is in bold are more important for passing the exam.
Larger bold type indicates it is extremely important. Information presented in italics (and sections
whose titles are in italics) should not be needed to directly answer exam questions and should be
skipped on first reading. It is provided to aid the readerʼs overall understanding of the subject, and to
be useful in practical applications.
Highly Recommended problems are double underlined.
Recommended problems are underlined.1
Section #
Pages
1
2
3
4
5
6
7
8
9
10
10-20
21-30
31-54
55-69
70-79
80-91
92-97
98-102
103-108
109-134
Introduction
European Options
Properties of Premiums of European Options
11
12
13
14
15
16
17
18
19
20
135-149
150-159
160-171
172-199
200-206
207-213
214-226
227-232
233-242
243-250
Replicating Portfolios
Risk Neutral Probabilities
Utility Theory and Risk Neutral Pricing
Binomial Trees, Risk Neutral Probabilities
Binomial Trees, Valuing Options on Other Assets
Other Binomial Trees
Binomial Trees, Actual Probabilities
Jensen's Inequality
Normal Distribution
LogNormal Distribution
A
B
C
D
E
Section Name
Put-Call Parity
Bounds on Premiums of European Options
Options on Currency
Exchange Options
Futures Contracts
Synthetic Positions
American Options
The Table of Contents is continued on the next page.
1
Note that problems include both some written by me and some from past exams. The latter are copyright by the
Society of Actuaries and the Casualty Actuarial Society and are reproduced here solely to aid students in studying
for exams. The solutions and comments are solely the responsibility of the author; the SOA and CAS bear no
responsibility for their accuracy. While some of the comments may seem critical of certain questions, this is intended
solely to aid you in studying and in no way is intended as a criticism of the many volunteers who work extremely long
and hard to produce quality exams.
HCMSA-2012-MFE/3F,
Section #
G
H
I
J
K
L
M
N
O
P
Pages
Financial Economics,
12/14/11,
Page 2
Section Name
21
22
23
24
25
26
27
28
29
30
251-254
255-283
284-313
314-318
319-321
322-326
327-339
340-349
350-360
361-375
Limited Expected Value
A LogNormal Model of Stock Prices
31
32
33
34
35
36
37
38
39
40
376-402
403-409
410-415
416-429
430-438
439-441
442-445
446-452
453-454
455-481
Option Greeks
Delta-Gamma Approximation
Option Greeks in the Binomial Model
Profit on Options Prior to Expiration
Elasticity
Volatility of an Option
Risk Premium of an Option
Sharpe Ratio of an Option
Market Makers
Delta Hedging
41
42
43
44
45
46
47
48
49
50
482-491
492-493
494
495-511
512-532
533-559
560-573
574-581
582-585
586-592
Gamma Hedging
Relationship to Insurance
Exotic Options
Asian Options
Barrier Options
Compound Options
Gap Options
Valuing European Exchange Options
Forward Start Options
Chooser Options
51
52
53
54
55
56
57
58
59
60
593-597
598-607
608-619
620-624
625-638
639-651
652-663
664-694
695-723
724-739
Options on the Best of Two Assets
Cash-or-Nothing Options
Asset-or-Nothing Options
Random Walks
Standard Brownian Motion
Arithmetic Brownian Motion
Geometric Brownian Motion
Geometric Brownian Motion Model of Stock Prices
Ito Processes
Ito's Lemma
Black-Scholes Formula
Black-Scholes, Options on Currency
Black-Scholes, Options on Futures Contracts
Black-Scholes, Stocks Paying Discrete Dividends
Using Historical Data to Estimate Parameters of the Stock Price Model
Implied Volatility
Histograms
Normal Probability Plots
The Table of Contents is continued on the next page.
HCMSA-2012-MFE/3F,
Section #
S
T
U
V
W
Pages
Financial Economics,
Section Name
61
62
63
64
65
66
67
68
69
70
740-752
753-765
766-772
773-778
779-787
788-793
794-812
813-831
832-833
834-855
Valuing a Claim on S^a
Black-Scholes Equation
Simulation
Simulating Normal and LogNormal Distributions
Simulating LogNormal Stock Prices
Valuing Asian Options via Simulation
Improving Efficiency of Simulation
Bonds and Interest Rates
The Rendelman-Bartter Model
The Vasicek Model
71
72
73
74
75
76
856-882
883-889
890-901
902-909
910-938
939-981
The Cox-Ingersoll-Ross Model
The Black Model
Interest Rate Caps
Binomial Trees of Interest Rates
The Black-Derman-Toy Model
Important Formulas and Ideas
982-1497
Solutions to Problems
My practice exams are sold separately.
12/14/11,
Page 3
HCMSA-2012-MFE/3F,
Financial Economics,
Chapter of Derivative Markets
9
102
11.1 – 11.43
12.1-12.54
135
14
18
19.1-19.5
20.1-20.76
21.1–21.37
22.18
23.1–23.29
24.1-24.510
Appendix B.1
Appendix C
Sections of Study Guide
3-10
11, 12, 14, 15.
10, 13, 16-17, 27, 54
23-26, 28, 31, 34-38
32-33, 39-42
43-51
19-22, 27, 29-30
63-67
55-61
62
52-53
27, 28, 58
68-75
1
18
12/14/11,
Page 4
I have included in my early sections, the 9 questions from the 2007 FM Sample Exam for
Derivatives Markets, based on earlier chapters of the textbook.
Unless otherwise stated chapter appendices are not included in the required readings from this text.
2
Excluding “Options on Commodities” on page 334.
Including Appendices 11.A and 11.B.
4
Including Appendix 12.A.
5
Including Appendix 13.B.
6
Sections 20.1–20.6 (up to but excluding “Multivariate Itôʼs Lemma” on pages 665-666) and 20.7 (up to but
excluding “Valuing a Claim on SaQb on pages 670-672 and excluding “Finding the lease rate” on top one-half of
page 669).
7
Sections 21.1–21.2 (excluding “What If the Underlying Asset Is Not and Investment Asset” on pages 688-690)
and 21.3 (excluding “The Backward Equation” on pages 691-692, and excluding the paragraph on page 692 that
begins “If a probability…” and through the end of the section).
8
But with only those definitions in Tables 22.1 and 22.2 that are relevant to Section 22.1.
9
Up to but excluding “Exponentially Weighted Moving Average” on page 746 and through the end of the section.
10
Up to but excluding “Forward rate agreements” on pages 806-808.
3
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 5
This study guide covers all of the material on the Joint Exam: CAS 3F and SOA MFE.11
The syllabus consists of various sections of the 2nd edition of Derivative Markets by Robert L.
McDonald, plus a short study note “Some Remarks On Derivative Markets” by Elias S. W. Shiu.12
Unless stated otherwise in a question assume:
•
The market is frictionless. There are no taxes, transaction costs, bid/ask spreads, or restrictions
on short sales. All securities are perfectly divisible. Trading does not affect prices. Information
is available to all investors simultaneously. Every investor acts rationally
(i.e., there is no arbitrage.)
•
The risk-free rate is constant
•
The notation is the same as used in Derivative Markets by Robert L. McDonald.
•
The MFE/3F CBT exam will provide a formula document as well as a normal
distribution calculator that will be available during the test by clicking buttons on
the item screen. Details are available on the Prometric Web Site.
http://www.prometric.com/SOA/MFE3F_calculator.htm
“Similar to other exam reference buttons, the normal distribution calculator button will be available
throughout the exam in the top right corner of every item screen. Click the button to call up the
calculator and calculate cumulative normal distribution and inverse cumulative normal distribution
values. Use these values to answer the question as needed.”
“When using the normal distribution calculator, values should be entered with five decimal places.
Use all five decimal places from the result in subsequent calculations.”
The normal distribution calculator button replaces the Normal Table.
The previous rule on rounding no longer applies.13
You can try the normal distribution calculator button at the Prometric Web Site.
You will benefit from using it at least part of the time when you are studying.
The formula sheet contains the same information about the Normal and LogNormal distributions as
was provided in the past.
11
In 2007 the CAS and SOA gave separate exams. Starting in 2008 they gave a joint exam.
The study note is available on the CAS and SOA webpages.
13
Unfortunately, my solutions were written up using the prior rule: “On Joint Exam 3F/MFE, when using the normal
distribution, choose the nearest z-value to find the probability, or if the probability is given, choose the nearest
z-value. No interpolation should be used. For example, if the given z-value is 0.759, and you need to find
Pr(Z < 0.759) from the normal distribution table, then chose the probability for z-value = 0.76:
Pr(Z < 0.76) = 0.7764.”
This should not make a significant difference.
12
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 6
Changes to the Syllabus from 2007:14
Section 12.6 of Derivative Markets, Perpetual American Options, is no longer on the syllabus.15
The final portion of Section 20.6, Multivariate Itoʼs Lemma, is no longer on the syllabus.16
The final portion of Section 24.5, Forward Rate Agreements, is no longer on the syllabus.
Added, the first portion of Section 20.7, Valuing a Claim on Sa , up to but excluding the
last subsection on “Valuing a Claim on Sa Q b .”
Changes to the Syllabus for Spring 2009:
Chapter 10 of Derivative Markets, exclude “Options on Commodities” on page 334.
Exclude Section 11.5 of Derivative Markets, on Binomial Trees, Discrete Dividends.
Add Appendices 11.A and 11.B of Derivative Markets.
Add Appendix 13.B of Derivative Markets.
Chapter 20 of Derivative Markets: exclude “Finding the lease rate” on top one-half of page 669.
Add parts of Chapter 21: Sections 21.1–21.2 (excluding “What If the Underlying Asset Is Not and
Investment Asset” on pages 688-690) and 21.3 (excluding “The Backward Equation” on pages
691-692, and excluding the paragraph on page 692 that begins “If a probability…” and
through the end of the section).
Add parts of Chapter 22: Section 22.1 (but with only those definitions in Tables 22.1 and 22.2 that
are relevant to Section 22.1.)
Add parts of Chapter 23: Sections 23.1–23.2 (up to but excluding “Exponentially Weighted
Moving Average” on page 746 and through the end of the section.)
Add Appendix B.1.
Add Appendix C.
Changes to the Syllabus for Fall 2009:17
Add Chapter 18 of Derivative Markets, about the LogNormal Stock Price Model.
Add Chapter 19.1–19.5 of Derivative Markets, about Monte Carlo Valuation, in other words
simulation.
Changes for 2011:
Computer based testing.
3 hours and approximately 30 questions.
14
Starting in 2008 this is a joint exam, 3F/MFE.
In 2007 Section 12.6 was on SOA MFE, but not CAS 3.
16
In 2007 this final portion of Section 20.6 was on CAS 3, but not SOA MFE.
17
Material was moved from Exam 4/C onto Exam 3F/MFE. Exam 3F/MFE was extended from 2 hours to 2.5 hours
and will consist of approximately 25 multiple-choice questions.
15
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 7
Exam Questions by Section of This Study Guide:
Section Section Name
A
B
B
1
2
3
4
5
6
C 7
8
9
C 10
D
11
12
13
14
D
E 15
16
17
18
F 19
E 20
21
22
23
G 24
25
H
26
27
28
29
30
Introduction
European Options
Prop. of Prems. of Euro. Options
Put-Call Parity
MFE CAS 3
Sample 5/07
MFE
5/07
CAS 3
11/07
MFE/3F
5/09
25
2
1
3, 4
1, 4
14, 16
12
15
9
Bounds on Premiums of Euro. Options
Options on Currency
Exchange Options
Futures Contracts
Synthetic Positions
American Options
Replicating Portfolios
Risk Neutral Probabilities
Utility Theory and Risk Neutral Pricing
Bin. Trees, Risk Neutral Probs.
Binomial Trees, Options on Other Assets
Other Binomial Trees
Binomial Trees, Actual Probabilities
Jensen's Inequality
Normal Distribution
LogNormal Distribution
Limited Expected Value
A LogNormal Model of Stock Prices
Black-Scholes Formula
Black-Scholes, Options on Currency
Black-Scholes, Options on Futures
Black-Scholes, Discrete Dividends
Historical Data to Estimate Parameters
Implied Volatility
Histograms
Normal Probability Plots
26
13
12
13
16
17
3
18, 19, 23
1
27
4, 49
14
15, 17
11
5, 46
14
2
50
6
7
55
20, 21
3, 8
7
20
21
15
17, 51
The SOA did not release its 11/07 exam MFE.
The CAS/SOA did not release the 5/08, 11/08,11/09, and subsequent exams 3F/MFE.
Continued on the next page
19
HCMSA-2012-MFE/3F,
Financial Economics,
Section Section Name
AI
B
J
31
32
33
34
35
36
37
38
K 39
C 40
L
D
41
42
43
44
45
Option Greeks
Delta-Gamma Approximation
Option Greeks in the Binomial Model
Profit on Options Prior to Expiration
Elasticity
Volatility of an Option
Risk Premium of an Option
Sharpe Ratio of an Option
Market Makers
Delta Hedging
Gamma Hedging
Relationship to Insurance
Exotic Options
Asian Options
Barrier Options
MFE
Sample
CAS 3
5/07
MFE
5/07
CAS 3 MFE/3F
11/07
5/09
8, 31
17
20
19
44, 45, 69
40
20, 41
13
22
5
29
9, 47, 65
32
33
10
24
27
42
46
47
M 48
49
E
50
N 51
52
53
54
O 55
Compound Options
Gap Options
Valuing European Exchange Options
Forward Start Options
56
57
58
Arithmetic Brownian Motion
Geometric Brownian Motion
Geo. Brown. Mot. Model Stock Pr.
59
Ito Processes
12, 23, 48, 61
63, 66, 67, 70
60
Ito's Lemma
13, 24, 35, 43
64, 68, 73
Chooser Options
Options on the Best of Two Assets
Cash-or-Nothing Options
Asset-or-Nothing Options
Random Walks
Standard Brownian Motion
Page 8
12/14/11,
34
18
2
17
28
26
19, 33
25
54
28, 53
6
4
34
10, 11, 32, 37
56, 74
16
P
Q
Continued on the next page
35
18
10, 18
12
6
HCMSA-2012-MFE/3F,
Financial Economics,
Section Section Name
R
A
61
62
63
64
B
S 65
Valuing a Claim on S^a
Black-Scholes Equation
Simulation
Simulating Normals & LogNormals
Simulating LogNormal Stocks
66
T 67
68
69
C 70
Valuing Asian Options via Sim.
Improving Efficiency of Simulation
Bonds and Interest Rates
The Rendelman-Bartter Model
The Vasicek Model
71
U 72
73
74
D
V 75
The Cox-Ingersoll-Ross Model
The Black Model
Interest Rate Caps
Binomial Trees of Interest Rates
The Black-Derman-Toy Model
MFE
Sample
12/14/11,
CAS 3
5/07
MFE
5/07
16, 62, 71, 72
36
Page 9
CAS 3 MFE/3F
11/07
5/09
11
8
52
57, 58, 59, 75
39
14, 22
36
13
15
21, 38, 60
7
3
15, 29, 30, 76
37
9
5
14
Questions no longer on the syllabus: MFE, 5/07, Q. 16.
In August 2010, the SOA/CAS updated the file of MFE Sample Exam questions.
There are now a total of 76 sample questions.
Check the SOA webpage to see if any additional Sample Exam questions have been
added.
Valuing a Claim on Sa was added to the syllabus in Spring 2008.
Material on simulation was moved here from exam 4/C in Fall 2009.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 10
Section 1, Introduction
Earlier Chapters of Derivative Markets by McDonald are on Exam 2/FM.18
Some of the ideas covered in those chapters are used in the chapters on your exam.
Derivatives:19
A derivative is an agreement between two people that has a value determined by the price of
something else.
For example, Alan gives Bob the right to buy from Alan a share of IBM stock one year from now at
a price of $120. This is an example of a stock option. The value of this option depends on the price
of IBM stock one year from now.
Options:
A call is an option to buy.
For example, Bob purchased a call option on IBM stock from Alan.
A put is an option to sell.
For example, if Debra purchased a put option on IBM stock from Carol, then Debra will have the
option in the future to sell a share of IBM stock to Carol at a specified price.
Continuously Compounded Risk Free Rate:20
If r is the continuously compounded annual risk free rate, then the present value of $1 T years in the
future is: e-rT.
r as used by McDonald is what an actuary would call the force of interest.
Effective Annual Rate:21
If r is the effective annual risk free rate, then the present value of $1 T years in the future is: 1/(1+r)T.
An effective annual rate is what an actuary would call the rate of interest.
Effective annual rate will be used in Interest Rate Caps and the Black-Derman-Toy Model, to be
discussed in subsequent sections. Otherwise, we will use continuously compounded rates.
18
Sections 1.1-1.4, 2.1-2.6, Appendix 2.A, 3.1-3.5, 4.1-4.4, 5.1-5.4, Appendix 5.B, 8.1-8.2.
Warren E. Buffett has said, “Derivatives are financial weapons of mass destruction, carrying dangers that, while
now latent, are potentially lethal.”
20
See Appendix B.1 of Derivative Markets by McDonald.
21
See Appendix B.1 of Derivative Markets by McDonald.
19
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 11
Selling Short:
If we sell a stock short, then we borrow a share of stock and sell it for the current market price. We will
give this person a share of stock at the designated time in the future. We also must pay this person
any stock dividends they would have gotten on the stock, when they would have gotten them.
Forward Contracts:
A forward contract is an agreement that sets the terms today, but the buying or selling
of the asset takes place in the future.
For example, Ed will be moving in a month, and his friend Fred agrees to buy Edʼs TV one month
from now for $200.
The purchaser of an option has bought the right to do something in the future, but has no obligation
to do anything. In contrast, in a forward contract both parties are obligated to fulfill their parts of the
contract.
Value of a Forward Contract:
F0,T = forward price at time T in the future.
For example, if Joe buys a forward contract to buy one share of ABC stock in two years at $120,
then F0,2 = $120. At time 2 years, Joe pays $120 and gets one share of stock.22
PV[F0,T] is the present value at time 0 of a forward contract to be executed at time T.
PV[F0,T] = F0,T e-rT.
Let us assume the current price of XYZ stock is S0 .
Assume XYZ stock pays no dividends.
Charlie can buy a forward contract to buy one share of XYZ stock in exchange for paying F0,T at
time T. If Charlie invests F0,T e-rT at the risk free rate, then at time T he will have F0,T.23
He uses that amount to fulfill his forward contract and at time T Charlie has one share of XYZ Stock.
Lucy can instead buy one share of XYZ stock now, for the current market price of S0 , and hold onto
the share of stock until at least time T.
22
This differs from the prepaid price. Joe might instead be able to pay $110 now and get a share of stock 2 years
from now. This is an example of a prepaid futures contract.
23
Charlie could invest in a Treasury Bond.
Financial Economics,
HCMSA-2012-MFE/3F,
12/14/11,
Page 12
Both Lucy and Charlie end up in the same situation, with one share of XYZ stock at time T.
Therefore, their investments must have equal present value.
S 0 = F0,T e-rT.
F0,T = S0 erT, in the absence of dividends.
If instead XYZ stock pays dividends, then Lucy would have collected any dividends paid from time
0 to T, while she owned the stock. Charlie would not. Thus Lucyʼs position is equal to Charlieʼs
position plus a receipt of dividends.
Therefore, S0 - PV[Div] = F0,T e-rT.
F0 , T = S0 er T - PV[Div] er T.
If the dividends are paid at discrete points in time, with amount Dti paid at time ti, then
F0 , T = S0 er T -
∑er(T -
ti)
Dt i .24
Exercise: The current price of a stock is $100.
It will to pay a dividend 3 months from now, a dividend 6 months from now, a dividend 9 months
from now, and a dividend 12 months from now. Each dividend is of size $2. r = 6%.
Determine the the forward price for a share of stock one year from now
[Solution: F0,1 = (100) e.06 - (2)(e.045 + e.03 + e.015 + e0 ) = $98.00.
Comment: Both sides of the equation are valued one year from now.]
Futures Contracts:
A futures contract is similar to a forward contract except:
A futures contract is typically traded on an exchange.
A futures contract is marked to market periodically.25
The buyer and the seller post margin.26
24
See Equation 5.7 in Derivative Markets by McDonald.
Marked-to-market means the item is revalued to reflect current market prices.
26
A deposit which compensates the other party to a futures contract in case one of the parties does not fulfill its
obligation.
25
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 13
Forward Contracts Versus Futures Contracts:27
Type of Market
Liquidity
Contract Form
Performance Guarantee
Transaction Costs
Forward Contract
Dealer or Broker
Low
Customized
Creditworthiness
Bid-ask spread
Futures Contract
(Commodities) Exchange
High
Standard
Mark-to-Market
Fees or Commissions
Continuous Dividends:
We often assume that dividends are paid at a continuous rate δ.28 Over a short period of time dt,
stock dividends of: δ S(t) dt are paid, where S(t) is the stock price at time t.
So that if one buy a share of stock at time 0, and reinvests the dividends in the stock, at time T one
would have eTδ shares of the stock.29
Exercise: One buys 1 million shares of a stock that pays dividends at the continuous annual rate of
2%. The dividends are reinvested in that stock.
After 3 years how many shares of the stock does one own?
[Solution: (1 million)e(3)(.02) = 1,061,837 shares.]
If instead of discrete dividends XYZ stock pays continuous dividends, then Lucy would have eTδ
shares of the stock at time T. If Charlieʼs future contract were for eTδ shares of the stock, then his
position would be equal to Lucyʼs. F0,T eTδ = S0 erT.
Therefore, in the case of dividends paid continuously: F0 , T = S0 eT(r - δ) . 30
Prepaid Forward Price:
The forward price is the price we would pay in the future for a forward contract. In contrast, the
P , is the price we would pay today for a forward contract.
prepaid forward price, F0,T
P =F
F0,T
0 , T e-rT.
27
Taken from Table 2.2 Financial Economics, Harry H. Panjer, editor.
This is a good approximation for a stock index fund.
29
δ acts similarly to a force of interest.
30
See Equation 5.8 in Derivative Markets by McDonald. This is like the accumulated value for a force of interest.
28
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 14
For example, let us assume we are paying today in order to own a share stock at time 3. Then the
P
prepaid forward price is F0,3
(S).
We would pay this price at time 0 in exchange for receiving the stock at time 3.
However, we would not receive any dividends the stock would pay between time 0 and 3.
P (S) = S - PV[Div].
Therefore, in the case of discrete dividends, F0,T
0
Exercise: The current price of a stock is 120. The stock will pay a dividend of 3 in 2 months.
What is the 5 month prepaid forward price of the stock? r = 6%.
[Solution: S0 - PV[Div] = 120 - 3e-(2/12)(6%) = 117.03.]
P (S) = S e-δT .
In the case of continuous dividends, F0,T
0
Exercise: The current price of a stock is 80. The stock pays dividends at a continuous rate of 1%.
What is the 5 month prepaid forward price of the stock?
[Solution: S0 e-δT = 80e-(5/12)(1%) = 79.67.]
If we pay S0 e-δT in order to buy e-δT shares of stock today and reinvest the dividends we would
have one share of stock at time T. Thus S0 e-δT is the price we would pay today to own one share
of stock at time T.
In the case of continuous dividends, somewhat more generally, FtP, T (S) = St e-δ(T-t).
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 15
Named Positions:31
One can buy various combinations of options and stock.
The more common such positions have been given names.
Bear Spread: The sale of an option together with the purchase of an otherwise identical
option with a higher strike price. Can construct a bear spread using either puts or calls.
The owner of the Bear Spread hopes that the stock price moves down.
Box Spread: Buy a call and sell a put at one strike price, plus at another (higher) strike
price sell a call and buy a put.32
Bull Spread: The purchase of an option together with the sale of an otherwise identical
option with a higher strike price. Can construct a bull spread using either puts or calls.
The owner of the Bull Spread hopes that the stock price moves up.
Butterfly Spread: Buying a K strike option, selling two K + ΔK strike options,
and buying a K + 2ΔK strike option.
Collar: Purchase a put and sell a call with a higher strike price.
Ratio Spread: Buying m of an option and selling n of an otherwise identical option at a
different strike.
Straddle: Purchase a call and the otherwise identical put.
Strangle: The purchase of a put and a higher strike call with the same time until
expiration.
For example, Gene Green buys a Straddle with K = 80.
He buys an 80-strike call and a similar 80-strike put.
His payoff at expiration is: Max[0, ST - 80]+ + Max[0, 80 - ST] = |ST- 80|.
The further the stock price at expiration is from 80, the larger Geneʼs payoff.
Gene is hoping there is a large movement in the stock price.33
31
See Chapter 3 of Derivative Markets by McDonald, on the syllabus of Exam 2/FM.
For European options, the box spread is equivalent ot a zero-coupon bond.
33
In other words, Gene is betting that the stockʼs volatility is high.
In contrast, the seller of a straddle is betting that the stockʼs volatility is low.
32
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 16
For example, Vanessa buys a 90 strike call and sells an otherwise identical 100 strike call.
This is an example of a Call Bull Spread. Vanessa hopes the stock price increases.
If Vanessa bought her 90 strike call from Nathan and sold her 100 strike call to Nathan, than Nathan
owns a Call Bear Spread. Nathan hopes the stock price declines.
Long and Short Positions:
Entering into a long position is buying.
Entering into a short position is selling or writing.
For example if you long one call option and long the similar put option, then you bought the call and
put, and you have purchased a straddle. If instead you short a call option and the similar put option,
then you have written (sold) a straddle.
If you short a 60-strike 3-month call and long a 80-strike 3-month call, then you have purchased a
Bear Spread.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 17
Actuarial Present Values:34
Let us assume that one year from now an insurer will pay either $50 with probability 70% or $100
with probability 30%. Then the expected payment in one year is: (0.7)(50) + (0.3)(100) = $65.
Assume that the continuously compound annual rate of interest is now 5%.
Then the actuarial present value of the insurerʼs payment is: 65 e-0.05 = $61.83.35
In general in order to calculate an actuarial present value, one takes a sum of the expected
payments at each point in time each multiplied by the appropriate discount factor. The discount
factor adjusts for the difference between the time value of money at the present and at the time
when the payment is made.
Exercise: In addition to the payments one year from now, the insurer will pay two years from now
either $50 with probability 50%, $100 with probability 40%, or $200 with probability 10%.
Assume that one year from now the continuously compound annual rate of interest will be 6%.
Determine the actuarial present value of the insurerʼs total payments, including those made one year
from now and two years from now.
[Solution: The expected payment in two years is: (0.5)(50) + (0.4)(100) + (0.1)(200) = $85.
Discounting back to the present: 85 exp[-0.05 - 0.06] = $76.15.
Adding in the actuarial present value of the payments made in one year, the actuarial present value
of the insurerʼs total payments is: $61.83 + $76.15 = $137.98.]
34
35
Covered extensively on CAS Exam 3L and SOA Exam MLC.
If instead the 5% were an effective annual rate, then the actuarial present value would be: 65/1.05 = $61.90.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 18
Problems:
1.1 (1 point) A stock price is 160. Assume r = 0.08 and there are no dividends.
What is the 4-year forward price?
A. less than 200
B. at least 200 but less than 210
C. at least 210 but less than 220
D. at least 220 but less than 230
E. at least 230
1.2 (1 point) A stock has a current price of 120.
The stock pays dividends at a continuously compounded rate of 1.5%. r = 0.08.
What is the 4-year prepaid forward price?
A. 113
B. 115
C. 117
D. 119
E. 121
1.3 (1 point) A stock has a two-year forward price of 99.66.
The stock pays dividends at a continuously compounded rate of 3%. r = 7%.
What is the current price of this stock?
A. 90
B. 92
C. 94
D. 96
E. 98
1.4 (1 point) A stock has a current price of 90.
The stock pays dividends at a continuously compounded rate of 2%. r = 0.06.
What is the 5-year forward price?
A. 100
B. 105
C. 110
D. 115
E. 120
1.5 (1 point) A stock has a current price of $100.
In 3 months the stock will pay a dividend of $2. r = 0.04.
What is the 4-month prepaid forward price?
1.6 (1 point) A stock has a four-year forward price of 89.43.
The stock pays dividends at a continuously compounded rate of 0.8%. r = 5.2%.
What is the current price of this stock?
A. 65
B. 70
C. 75
D. 80
E. 85
1.7 (5B, 11/98, Q.10) (1 point) Which of the following are true regarding financial derivatives?
1. Firms typically issue derivatives to raise money on short notice.
2. A forward contract may be traded on an organized exchange.
3. A warrant is a derivative.
A. 1
B. 3
C. 1, 3
D. 2, 3
E. 1, 2, 3
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 19
1.8 (FM Sample Exam, Q.4) Zero-coupon risk-free bonds are available with the following
maturities and yield rates (effective, annual):
Maturity (years)
Yield
1
0.06
2
0.065
3
0.07
You need to buy corn for producing ethanol. You want to purchase 10,000 bushels one year from
now, 15,000 bushels two years from now, and 20,000 bushels three years from now. The current
forward prices, per bushel, are 3.89, 4.11, and 4.16 for one, two, and three years respectively.
You want to enter into a commodity swap to lock in these prices.
Which of the following sequences of payments at times one, two, and three will NOT be acceptable
to you and to the corn supplier?
A. 38,900, 61,650, 83,200
B. 39,083, 61,650, 82,039
C. 40,777, 61,166, 81,554
D. 41,892, 62,340, 78,997
E. 60,184, 60,184, 60,184
1.9 (FM Sample Exam, Q.6) The current price of one share of XYZ stock is 100. The forward
price for delivery of one share of XYZ stock in one year is 105. Which of the following statements
about the expected price of one share of XYZ stock in one year is TRUE?
A. It will be less than 100
B. It will be equal to 100
C. It will be strictly between 100 and 105
D. It will be equal to 105
E. It will be greater than 105.
1.10 (FM Sample Exam, Q.7) A non-dividend paying stock currently sells for 100.
One year from now the stock sells for 110.
The risk-free rate, compounded continuously, is 6%.
The stock is purchased in the following manner:
• You pay 100 today
• You take possession of the security in one year
Which of the following describes this arrangement?
A. Outright purchase
B. Fully leveraged purchase
C. Prepaid forward contract
D. Forward contract
E. This arrangement is not possible due to arbitrage opportunities
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 20
1.11 (FM Sample Exam, Q.8) You believe that the volatility of a stock is higher than indicated by
market prices for options on that stock. You want to speculate on that belief by buying or selling
at-the-money options. What should you do?
A. Buy a strangle
B. Buy a straddle
C. Sell a straddle
D. Buy a butterfly spread
E. Sell a butterfly spread
1.12 (CAS3, 11/07, Q.25) (2.5 points) On January 1, 2007, the Florida Property Company
purchases a one-year property insurance policy with a deductible of $50,000. In the event of a
hurricane, the insurance company will pay the Florida Property Company for losses in excess of the
deductible. Payment occurs on December 31, 2007. For the last three months of 2007, there is a
20% chance that a single hurricane occurs and an 80% chance that no hurricane occurs. If a hurricane
occurs, then the Florida Property Company will experience $1,000,000 in losses. The continuously
compounded risk-free rate is 5%. On October 1, 2007, what is the risk-neutral expected value of
the insurance policy to the Florida Property Company?
A. Less than $185,000
B. At least $185,000, but less than $190,000
C. At least $190,000, but less than $195,000
D. At least $195,000, but less than $200,000
E. At least $200,000
1.13 (8, 5/09, Q.21) (2.25 points) Given the following information about a box spread related to
1,000 shares of Company XYZ stock:
• The current price of Company XYZ's stock is $100.
• The strike prices of the European call options underlying the box spread are $110 and $120.
• The time to maturity of the box spread is 1 year.
• The continuously compounded risk-free rate is 5% per annum.
Investor A is willing to purchase the box spread from you for $9,750.
Investor B is willing to sell the box spread to you for $9,750.
Assume there are no taxes or transaction costs and you can borrow or lend at the risk-free rate.
a. (1.5 point) Explain whether you purchase or sell the box spread.
Calculate the profit you earn. Show all work.
b. (0.75 point) Assume the options underlying the box spread are American instead of
European. The investor with whom you entered into the box spread transaction in part a.
above believes the price of Company XYZ will not decrease.
Explain the investor's expected actions immediately after entering the box spread transaction
with you.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 21
Section 2, European Options
There are various types of options. The simplest and the most important type for this exam are
European Options. The adjective “European” does not refer to where the option is bought.
Call Options:
ABC Stock is currently selling for $100. Dick buys from Jane an option to buy one year from today
a share of ABC Stock for $150.36 If one year from now ABC Stock has a market price of more than
$150 dollars, then Dick should use this option to buy a share of ABC Stock from Jane at $150. Dick
could then sell this share of ABC Stock for the market price and make a profit.
This is an example of a European Call Option.
A European Call Option gives the buyer the right to buy one share of a certain stock at a
strike price (exercise price) upon expiration. A European option may only be exercised
on one specific day.37 A call is an option to buy.
Dick has purchased a 1 year European Call Option on ABC Stock, with a strike price of $150.
Payoff on a Call Option:
The eventual value to Dick of this option, depends on the price of ABC Stock one year from now.
For example, if ABCʼs market price turned out to be $180 per share one year from today, then Dick
could buy a share of ABC from Jane for the $150 strike price, and turn around and sell that share for
$180. Dick would make a profit of $30, ignoring what he originally paid Jane to buy the option.38
If the future price of ABC is $150 or less, then Dick would not exercise his option.39 In that case, his
option turns out to have no value to Dick.
Future Price of ABC
$120
$140
$160
$180
36
Payoff on the Option to Dick
0
0
$10
$30
While for simplicity I have used in the example one share, one could buy an option for 100 shares or 1000 shares.
There are other exercise styles. See page 32 of Derivatives Markets by McDonald.
Others will be discussed subsequently.
38
And ignoring any transaction costs.
39
Dick has not agreed to buy a share from Jane. Dick does not have an obligation to buy, rather Dick has purchased
the right to buy a share if he wishes to.
37
Financial Economics,
HCMSA-2012-MFE/3F,
12/14/11,
Page 22
When the stock price is greater than the strike price, exercising the option makes money; this call
option is “in the money”. If the stock price and strike price are equal, then the option is “at the
money.” If the stock price is less than the strike price, then the call option is “out of the money.”
⎧Y if Y ≥ 0 40
Let Y+ = Max[0, Y] = ⎨
.
⎩0 if Y < 0
Then the eventual payoff on Dickʼs call option is (S1 - 150)+, where S1 is the price of ABC Stock
one year from now. Here is a graph of the future value of Dickʼs call option:
Call Payoff
140
120
100
80
60
40
20
50
100
150
200
250
300
Stock Price
In general, the future value of a European call option is: (ST - K)+, where ST is the price of the
stock on the expiration date of the call and K is the strike price of the call.
Put Options:
XYZ Stock is currently selling for $200. Mary buys from Rob an option to sell one year from today
a share of XYZ Stock for $250. If one year from now XYZ Stock has a market price of less than
$250, then Mary should buy a share of XYZ Stock at the market price and then use her option to
sell a share of XYZ Stock to Rob for $250, making a profit.
This is an example of a European Put Option.
A European Put Option gives the buyer the right to sell one share of a certain stock at a
strike price (exercise price) upon expiration. A put is an option to sell.
Mary has purchased a 1 year European Put Option on XYZ Stock, with a strike price of $250.
40
This very useful actuarial notation is not on the syllabus of this exam.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 23
Payoff on a Put Option:
The eventual payoff to Mary of her option, depends on the price of XYZ Stock one year from now.
For example, if XYZ market price turned out to be $220 per share one year from today, then Mary
could buy a share of XYZ for $220 and turn around and sell that share for $250 to Rob. Mary would
make a profit of $30, ignoring what she originally paid Rob to buy the option.41
If the future price of XYZ Stock is $250 or more, then Mary would not exercise her option.42 In this
case, her option turns out to have no value to Mary.
Future Price of XYZ
$220
$240
$260
$280
Payoff on the Option to Mary
$30
$10
0
0
The eventual value of Maryʼs put option is (250 - S1 )+, where S1 is the price of XYZ Stock one
year from now.43 Here is a graph of the future value of Maryʼs put option:
Put Payoff
250
200
150
100
50
100
200
300
400
500
Stock Price
In general, the future value of a European put option is: (K - ST ) +, where ST is the price of the
stock on the expiration date and K is the strike price.
41
And ignoring any transaction costs.
Mary has not agreed to sell a share to Rob. Mary does not have an obligation to sell, rather Mary has purchased the
right to sell a share if she wishes to.
43
Y+ is 0 if Y < 0, and Y is Y ≥ 0.
42
Financial Economics,
HCMSA-2012-MFE/3F,
12/14/11,
Page 24
When the stock price is less than the strike price, exercising the put makes money; this put option is
in the money. If the stock price and strike price are equal, then the option is at the money.
If the stock price is greater than the strike price, then the put option is out of the money.
Expected Future Value of an Option:
The future payoff on a European call option is: (ST - K)+, where ST is the price of the stock on the
expiration date and K is the strike price. The future payoff on a European put option is: (K - ST)+.
Of course, at the time one could purchase an option, one does not know the future price of the stock.
The future price of the stock is a random variable. The expected value of the option can be obtained
by averaging using the distribution of future stock prices.
The expected future value of a European call option is: E[(ST - K)+].
The expected future value of a European put option is: E[(K - ST)+].
If one knew the distribution of ST, then
E[(ST - K)+] = E[ST - K | ST > K] Prob[ST > K] + (0)Prob[ST ≤ K]
= (E[ST | ST > K] - K) Prob[ ST > K].
Similarly,
E[(K - ST)+] = (0)Prob[ST > K] + E[K - ST | ST ≤ K] Prob[ ST ≤ K]
= (K- E[ST | St ≤ K]) Prob[ ST ≤ K].
Limited Expected Values:44
Let X ∧ K = Min[X, K].
Then the limited expected value is: E[X
E[(ST - K)+] = E[ST] - E[ST
E[(K - ST)+] = K - E[ST
∧
∧
∧
K].
K].
K].
This manner of writing the expected future value can be useful if for example the distribution of future
prices is LogNormal and if one had a formula for the limited expected value of a LogNormal
Distribution.45
44
45
See “Mahlerʼs Guide to Loss Distributions” or Loss Models, covering material on the syllabus of Exam 4/C.
If the distribution of St were LogNormal, this would lead to the Black-Scholes formula for valuing a put option.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 25
Problems:
Use the following information for the next 3 questions:
The price of the stock of the Daily Planet Media Company 1 year from now has the following
distribution:
Price Probability
60 20%
80 30%
100 30%
120 20%
2.1 (1 point) Determine the expected stock price of Daily Planet Media Company one year from
now.
2.2 (1 point) Determine the expected payoff of a 1 year European call option on one share of Daily
Planet Media Company, with a strike price of 85.
(A) Less than 7
(B) At least 7, but less than 9
(C) At least 9, but less than 11
(D) At least 11, but less than 13
(E) At least 13
2.3 (1 point) Determine the expected payoff of a 1 year European put option on one share of Daily
Planet Media Company, with a strike price of 85.
A) Less than 7
(B) At least 7, but less than 9
(C) At least 9, but less than 11
(D) At least 11, but less than 13
(E) At least 13
2.4 (2 points) Graph the future value of a European call option with a strike price of 100, as a function
of the future stock price.
2.5 (2 points) Graph the future value of a European put option with a strike price of 100, as a
function of the future stock price.
2.6 (2 points) Graph the payoff on a European call option with a strike price of 100 plus the
corresponding put, as a function of the future stock price. This position is called a straddle.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 26
2.7 (3 points) Christopher buys a $60 strike European call, sells two $70 strike European calls,
and buys an $80 strike European call.
The options are on the same stock and have the same expiration date.
This position is called a Butterfly Spread.
Graph the payoff on this portfolio as a function of the future price of the stock.
2.8 (2 points) Jason buys a $100 strike European put, and sells a $120 strike European put.
The puts are on the same stock and have the same expiration date.
This position is called a Put Bull Spread.
Graph the payoff on this portfolio as a function of the future price of the stock.
2.9 (3 points) Melissa buys a $90 strike European call, sells a $90 strike European put,
sells a $130 strike European call, and buys a $130 strike European put.
The options are on the same stock and have the same expiration date.
This position is called a Box Spread.
Graph the payoff on this portfolio as a function of the future price of the stock.
2.10 (3 points) Amanda buys two $100 strike European call, sells three $110 strike European calls,
and buys a $130 strike European call.
The options are on the same stock and have the same expiration date.
This position is called a Asymmetric Butterfly Spread.
Graph the payoff on this portfolio as a function of the future price of the stock.
2.11 (3 points) Robert buys 1000 calls on a stock with a strike price of $120.
The premium per call is $8. Robert also pays a total commission of $100.
Determine the stock price at expiration at which Robert will break even.
Graph Robertʼs profit as a percent of his initial investment, as a function of the stock price at
expiration of the call. (Ignore the time value of money.)
2.12 (2 points) Tiffany buys a $90 strike European call and sells a $90 strike European put.
The options are on the same stock and have the same expiration date.
Graph the payoff on this portfolio as a function of the future price of the stock.
2.13 (3 points) Heather buys a $70 strike European put and sells a $90 strike European call.
The options are on the same stock and have the same expiration date.
This position is called a Collar.
Graph the payoff on this portfolio as a function of the future price of the stock.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 27
2.14 (1 point) ABC stock costs $78.
ABC stock does not pay dividends.
Harold enters into a long position on a $80-strike two-year European call on ABC,
and enters into a short position on a $80-strike two-year European put on ABC.
Harold pays a net of $6.33 for entering these positions.
What is the continuously compounded risk free rate?
A. 5.0%
B. 5.5%
C. 6.0%
D. 6.5%
E. 7.0%
2.15 (3 points) Allen buys a $70 strike European put, sells four $100 strike European puts,
and buys three $110 strike European puts.
The options are on the same stock and have the same expiration date.
This position is called a Asymmetric Butterfly Spread.
Graph the payoff on this portfolio as a function of the future price of the stock.
2.16 (3 points) Kimberly buys 100 puts on a stock with a strike price of $80.
The premium per put is $5. Kimberly also pays a total commission of $60.
Determine the stock price at expiration at which Kimberly will break even.
Graph Kimberlyʼs profit as a percent of her initial investment, as a function of the stock price at
expiration of the put. (Ignore the time value of money.)
2.17 (2 points) Nicholas buys a share of stock, sells a $110 strike European call on that stock, and
buys a $110 strike European put on that stock. The options have the same expiration date.
Graph the value of this portfolio when the options expire as a function of the future price of the stock.
2.18 (2 points) Let S(t) be the price of a stock at time t.
The stock pays dividends at the continuously compounded rate δ.
The continuously compounded risk free rate is r.
Assume a contract is purchased at time 0 and pays at time T: Max[S(T), 100 ].
Determine the premium for this contract in terms of the premium of a European option and other
known quantities.
2.19 (2 points) Kevin writes (sells) a 60 strike call and a 60 strike put. The options have the same
expiration date. Graph the value of this portfolio when the options expire as a function of the future
price of the stock. This position is called a written straddle.
2.20 (2 points) Aaron owns a share of stock of the Charming Prints Company.
The current price of Charming Prints Company stock is $100.
Briefly discuss why might Aaron buy a Collar.
Financial Economics,
HCMSA-2012-MFE/3F,
12/14/11,
Page 28
2.21(1 point) Several years ago Warren bought 1000 shares of XYZ stock.
Fortunately for Warren the value of XYZ stock has increased substantially since then.
However, for tax reasons Warren does not wish to sell his XYZ stock and realize his capital gains.
Rather Warren plans to sell his XYZ stock one year from now.
Warren is worried that by time he is ready to sell his stock his capital gains may decrease or vanish.
Briefly describe how Warren could purchase a European option to hedge this risk.
2.22 (2 points) Lauren buys a 70 strike put and a 90 strike call. The options have the same
expiration date. Graph the value of this portfolio when the options expire as a function of the future
price of the stock. This position is called a strangle.
2.23 (2 points) Let Y+ equal the maximum of Y and zero.
Let S and Q be two random variables.
(a) Determine (S - Q)+ + (Q - S)+.
(b) Determine (S - Q)+ - (Q - S)+.
2.24 (1 point) The Rich and Fine Stock Index has a current price of 800.
An insurer offers a contract that will pay the value of the Rich and Fine Stock Index two years from
now; however, the contract will pay a minimum of 750. The insurer buys the index.
Briefly describe how the insurer could purchase a European option to hedge its risk.
2.25 (5B, 11/98, Q.15) (1 point) What combination of stocks, options and borrowing/lending could
be represented by the following position diagram?
Valueof Position
50
100
150
200
Share Price
- 20
- 40
- 60
- 80
- 100
1. Sell one share of stock short and borrow the present value of $100.
2. Sell one call with exercise price of $100 and sell one put with exercise price of $100.
3. Sell one share of stock short, sell two puts with exercise price of $100, and lend the present
value of $100.
A. 1
B. 2
C. 3
D. 2, 3
E. 1, 2, 3
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 29
2.26 (5B, 11/99, Q.30) (2 points) ABC Insurance Company has purchased a reinsurance contract
from Reliable Reinsurer providing coverage for $10 million in excess of $20 million. In other words,
Reliable Reinsurer has agreed to pay up to, but no more than, $10 million beyond the initial $20
million in loss dollars retained by ABC.
a. (1 point) Draw a position diagram showing the payoff to ABC from the reinsurance as a function of
the amount of ABC's total loss. Label both axes.
b. (1 point) If we think of ABC's total loss as the "underlying asset," we can model this reinsurance
contract as a mixture of simple options.
Describe the option position that replicates the payoffs from the reinsurance contract.
2.27 (5B, 11/99, Q.31) (2 points) Norbert Corporation owns a vacant lot with a book value of
$50,000. By a stroke of luck, Norbert finds a buyer willing to pay $200,000 for the lot. However,
Norbert must also give the buyer a put option to sell the lot back to Norbert for $200,000 at the end
of two years. Moreover, Norbert agrees to pay the buyer $40,000 for a call option to repurchase
the lot for $200,000 at the end of two years.
a. (1 point) What would likely happen if the lot is worth more than $200,000 at the end of
two years? What if it is worth less than $200,000? Why?
b. (1 point) In effect, Norbert has borrowed money from the buyer.
What is the effective annual interest rate per year on the loan? Show all work.
2.28 (FM Sample Exam, Q.3) Happy Jalapenos, LLC has an exclusive contract to supply
jalapeno peppers to the organizers of the annual jalapeno eating contest. The contract states that
the contest organizers will take delivery of 10,000 jalapenos in one year at the market price. It will
cost Happy Jalapenos 1,000 to provide 10,000 jalapenos and todayʼs market price is 0.12 for one
jalapeno. The continuously compounded risk-free interest rate is 6%.
Happy Jalapenos has decided to hedge as follows (both options are one-year, European):
Buy 10,000 0.12-strike put options for 84.30 and sell 10,000 0.14-strike call options for 74.80.
Happy Jalapenos believes the market price in one year will be somewhere between 0.10 and
0.15 per pepper. Which interval represents the range of possible profit one year from now for
Happy Jalapenos?
A. –200 to 100
B. –110 to 190
C. –100 to 200
D. 190 to 390
E. 200 to 400
Financial Economics,
HCMSA-2012-MFE/3F,
12/14/11,
Page 30
2.29 (FM Sample Exam, Q.9) You are given the following information:
• The current price to buy one share of ABC stock is 100
• The stock does not pay dividends
• The risk-free rate, compounded continuously, is 5%
• European options on one share of ABC stock expiring in one year have the following
prices:
Strike Price
Call option price
Put option price
90
14.63
0.24
100
6.80
1.93
110
2.17
6.81
A butterfly spread on this stock has the following profit diagram.
6
4
2
85
90
95
100
105
110
115
120
-2
Which of the following will NOT produce this profit diagram?
A. Buy a 90 put, buy a 110 put, sell two 100 puts
B. Buy a 90 call, buy a 110 call, sell two 100 calls
C. Buy a 90 put, sell a 100 put, sell a 100 call, buy a 110 call
D. Buy one share of the stock, buy a 90 call, buy a 110 put, sell two 100 puts
E. Buy one share of the stock, buy a 90 put, buy a 110 call, sell two 100 calls.
Financial Economics,
HCMSA-2012-MFE/3F,
12/14/11,
Page 31
Section 3, Properties of Premiums of European Options
The “premium” of an option is its current price, the actuarial present value of its future payoffs.
Actuarial Present Value of a Call Option:
The actuarial present value or premium of a European call option is: E[(ST - K)+ ] e-rT.
If we let C(S0 , K, T) be the actuarial present value of the call on a stock with current price S0 ,
strike price K, and time until expiration T: CEur(S0 , K, T) = E[(ST - K)+] e-Tr.
Exercise: The price of a stock two years from now has the following distribution:
$50 @ 20%, $100 @ 40%, $150 @30%, $200 @10%.
The continuously compounded annual risk free rate is: r = 4%.
Determine the actuarial present value of a European call option on this stock with 2 years to
expiration and a strike price of $90.
[Solution: e-0.08 {(20%)(0) + (40%)(10) + (30%)(60) + (10%)(110)} = $30.46.]
Exercise: In the previous exercise, change the strike price to $100.
[Solution: e-0.08 {(20%)(0) + (40%)(0) + (30%)(50) + (10%)(100)} = $23.08.]
The actuarial present value of this European call option as a function of the strike price:
PV
100
80
60
40
20
50
100
150
200
K
The actuarial present value, in other words premium, of the call decreases as the strike
price increases and the curve is concave upwards.
These are general properties for the behavior of a call as one varies the strike price.46
46
See pages 292, 299, and 300 in Derivative Markets by McDonald.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 32
General Properties of European Call Premiums:
The value of an call option with strike price $90 is greater than an otherwise similar call with a strike
price of $100. It is less valuable to have the option to buy something at $100 than it is to have the
option to buy that same thing at $90. Call premiums decrease as the strike price increases:
∂C
≤ 0.
∂K
For K1 < K2 , C(K1 ) ≥ C(K2 ).47 48
For example, assume the future price of a stock at expiration of a call has the following distribution:
$50 @ 20%, $100 @ 40%, $150 @30%, $200 @10%.
Stock
Price
50
100
150
200
Payoff on Call
with K = 90
0
10
60
110
Payoff on Call
with K = 100
0
0
50
100
Difference
0
10
10
10
The difference in payoffs is at most the difference in strike prices, 10.
Being able to buy at $90 is worth at most $10 more than being able to buy at $100.49
Therefore the difference in call premiums is at most the difference in strike prices:
For K1 < K2 , C(K1 ) - C(K2 ) ≤ K2 - K1 .50 51
∂C
≥ -1.
∂K
Since European options can only be exercised at expiration, the difference in option premiums
cannot be more than the present value of the difference in payoffs. Thus:
For K1 < K2 , C(K1 ) - C(K2 ) ≤ e-rT (K2 - K1 ).52
47
See equation 9.13 in Derivative Markets by McDonald.
This equation holds both for European and American options, to be discussed subsequently.
49
Sometimes being able to buy at $90 and being able to buy at $100 are both worth nothing.
50
See equation 9.15 in Derivative Markets by McDonald.
51
This equation holds both for European and American options, to be discussed subsequently.
52
See Appendix 9.B in McDonald, not on the syllabus.
48
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 33
The third property is referred to as the convexity of the option price with respect to the strike price.53
For K1 < K2 < K3 ,
C(K1 ) - C(K2 ) C(K2 ) - C(K3 ) 54 55
≥
.
K 2 - K1
K 3 - K2
∂2C
≥ 0.
∂K2
For example, let us assume that a call with strike price $90 is worth $7 more than a similar call with a
strike price of $100. Then a call with a strike price of $100 exceeds that of a similar call with a strike
price of $110, but by less than or equal to $7.
Convexity follows from the fact that the second derivative of the call premium with respect to K is
positive:
∂2C
≥ 0. C decreases as K increases and the slope is negative; as K increases the slope
∂K2
increases, in other words gets closer to zero. As K decreases the slope decreases; however, the
slope can not get less than -1.
As mentioned before, the graph of C as a function of K is concave upwards.
53
A convex function has a curve that is concave upwards, shaped like a bowl.
See equation 9.17 in Derivative Markets by McDonald. This equation holds both for European and American
options, to be discussed subsequently.
55
Increased Limits Factors share this same property for the same underlying mathematical reason.
See Sheldon Rosenbergʼs review of “”On the Theory of Increased Limits and Excess of Loss Pricing,” PCAS 1977.
54
Financial Economics,
HCMSA-2012-MFE/3F,
12/14/11,
Page 34
Mathematically the three conditions for call premiums could be summarized as:
-1 ≤
∂C
∂2C
≤ 0, and
≥ 0.
∂K
∂K2
This is illustrated in the following graph:56
C
100
80
60
40
20
50
100
150
200
K
As K approaches zero, the slope of the above curve approaches -e-rT, which is close to but more
than minus one. As K approaches infinity, the slope of the above curve approaches 0.
56
The put premiums are computed via the Black-Scholes formula to be discussed subsequently.
This is for a 2-year European call, and σ = 40%, r = 6%, δ = 0.
Financial Economics,
HCMSA-2012-MFE/3F,
12/14/11,
Page 35
Explaining the Behavior of the Value of Calls as a Function of K:
Let F(S) be the distribution of the future price of the stock.
The expected value of the call is:
∞
E[(ST - K)+] =
∫ {1 - F(x)} dx .
57
K
∞
Therefore, C(K) =
e-rT
∫ {1 - F(x)} dx .
K
∂C
= -e-rT{1 - F(K)} ≤ 0.
∂K
Therefore, C declines as K increases.
C(K1 ) - C(K2 ) =
e-rT
K2
∫ {1 -
F(x)} dx ≤ e-rT(K2 - K1 ) {1 - F(K1 )} ≤ e-rT(K2 - K1 ) ≤ K2 - K1 .
K1
∂2C
= e-rT f(K) ≥ 0.
∂K2
Therefore, C(K) is concave upwards.
57
See “Mahlerʼs Guide to Loss Distributions,” covering material on Exam 4/C.
This is an expression for the expected excess losses.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 36
Lee Diagrams and Call Premiums:58
One can present the same ideas graphically via “Lee Diagrams”.59 Lee Diagrams have the x-axis
correspond to probability, while the y-axis corresponds to size of loss. Here we will graph the
distribution function of the future price of the stock, with the horizontal axis corresponding to
probability and the vertical axis corresponding to the stock price at expiration of the option.
The expected payoff of a European Call, is equal to E[(ST - K)+].
E[(X - K)+] is the expected losses excess of K, and corresponds to the area on the Lee Diagram
above the horizontal line at height K and also below the curve graphing F(x).
As K increases, the area above the horizontal line at height K decreases; in other words, the value of
the call decreases as K increases.
58
Not on the syllabus of your exam!
See “The Mathematics of Excess of Loss Coverage and Retrospective Rating --- A Graphical Approach”,
by Y.S. Lee, PCAS LXXV, 1988. Currently on the syllabus of CAS Advanced Ratemaking Exam.
See also “Mahlerʼs Guide to Loss Distributions,” covering material on Exam 4/C.
59
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 37
For an increase in K of ΔK, the value of the call decreases by Area A in the following Lee Diagram:
Stock Price
K + ΔK
A
K
1
P r o b.
The absolute value of the change in the value of the call, Area A, is smaller than a rectangle of height
ΔK and width 1 - F(K). Thus Area A is smaller than ΔK {1 - F(K)} ≤ ΔK. Thus a change of ΔK in the
strike price results in a absolute change in the value of the call option smaller than ΔK.
The following Lee Diagram shows the effect of raising the strike price by fixed amounts:
The successive absolute changes in the value of the call are represented by Areas A, B, C, and D.
We see that the absolute changes in the value of the call get smaller as the strike price increases.
Financial Economics,
HCMSA-2012-MFE/3F,
12/14/11,
Page 38
Actuarial Present Value of a Put Option:
The actuarial present value of a European put option is: E[(K - ST ) + ] e-rT.
If we let P(S0 , K, T) be the actuarial present value of the put on a stock with current price S0 ,
strike price K, and time until expiration T: PEur(S0 , K, T) = E[(K - ST)+] e-Tr.
Exercise: The price of a stock two years from now has the following distribution:
$50 @ 20%, $100 @ 40%, $150 @30%, $200 @10%.
r = 4%. Determine the actuarial present value of a European put option on this stock with 2 years to
expiration and a strike price of $120.
[Solution: e-0.08 {(20%)(70) + (40%)(20) + (30%)(0) + (10%)(0)} = $20.31.]
The actuarial present value of the above European put option as a function of the strike price:
PV
80
60
40
20
60
80
100
120
140
160
180
200
K
The actuarial present value of the put increases as the strike price increases and the
curve is concave upwards.
These are general properties for the behavior of a put as one varies the strike price.60
60
See pages 292, 299, and 300 in Derivative Markets by McDonald.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 39
General Properties of European Put Premiums:
For K1 < K2 , P(K1 ) ≤ P(K2 ) .61 62
∂P
≥ 0.
∂K
For example, the value of an put option with strike price $100 is greater than an otherwise similar put
with a strike price of $90. It is more valuable to have the option to sell at $100 than to have the
option to sell at $90.
For K1 < K2 , P(K2 ) - P(K1 ) ≤ K2 - K1 .63 64
∂P
≤ 1.
∂K
For example, the value of a put option with strike price $100 is greater than an otherwise similar put
with a strike price of $90 by at most $10. Being able to sell at $100 is worth at most $10 more than
being able to sell at $90.65
For K1 < K2 < K3 ,
P(K2 ) - P(K1 ) P(K3 ) - P(K2 ) 66
≤
.
K 2 - K1
K 3 - K2
∂2P
≥ 0.
∂K2
For example, let us assume that a put with strike price $100 is worth $6 more than a similar put with a
strike price of $90. Then a put with a strike price of $110 exceeds that of a similar put with a strike
price of $100 by at least $6.
This is referred to as the convexity of the option price with respect to the strike price.67 It follows from
the fact that the second derivative of the value of the put with respect to K is positive:
∂2P
≥ 0.
∂K2
P increases as K increases and the slope is positive; as K increases the slope increases, in other
words gets further from zero. However, the slope can not exceed one.
As mentioned before, the graph of P as a function of K is concave upwards.68
61
See equation 9.14 in Derivative Markets by McDonald.
This equation holds both for European and American options, to be discussed subsequently.
63
See equation 9.16 in Derivative Markets by McDonald.
64
This equation holds both for European and American options, to be discussed subsequently. In the case of
European options, the righthand side could be instead the present value of the difference in strike prices.
65
Sometimes being able to sell at $90 and being able to sell at $100 are both worth nothing.
66
See equation 9.18 in Derivative Markets by McDonald. This equation holds both for European and American
options, to be discussed subsequently.
67
A convex function has a curve that is concave upwards, shaped like a bowl.
68
The graph of a function is concave upwards if it is shaped like a bowl. If the second derivative is positive, then the
graph of a function is concave upwards. A convex function is such that for any 0 ≤ t ≤ 1, and x ≤ y,
f(tx + (1-t)y) ≤ tf(x) + (1 -t)f(y). A convex function has a graph that is concave upwards.
62
Financial Economics,
HCMSA-2012-MFE/3F,
12/14/11,
Page 40
Mathematically the three conditions for put premiums could be summarized as:
1≥
∂P
∂2P
≥ 0, and
≥ 0.
∂K
∂K2
This is illustrated in the following graph:69
P
15
10
5
40
60
80
100
K
As K approaches infinity, the slope of the above curve approaches e-rT, which is close to but less
than one. As K approaches zero, the slope of the above curve approaches 0.
69
The put premiums are computed via the Black-Scholes formula to be discussed subsequently.
This is for a 2-year European put, and σ = 40%, r = 6%, δ = 0.
Financial Economics,
HCMSA-2012-MFE/3F,
12/14/11,
Explaining the Behavior of the Value of Puts as a Function of K:
As before, let F(S) be the distribution of the future price of the stock.
The expected value of the put is:70
K
E[(K - ST)+] =
∫ F(x) dx .
0
K
Therefore, P(K) = e-rT
∫ F(x) dx .
0
∂P
= e-rT F(K) ≥ 0.
∂K
Therefore, P increases as K increases.
P(K2 ) - P(K1 ) =
e-rT
K2
∫ F(x) dx ≤ e-rT(K2 - K1) F(K1) ≤ e-rT(K2 - K1) ≤ K2 - K1.
K1
∂2P
= e-rT f(K) ≥ 0.
∂K2
Therefore, P(K) is concave upwards.
70
See “Mahlerʼs Guide to Loss Distributions,” covering material on Exam 4/C.
This is an expression for the expected amount by which X is less than K.
Page 41
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 42
Lee Diagrams and Put Premiums:71
The expected payoff of a European put is E[(K - ST)+], which corresponds to Area P below the
horizontal line at height K and also above the curve graphing F(x) in the following Lee Diagram:
As K increases, the area below the horizontal line at height K increases; in other words, the value of
the put increases as K increases.
71
Not on the syllabus of your exam!
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 43
For an increase in K of ΔK, the value of the put increases by Area A in the following Lee Diagram:
The change in the value of the put, Area A, is smaller than a rectangle of height ΔK and width
F(K +ΔK). Thus Area A is smaller than ΔK F(K +ΔK) ≤ ΔK. Thus a change of ΔK in the strike price
results in a change in the value of the put option smaller than ΔK.
The following Lee Diagram shows the effect of raising the strike price by fixed amounts:
The successive changes in the value of the put are represented by Areas A, B, C, and D.
We see that the changes in the value of the put get larger as the strike price increases.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 44
Arbitrage:
If there is a possible combination of buying and selling with no net investment that has
no risk but generates positive (or at least nonnegative) cashflows, this is an arbitrage
opportunity. Taking advantage of such an opportunity is called arbitrage. In other words,
arbitrage is free money.
If an arbitrage opportunity existed, clever traders would take advantage of it. Relatively quickly, the
prices would adjust so as to remove this opportunity for arbitrage.
Generally, we assume prices should be such that they do not permit arbitrage. In other words, we
assume that there is no free lunch. This is called “no-arbitrage” pricing.
Arbitrage Opportunities with Two Calls:
If one of the general properties of option premiums that has been discussed is violated, that creates
an opportunity for arbitrage.
For example, two otherwise similar calls have the following premiums:
100 strike call costs 10
110 strike call costs 12.
This violates the principal that call premiums should not increase as the strike price increases.
The 100 strike call is cheap relative to the 110 strike call; the 110 strike call is expensive relative to
the 100 strike call.
We can buy a 100 strike call and sell a 110 strike call.
We make 12 - 10 = 2 from this set of transactions.
We invest the 2 at the risk free rate, r.
At expiration of the calls at time T, we have:
If S ≤ 100: 2erT > 0
If 110 ≥ S > 100: (S - 100) + 2erT > 0.
If S > 110: (S - 100) - (S - 110) + 2erT = 10 + 2erT > 0.
Thus we always end up with a positive (or at least nonnegative) position, having taken no risk.
This demonstrates arbitrage.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 45
In another example, two otherwise similar calls have the following premiums:
100 strike call costs 20
105 strike call costs 13.
This violates the principal that call premiums should not decrease by more than the increase of the
strike price. The 105 strike call is cheap relative to the 100 strike call.
We can sell a 100 strike call and buy a 105 strike call.
We make 20 - 13 = 7 from this set of transactions.
We invest the 7 at the risk free rate, r.
At expiration of the calls at time T, we have:
If S ≤ 100: 7erT > 0
If 105 ≥ S > 100: -(S - 100) + 7erT ≥ 7erT - 5 > 0.
If S > 105: (S - 100) - (S - 105) + 7erT = 7erT - 5 > 0.
Thus we always end up with a positive (or at least nonnegative) position, having taken no risk.
This demonstrates arbitrage.
If the principals for put premiums are violated, one can set up similar opportunities for arbitrage to
those illustrated for calls.
Arbitrage Opportunities When Convexity is Violated by Call Premiums:
Three otherwise similar calls have the following premiums:
100 strike call costs 20
110 strike call costs 17.
140 strike call costs 7.
(20 - 17) / (110 - 100) = 0.3.
(17 - 7) / (140 - 110) = 0.333.
0.333 > 0.3. This violates the convexity of the call premium with respect to the strike price.
The arbitrage opportunity in such situations involves buying some of the low strike and high strike
calls, while selling some of the medium strike calls. While there are many possible positions that
demonstrate arbitrage when convexity is violated, McDonald has a technique of coming up with one
such a portfolio.
HCMSA-2012-MFE/3F,
Let λ =
Financial Economics,
12/14/11,
Page 46
K3 - K2 72
.
K 3 - K1
λ is the amount of the distance between the high and medium strike prices as a fraction of the
distance between the high and low strike prices. In this example, λ = (140 - 110) / (140 -100) = 3/4.
We note that K2 = λ K1 + (1 - λ)K3 , a weighted average of K1 and K3 with weights λ and 1 - λ.
In this example, 110 = (3/4)(100) + (1 - 3/4)(140).
The convexity relationship for call premiums was:
For K1 < K2 < K3 , {C(K1 ) - C(K2 )} / {K2 - K1 } ≥ {C(K2 ) - C(K3 )} / {K3 - K2 }.
⇒ {C(K1 ) - C(K2 )}{K3 - K2 } ≥ {C(K2 ) - C(K3 )}{K2 - K1 }.
⇒ (K3 - K2 )C(K1 ) + (K2 - K1 )C(K3 ) ≥ (K3 - K1 )C(K2 ). ⇒ C(K2 ) ≤ λ C(K1 ) + (1 - λ) C(K3 ).
For convexity to hold, we require C(K2 ) ≤ λ C(K1 ) + (1 - λ) C(K3 ).73
In this example, we require:
C(110) ≤ (3/4) C(100) + (1 - 3/4) C(140) = (3/4)(20) + (1/4)(7) = 16.75.
The height of the line between the points (K1 , C(K1 )) and (K3 , C(K3 )), at the strike price K is:
y = βC(K1 ) + (1 - β)C(K3 ), where β =
K3 - K
.
K3 - K 1
Thus, we require that the point (K2 , C(K2 )) not be above this line.
72
See equation 9.19 in Derivative Markets by McDonald.
It is somewhat arbitrary that McDonald has a numerator of K3 - K2 rather than K2 - K1 .
73
See equation 9.20 in Derivative Markets by McDonald.
Financial Economics,
HCMSA-2012-MFE/3F,
12/14/11,
Page 47
For this example, here is the line, as well as the premiums for the three calls:
C
20
17
7
100
110
140
K
Convexity is violated because the point (110, 17) is above the line.74
In other words, the curve of option premium as a function of strike price is not concave upwards.75
74
75
The point (110. 16.75) is on the line.
If concave upwards, the curve should be below any chord drawn between two points on the curve.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 48
In order to demonstrate arbitrage when convexity is violated, in general, one can buy λ of the
lowest strike call, buy 1 - λ of the high strike call, and sell 1 of the medium strike call.76 77
In this example, we buy 3/4 of the 100 strike calls, buy 1/4 of the 140 strike calls, and sell 1 of the
110 strike calls. Equivalently, we can buy 3 of the 100 strike calls, buy 1 of the 140 strike calls, and
sell 4 of the 110 strike calls.
When we set up this portfolio, we get: (-3)(20) + (4)(17) + (-1)(7) = 1.
We invest this 1 at the risk free rate r.
At expiration of the calls at time T, we have:
If S ≤ 100: erT > 0
If 110 ≥ S > 100: 3(S - 100) + erT > 0.
If 140 ≥ S > 110: 3(S - 100) - (4)(S - 110) + erT = 140 - S + erT > 0.
If S > 140: 3(S - 100) - (4)(S - 110) + (S - 140) + erT = erT > 0.
Thus we always end up with a positive (or at least nonnegative) position, having taken no risk.
This demonstrates arbitrage.78
Arbitrage Opportunities When Convexity is Violated by Put Premiums:
If convexity for put premiums are violated, one can set up similar opportunities for arbitrage.
For example, three otherwise similar puts have the following premiums:
80 strike put costs 12
100 strike put costs 18.
110 strike put costs 20.
(18 - 12) / (100 - 80) = 0.3.
(20 - 18) / (110 - 100) = 0.2.
0.2 < 0.3. This violates the convexity of the put premium with respect to the strike price.
As with calls, the arbitrage opportunity in such situations involves buying some of the low strike and
high strike option, while selling some of the medium strike option. While there are many possible
positions that demonstrate arbitrage when convexity is violated, one can use the same technique of
coming up with one such a portfolio as was discussed for calls.
76
One could multiply all of the amounts by a constant in order to make them integer.
Since convexity is violated, C(K2 ) > λ C(K1 ) + (1 -λ) C(K3 ). Therefore, the medium strike call is overpriced relative
to this weighted average of the low strike and high strike calls. Therefore, we sell the medium strike call.
78
Recall, that there are other portfolios that would also demonstrate arbitrage.
77
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 49
In this example, λ = (110 - 100) / (110 - 80) = 1/3.
Thus, we buy 1/3 of the 80 strike puts, buy 2/3 of the 110 strike puts,
and sell 1 of the 100 strike puts.
Equivalently, we can buy 1 of the 80 strike puts, buy 2 of the 110 strike puts,
and sell 3 of the 100 strike puts.
When we set up this portfolio, we get: (-1)(12) + (3)(18) + (-2)(20) = 2.
We invest this 2 at the risk free rate r. At time T we get back 2erT.
At expiration of the puts at time T, we have:
If S ≥ 110: 2erT > 0
If 110 > S ≥ 100: (2)(110 - S) + 2erT > 0.
If 100 > S ≥ 80: (2)(110 - S) - (3)(100 - S) + 2erT = S - 80 + erT > 0.
If S < 80: (2)(110 - S) - (3)(100 - S) + (80 - S) + 2erT = 2erT > 0.
Thus we always end up with a positive (or at least nonnegative) position, having taken no risk.
This demonstrates arbitrage.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 50
Chart of Some Arbitrage Opportunities:79
Strike Prices: K1 < K2 < K3 . C is the premium for a Call, while P is the premium for a put.
Condition
Arbitrage if the Condition is Violated
C(K1 ) ≥ C(K2 ).
Buy the K1 Call and Sell the K2 Call (Call Bull Spread)
C(K1 ) - C(K2 ) ≤ K2 - K1 .
Sell the K1 Call and Buy the K2 Call (Call Bear Spread)
C(K1) - C(K2 )
K2 - K1
Buy λ = (K3 - K2 )/(K3 - K1 ) of K1 , Sell 1 of K2 , Buy 1 - λ of K3 .80
≥
C(K2) - C(K3 )
.
K3 - K2
(Asymmetric butterfly spread)
P(K2 ) ≥ P(K1 ).
Sell the K1 Put and Buy the K2 Put (Put Bear Spread)
P(K2 ) - P(K1 ) ≤ K2 - K1 .
Buy the K1 Put and Sell the K2 Put (Put Bull Spread)
P(K2 ) - P(K 1)
K2 - K1
Buy λ = (K3 - K2 )/(K3 - K1 ) of K1 , Sell 1 of K2 , Buy 1 - λ of K3 .81
≤
79
P(K3 ) - P(K 2)
.
K3 - K2
(Asymmetric butterfly spread)
See page 300 of Derivative Markets by McDonald.
There are other sets of amounts of each call that would also demonstrate arbitrage.
81
There are other sets of amounts of each put that would also demonstrate arbitrage.
80
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 51
Problems:
Use the following information for the next 4 questions:
The price of the stock of the Willy Wonka Chocolate Company 6 months from now has the
following distribution:
Price Probability
100 20%
150 40%
200 30%
250 10%
The continuously compounded annual rate of interest is 5%.
3.1 (1 point) Determine the expected payoff of a 6 month European call option on one share of
Willy Wonka Chocolate Company, with a strike price of 180.
A. 12
B. 13
C. 14
D. 15
E. 16
3.2 (1 point) Determine the expected payoff of a 6 month European put option on one share of
Willy Wonka Chocolate Company, with a strike price of 160.
A. 12
B. 13
C. 14
D. 15
E. 16
3.3 (1 point) Determine the actuarial present value of a 6 month European call option on one share
of Willy Wonka Chocolate Company, with a strike price of 170.
A. 16.0
B. 16.2
C. 16.4
D. 16.6
E. 16.8
3.4 (1 point) Determine the premium of a 6 month European put option on one share of Willy
Wonka Chocolate Company, with a strike price of 170.
A. 19.5
B. 20.0
C. 20.5
D. 21.0
E. 21.5
3.5 (1 point) 3 European put options on a stock are otherwise similar except for their strike price.
A put with a strike price of 150 has a premium of 30, in other words costs 30.
A put with a strike price of 160 has a premium of 34, in other words costs 34.
A put with a strike price of 180 has a premium of 40, in other words costs 40.
What general property of the value of puts is violated?
3.6 (3 points) Briefly describe an opportunity for arbitrage presented by the situation in the
previous question.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 52
3.7 (1 point) Two European put options on a stock are otherwise similar except for their strike price.
A put with a strike price of 60 has a premium of 7.
A put with a strike price of 80 has a premium of 15.
Which of the following is true about the premium of a similar 70 strike put?
A. The smallest possible premium is 8
B. The smallest possible premium is 11
C. The largest possible premium is 8
D. The largest possible premium is 11
E. None of A, B, C, or D
3.8 (1 point) Two European call options on a stock are otherwise similar except for their strike price.
A call with a strike price of 100 has a premium of 27, in other words costs 27.
A call with a strike price of 110 has a premium of 15, in other words costs 15.
What general property of the value of calls is violated?
3.9 (2 points) Briefly describe an opportunity for arbitrage presented by the situation in the
previous question.
3.10 (1 point) Two European call options on a stock are otherwise similar except for their strike price.
A call with a strike price of 80 has a premium of 12.
A call with a strike price of 85 has a premium of 10.
Which of the following is true about the premium of a similar 100 strike call?
A. The smallest possible premium is 2
B. The smallest possible premium is 4
C. The largest possible premium is 2
D. The largest possible premium is 4
E. None of A, B, C, or D
3.11 (1 point) 2 European put options on a stock are otherwise similar except for their strike price.
A put with a strike price of 150 has a premium of 22, in other words costs 22.
A put with a strike price of 170 has a premium of 46, in other words costs 46.
What general property of the value of puts is violated?
3.12 (2 points) Briefly describe an opportunity for arbitrage presented by the situation in the
previous question.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 53
3.13 (3 points) European call and put prices for options on a given stock are available as follows:
Strike Price Call Price
Put Price
$50
$21
$3
$60
$16
$7
$80
$8
$14
All six options have the same expiration date, which is no more than two years from now.
After reviewing the information above, Danielle, Eric, and Felicia agree that arbitrage opportunities
arise from these prices.
Danielle believes that one could use the following portfolio to obtain arbitrage profit: Long two calls
with strike price 50; short four calls with strike price 60; long two calls with strike price 80; and either
lend or borrow some money.
Eric believes that one could use the following portfolio to obtain arbitrage profit: Long two puts with
strike price 50; short three puts with strike price 60; long one put with strike price 80; and either lend
or borrow some money.
Felicia believes that one could use the following portfolio to obtain arbitrage profit: Short one call with
strike price 40; long two calls with strike price 50; short one call with strike price 80; long one put with
strike price 40; short two puts with strike price 50; long one put with strike price 80; and either lend or
borrow some money.
Which of the following statements is true?
(A) Danielle and Eric are correct, while Felicia is not.
(B) Danielle and Felicia are correct, while Eric is not.
(C) Eric and Felicia are correct, while Danielle is not.
(D) All of them are correct.
(E) None of A, B, C, or D.
3.14 (1 point) 3 European call options on a stock are otherwise similar except for their strike price.
A call with a strike price of 100 has a premium of 25, in other words costs 25.
A call with a strike price of 110 has a premium of 20, in other words costs 20.
A call with a strike price of 115 has a premium of 16, in other words costs 16.
What general property of the value of calls is violated?
3.15 (3 points) Briefly describe an opportunity for arbitrage presented by the situation in the
previous question.
3.16 (1 point) Two European call options on a stock are otherwise similar except for their strike price.
A call with a strike price of 120 has a premium of 22, in other words costs 22.
A call with a strike price of 125 has a premium of 16, in other words costs 16.
What general property of the value of calls is violated?
3.17 (2 points) Briefly describe an opportunity for arbitrage presented by the situation in the
previous question.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 54
3.18 (1 point) Two European call options on a stock are otherwise similar except for their strike price.
A call with a strike price of 80 has a premium of 12, in other words costs 12.
A call with a strike price of 90 has a premium of 14, in other words costs 14.
What general property of the value of calls is violated?
3.19 (2 points) Briefly describe an opportunity for arbitrage presented by the situation in the
previous question.
3.20 (2 points) Two European call options are otherwise similar except for their strike prices.
A call with a strike price of 85 has a premium of 13.
A call with a strike price of 90 has a premium of 7.
In order to take advantage of arbitrage, you buy 1000 of the 90 strike calls and sell x of the 85 strike
calls. If r = 0%, what is the smallest possible value of x?
A. 800
B. 825
C. 850
D. 875
E. 900
3.21 (MFE Sample Exam, Q.2) Near market closing time on a given day, you lose access to
stock prices, but some European call and put prices for a stock are available as follows:
Strike Price Call Price
Put Price
$40
$11
$3
$50
$6
$8
$55
$3
$11
All six options have the same expiration date.
After reviewing the information above, John tells Mary and Peter that no arbitrage opportunities can
arise from these prices.
Mary disagrees with John. She argues that one could use the following portfolio to obtain arbitrage
profit: Long one call option with strike price 40; short three call options with strike price 50; lend $1;
and long some calls with strike price 55.
Peter also disagrees with John. He claims that the following portfolio, which is different from Maryʼs,
can produce arbitrage profit: Long 2 calls and short 2 puts with strike price 55; long 1 call and short 1
put with strike price 40; lend $2; and short some calls and long the same number of puts with strike
price 50.
Which of the following statements is true?
(A) Only John is correct.
(B) Only Mary is correct.
(C) Only Peter is correct.
(D) Both Mary and Peter are correct.
(E) None of them is correct.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 55
Section 4, Put-Call Parity
The value of an otherwise similar call and put option on the same asset are related.
Adam and Eve Example:
XYZ stock pays no dividends.
Adam buys a 2 year call on XYZ stock with strike price $100.
Adam also loans out $100 e-2r at the risk free rate.
In two years, Adam will have $100.
If XYZ stock has a price > $100, Adam will use $100 and his call to buy a share of XYZ.
Otherwise, Adam keeps his $100.
Two years from now, Adam ends up with a share of XYZ or $100, whichever is worth more.
Eve buys a 2 year put on XYZ stock with strike price $100 from Seth.
Eve also buys a share of XYZ stock.
In two years, if XYZ stock has a price < $100, Eve will use her put to sell a share of XYZ stock to
Seth for $100. Otherwise, Eve does not exercise her put.
Two years from now, Eve ends up with a share of XYZ or $100, whichever is worth more.
Future Stock Price
< $100
> $100
Adam
$100
Stock
Eve
$100
Stock
Two years from now Adam and Eve have the same position.
Therefore, Adam and Eveʼs initial positions must have the same price.
Call + K e-rT = Put + Stock.
If the stock had paid dividends, then Eve would have collected them, while Adam will not. If we
subtract the present value of these dividends, then their two positions would still be equal.
PV[F0,T] = S0 - PV[Div].
Setting equal the values of Adamʼs position and Eveʼs position minus any dividends we have:
Call + K e-rT = Put + PV[F0,T].
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 56
Put-Call Parity formula:
Thus we have the following put-call parity formula:82
C E u r(K, T) = PE u r(K, T) + PV[F0 , T] - PV[K].
Exercise: In the absence of stock dividends, determine the future value of the sum of:
(a) A share of stock.
(b) A European put option on that stock with a strike price of K.
(c) Having sold a call option on that stock with a strike price of K and the same expiration as the put.
[Solution: Future value is: ST + (K - ST)+ - (ST - K)+.
If S ≤ K, then this future value is: ST + K - ST - 0 = K.
If S ≥ K, then this future value is: ST + 0 - (ST - K) = K.
Comment: (X-d)+ - (d-X)+ = X - d. ⇒ X + (d-X)+ - (X-d)+ = d.]
Thus we again have put-call parity. As in the Adam and Eve example, in the absence of dividends:
Share of Stock at Time T + Put - Call = K.
⇔ Value of Call at Time T - Value of Put at Time T = Value of Stock at Time T - Strike Price.
Stocks With Discrete Dividends:
If dividends are paid at discrete times, then PV[F0,T] = S0 - PV[Div].83
C E u r(K, T) = PE u r(K, T) + S0 - PV[Div] - K e-rT.84
Stocks With Continuous Dividends:
If dividends are paid continuously, then PV[F0,T] = S0 e-δT.
C E u r(K, T) = PE u r(K, T) + S0 e-δT - K e-rT.85
82
See Equation 3.1 in Derivatives Markets by McDonald.
F0,T, the future value of the share of stock, is the expected value of owning a share at time T, excluding any
dividends that may have been paid from time 0 to time T.
84
See equation 9.2 in Derivatives Markets by McDonald.
85
See below equation 9.2 in Derivatives Markets by McDonald. In the case of no dividends, δ = 0.
83
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 57
Exercise: S0 = $100, r = 6%, δ = 2%, T = 1/2.
The present value of a European put option with a strike price of $120 is $30.
Determine the present value of a European call option with the same strike price of $120.
[Solution: $30 + $100 e-0.01 - $120 e-0.03 = $12.55.]
Thus given r, δ, S0 , K and T, if we know the value of either the put or call, then we know the value of
the other one.
Exercise: S0 = $100, r = 6%, δ = 2%, T = 1/2.
The present value of a European call option with a strike price of $110 is $15.
Determine the present value of a European put option with the same strike price of $110.
[Solution: $15 - $100 e-0.01 + $110 e-0.03 = $22.74.]
Exercise: On a stock that pays continuous dividends, at what strike price are a European put and call
worth the same?
[Solution: 0 = CEur(K, T) - PEur(K, T) = S0 e-δT - K e-rT. ⇒ K = S0 e(r-δ)T.
Comment: In a risk-neutral environment, the expected stock price at time T is: S0 e(r-δ)T.]
Strike Price Equal to Current Stock Price:86
If the stock pays no dividends, then CEur(K, T) - PEur(K, T) = S0 - K e-rT.
Let us assume S0 = K = $100. Then C - P = $100(1 - e-rT).
Let us assume we have $100 today available to spend.
If we buy the stock now for $100, we can hold it until time T.
If we instead buy the call and sell the put, then this will cost $100(1 - e-rT).
Subtracting this from our $100, we would have $100 e-rT left.
Assuming we invest $100 e-rT at the risk free rate, when time T comes we would have $100.
At time T if ST ≥ 100, we can buy a share for 100 using our call.
At time T if ST < 100, then we buy a share for 100 from the person who bought our put.
In both situations, we end up with the stock and have invested $100 at time 0 for it.
By buying the call and selling the put, we have deferred the payment of $100 until T.
This deferral costs interest on the $100 of: $100(1 - e-rT).
Thus the option premiums differ by interest on the deferral of the payment for the stock.
86
See Example 9.1 in Derivative Markets by McDonald. When S = K, the option is said to be at the money.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 58
Derivation of Put-Call Parity:
The payoff on a call is (S - K)+.
The payoff on a put is (K - S)+.
If one buys a call and sells a similar put, the payoff is: (S - K)+ - (K - S)+ = S - K.
Therefore, taking the prepaid forward prices, in other words the actuarial present values:
P
C - P = FP
0,T [ST] - F0,T [K] = PV[F0,T] - PV[K].
Bonds:
A bond pays coupons to its owner. These coupon payments act mathematically like stock
dividends paid at discrete times. Therefore, if B0 is the current price of the bond:87
C E u r(K, T) = PE u r(K, T) + B0 - PV[Coupons] - K e-rT.
Summary:88
As will be discussed in subsequent sections, there are similar relationships for other assets.
Asset
Parity Relationship
Stock, No Dividends
S 0 = C - P + e-rT K
Stock, Discrete Dividends
S 0 - PV[Div] = C - P + e-rT K
Stock, Continuous Dividends
e-δT S0 = C - P + e-rT K
Bond
B 0 - PV[Coupons] = C - P + e-rT K.
Futures Contract
e-rT F0,T = C - P + e-rT K
Currency
exp[-rf T] x0 = C - P + e-rT K
Exchange Options
P (S ) = C - P + F P (Q ) ⇔
F0,T
0
0
0,T
exp[-δS T] S0 = C - P + exp[-δQ T] Q0
87
While listed by McDonald, he does not discuss further this formula involving bonds.
In this formula, e-rT is the price of a zero-coupon bond that pays 1 at time T.
88
See Table 9.9 at page 305 of Derivatives Markets by McDonald.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 59
Problems:
4.1 (2 points) Jay Corp. common stock is priced at $80 per share.
The company just paid its $1 quarterly dividend. Interest rates are 6.0%.
A $75 strike European call, maturing in 8 months, sells for $10.
What is the price of a 8-month, $75 strike European put option?
A. 3
B. 4
C. 5
D. 6
E. 7
4.2 (2 points) A bond currently costs $830. The bond has $10 quarterly coupons.
A coupon has just been paid. r = 4%. What is the difference in price between a 2-year $800 strike
European call option and the corresponding put?
A. 15
B. 20
C. 25
D. 30
E. 35
Use the following information for the next two questions:
The Rich and Fine stock index is priced at 1300. Dividends are paid at the rate of 2%.
The continuously compounded risk free rate is 5%.
4.3 (1 point) A 1800 strike European call, maturing in 4 years, sells for 196.
What is the price of a 4-year, 1800 strike European put option?
A. Less than 400
B. At least 400, but less than 450
C. At least 450, but less than 500
D. At least 500, but less than 550
E. At least 550
4.4 (1 point) A 1500 strike European put, expiring in 3 years, sells for 293.
What is the price of a 3-year, 1500 strike European call option?
A. 150
B. 175
C. 200
D. 225
E. 250
4.5 (2 points) For a stock that does not pay dividends, the difference in price between otherwise
similar European call and put options is 29.05. The current stock price is 100. r = 6%.
The time until expiration is 2 years. Determine the strike price.
A. 80
B. 85
C. 90
D. 95
E. 100
4.6 (2 points) 160 = current market price of the stock
130 = exercise price of the option
5% = annual risk free force of interest
18 months = time until the exercise date of the option
3% = (continuously compounded) annual rate at which dividends are paid on this stock
If a European put has an option premium of 13, determine the premium for a similar European call.
A. 35
B. 40
C. 45
D. 50
E. 55
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 60
Use the following information for the next four questions:
You have the following bid and ask prices on 3.25 month European options on HAL stock with a
strike price of $160.
Call Bid
Call Ask
Put Bid
Put Ask
$13.00
$13.40
$2.90
$3.20
You buy at the ask and sell at the bid.
Your continuously compounded lending rate is 1.9% and your borrowing rate is 2%.
The current price of HAL stock is $169.70.
Ignore transaction costs on the stock.
HAL will pay a $0.36 dividend one month from today.
4.7 (2 points) You sell the call, buy the put, buy the stock, and borrow the present value of the
strike price plus dividend. What is the cost?
4.8 (2 points) Briefly discuss whether parity is violated by the situation in the previous question.
4.9 (2 points) What is the cost if you buy the call, sell the put, short the stock, and lend the present
value of the strike price plus dividend?
4.10 (2 points) Briefly discuss whether parity is violated by the situation in the previous question.
4.11 (2 points) The price of a non-dividend paying stock is $85 per share. A 18 month, at the
money European put option is trading for $5. If the interest rate is 6.5%, what is the price of a
European call at the same strike and expiration?
A. 12
B. 13
C. 14
D. 15
E. 16
4.12 (2 points) As a function of the future stock price, graph the future value of the sum of a share of
the stock and a European put option on that same stock with a strike price of 100.
4.13 (2 points) A stock that pays continuous dividends at 1.5%, has a price of 130. r = 4%.
At what strike price are a 2 year European put and call worth the same?
A. Less than 126
B. At least 126, but less than 128
C. At least 128, but less than 132
D. At least 132, but less than 136
E. At least 136
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 61
4.14 (2 points) The prices for otherwise similar European puts and calls are:
Strike
100
120
Call premium
27
23
Put premium
25
26
Describe a spread position involving puts and calls that you can use to effect arbitrage, and
demonstrate that arbitrage occurs.
4.15 (2 points) Kay Corp. common stock is priced at $120 per share.
Yesterday, the company paid its $2 quarterly dividend.
The continuously compounded interest rate is 5.0%.
The company just paid its $2 quarterly dividend. Interest rates are 5.0%.
A $130 strike European put, maturing in 4 months, sells for $15.
What is the price of a 4-month, $130 strike European call option?
A. 3
B. 4
C. 5
D. 6
E. 7
4.16 (3 points) You are given the following premiums for European options:
Strike Price Call Price
Put Price
$60
$3.45
$80
$23.85
$9.88
$90
$19.56
$14.38
$110
$13.09
All of the options are on the same stock and have the same expiration date.
Determine the sum of the price of the 60 strike call plus the price of the 110 strike put.
A. 59.0
B. 59.5
C. 60.0
D. 60.5
E. 61.0
4.17 (2 points) Joe buys a share of a stock, buys a European put option on that stock with a strike
price of 100, and sells a European call option on that stock with a strike price of 100. Graph the
future value of Joeʼs investments as a function of the stock price at expiration of the options.
4.18 (2 points) 150 = current market price of the stock
140 = exercise price of the option
6% = annual risk free force of interest
6 months = time until the exercise date of the option
3% = (continuously compounded) annual rate at which dividends are paid on this stock
If a European call has an option premium of 23, determine the premium for a similar European put.
A. 7
B. 8
C. 9
D. 10
E. 11
4.19 (2 points) A European put and call on the same stock have the same time until maturity and the
same strike price. S0 = 100. K = 115. r = 6%. δ = 2%.
If the put and the call have the same premium, what is the time until maturity?
A. 1.5 years
B. 2.0 years
C. 2.5 years
D. 3.0 years
E. 3.5 years
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 62
4.20 (2 points) A stock is priced at $100 per share.
The stock is forecasted to pay dividends of $0.80, $1.20, and $1.50 in 3, 6, and 9 months,
respectively. r = 5.5%.
A $125 strike European call, maturing in 9 months, sells for $8.
What is the price of a 9-month, $125 strike European put option?
A. Less than 25
B. At least 25, but less than 30
C. At least 30, but less than 35
D. At least 35, but less than 40
E. At least 40
4.21 (2 points) Consider 6 month 80 strike European options on a nondividend paying stock.
Two months after being purchased, the call and the put would have the same value for a stock price
of 78.68. Determine r.
A. 4.0%
B. 4.5%
C. 5.0%
D. 5.5%
E. 6.0%
4.22 (2 points) Near market closing time on a given day, you lose access to stock prices, but some
European call and put prices for a stock are available as follows:
Strike Price Call Price
Put Price
$70
$11
$3
$80
$6
$7
$90
$3
All of the options have the same expiration date.
Determine the price of the 90 strike put.
A. 12.0
B. 12.5
C. 13.0
D. 13.5
E. 14.0
4.23 (2 points) A one year European put and call on the same stock with the same strike price have
the same premium.
The current stock price is 84. The stock will pay dividends of 2, one month from now, four months
from now, seven months from now, and ten months from now.
If r = 5%, what is the strike price?
A. 75
B. 80
C. 85
D. 90
E. 95
4.24 (2 points) A two-year Call Bull Spread has strike prices that differ by 10.
The premium for this Call Bull Spread is 4.19.
r = 5.6%.
Determine the premium for a similar Put Bear Spread.
A. 3.75
B. 4.00
C. 4.25
D. 4.50
E. 4.75
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 63
4.25 (2 points) You are given the following information on European options on a given stock index
that does not pay dividends.
All the options have the same strike price and all expire on July 1, 2008.
As of January 1, 2008
As of March 1, 2008
Stock Index Price
860
1020
Call Premium
66.67
154.64
Put Premium
86.64
21.24
Determine the continuously compounded risk rate of interest, r.
A. 4.5%
B. 5.0%
C. 5.5%
D. 6.0%
E. 6.5%
4.26 (2 points) A stock is priced at $93 per share.
The stock pays dividends at a continuously compounded rate δ.
r = 5.2%.
The premium for a $90-strike 2-year European call is $18.50.
The premium for a $90-strike 2-year European put is $9.36.
Determine δ.
A. 0.5%
B. 1.0%
C. 1.5%
D. 2.0%
E. 2.5%
4.27 (3 points) You observe the prices of various European call and put options all on the same
stock and all with the same expiration date:
Strike Price Call Price
Put Price
$60
$21
$6
$70
$17
$11
$90
$12
$25
After reviewing the information above, Andrew tells Brittany and Christopher that no arbitrage
opportunities can arise from these prices.
Brittany disagrees with Andrew. She argues that one could use a portfolio of various calls to obtain
arbitrage profit.
Christopher also disagrees with Andrew. He claims that one could use a portfolio of various puts to
obtain arbitrage profit.
Which of the following statements is true?
(A) Only Andrew is correct.
(B) Only Brittany is correct.
(C) Only Christopher is correct.
(D) Both Brittany and Christopher are correct.
(E) None of them is correct.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 64
4.28 (3 points) A nondividend paying stock has a current price of 76.
You observe the prices of various 2-year European options on this stock:
70-strike call has a premium of 12.56.
80-strike put has a premium of 14.70.
100-strike call has a premium of 38.22.
Determine the possible values of r.
4.29 (3 points) Consider European options all on the same stock and all with the same time until
expiration.
Dustin buys a 90-strike call and sells a 100-strike put. His net cost is 24.47.
Edith buys a 100-strike call and sells a 90-strike put. Her net cost is 16.85.
Frank buys a 120-strike call and sells a 120-strike put. His net cost is 1.61.
Gabrielle buys a 110-strike call and sells a 110-strike put.
What is Gabrielleʼs net cost?
A. 8.4
B. 8.6
C. 8.8
D. 9.0
E. 9.2
4.30 (2 points) Consider European options all on the same stock and all with the same time until
expiration.
Ashley buys a 60-strike call and sells an 80-strike put. Her net cost is 1.22.
Chelsea buys an 80-strike call and sells a 60-strike put. Her net cost is 4.52.
Benjamin buys an 70-strike call and sells a 70-strike put.
What is Benjaminʼs net cost?
A. 2.8
B. 2.9
C. 3.0
D. 3.1
E. 3.2
4.31 (2 points) A stock pays dividends at a continuously compounded rate of 2%.
The current price of the stock is 82.
The effective annual risk-free interest rate is 10%.
You buy a three-year 80-strike European call option on the stock and sell a similar put.
Determine the premium for this position.
(A) 16
(B) 17
(C) 18
(D) 19
(E) 20
4.32 (3 points)
Consider European options all on the same stock and all with the same time until expiration.
Let C(K) be the premium for a call with strike K.
Let P(K) be the premium for a put with strike K.
Let A(K) = C(K) - C(K + 10).
Let B(K) = P(K + 10) - P(K).
A(95) = 3.49.
B(95) = 5.36.
B(80) = 3.87.
Determine A(80).
A. 4.7
B. 4.8
C. 4.9
D. 5.0
E. 5.1
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 65
4.33 (5B, 5/99, Q.1) (1 point) You sell a 1-year European call option on Greystokes Inc. with an
exercise price of 110 and buy a 1-year European put option with the same exercise price and term.
The current continuously compounded risk-free rate is 12% and the value of your combined position
is zero.
Greystokes Inc. is a non-dividend-paying stock. What is the price of a share of Greystokes Inc.?
A. Less than 95.00
B. At least 95.00, but less than 97.50
C. At least 97.50, but less than 100.00
D. At least 100.00, but less than 102.50
E. At least 102.50
4.34 (IOA 109, 9/00, Q.1) (8.25 points)
(i) (1.5 points) State what is meant by put-call parity.
(ii) (4.5 points) Derive an expression for the put-call parity of a European option that has a dividend
payable prior to the exercise date.
(iii) (2.25 points) If the equality in (ii) does not hold, explain how an arbitrageur can make a riskless
profit.
4.35 (MFE Sample Exam, Q.1) Consider a European call option and a European put option on a
nondividend-paying stock. You are given:
(i) The current price of the stock is $60.
(ii) The call option currently sells for $0.15 more than the put option.
(iii) Both the call option and put option will expire in 4 years.
(iv) Both the call option and put option have a strike price of $70.
Calculate the continuously compounded risk-free interest rate.
(A) 0.039
(B) 0.049
(C) 0.059
(D) 0.069
(E) 0.079
4.36 (FM Sample Exam, Q.1) Which statement about zero-cost purchased collars is FALSE?
A. A zero-width, zero-cost collar can be created by setting both the put and call strike prices
at the forward price.
B. There are an infinite number of zero-cost collars.
C. The put option can be at-the-money.
D. The call option can be at-the-money.
E. The strike price on the put option must be at or below the forward price.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 66
4.37 (FM Sample Exam, Q.2) You are given the following information:
• The current price to buy one share of XYZ stock is 500.
• The stock does not pay dividends.
• The risk-free interest rate, compounded continuously, is 6%.
• A European call option on one share of XYZ stock with a strike price of K that expires in
one year costs 66.59.
• A European put option on one share of XYZ stock with a strike price of K that expires in
one year costs 18.64.
Using put-call parity, determine the strike price, K.
A. 449
B. 452
C. 480
D. 559
E. 582
4.38 (FM Sample Exam, Q.5) You are given the following information:
• One share of the PS index currently sells for 1,000.
• The PS index does not pay dividends.
• The effective annual risk-free interest rate is 5%.
You want to lock in the ability to buy this index in one year for a price of 1,025. You can do this by
buying or selling European put and call options with a strike price of 1,025.
Which of the following will achieve your objective and also gives the cost today of establishing this
position?
A. Buy the put and sell the call, receive 23.81
B. Buy the put and sell the call, spend 23.81
C. Buy the put and sell the call, no cost
D. Buy the call and sell the put, receive 23.81
E. Buy the call and sell the put, spend 23.81
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 67
4.39 (CAS3, 5/07, Q.3) (2.5 points) For a dividend paying stock and European options on this
stock, you are given the following information:
• The current stock price is $49.70.
• The strike price of options is $50.00.
• The time to expiration is 6 months.
• The continuous risk-free rate is 3% annually.
• The continuous dividend yield is 2% annually.
• The call price is $2.00.
• The put price is $2.35.
Using put-call parity, calculate the present value arbitrage profit per share that could be generated,
given these conditions.
A. Less than $0.20
B. At least $0.20 but less than $0.40
C. At least $0.40 but less than $0.60
D. At least $0.60 but less than $0.80
E. At least $0.80
4.40 (CAS3, 5/07, Q.4) (2.5 points)
The price of a European call that expires in six months and has a strike price of $30 is $2.
The underlying stock price is $29, and a dividend of $0.50 is expected in two months and in five
months. The term structure is flat, with all continuously compounded, risk-free rates being 10%.
Calculate the price of a European put option on the same stock with expiration in six months and a
strike price of $30.
A. 0.57
B. 1.06
C. 1.54
D. 2.02
E. 2.51
4.41 (MFE, 5/07, Q.1) (2.6 points) On April 30, 2007, a common stock is priced at $52.00.
You are given the following:
(i) Dividends of equal amounts will be paid on June 30, 2007 and September 30, 2007.
(ii) A European call option on the stock with strike price of $50.00 expiring in six months
sells for $4.50.
(iii) A European put option on the stock with strike price of $50.00 expiring in six months
sells for $2.45.
(iv) The continuously compounded risk-free interest rate is 6%.
Calculate the amount of each dividend.
(A) $0.51
(B) $0.73
(C) $1.01
(D) $1.23
(E) $1.45
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 68
4.42 (MFE, 5/07, Q.4) (2.6 points) For a stock, you are given:
(i) The current stock price is $50.00.
(ii) δ = 0.08
(iii) The continuously compounded risk-free interest rate is r = 0.04.
(iv) The prices for one-year European calls (C) under various strike prices (K) are shown below:
K
C
$40 $ 9.12
$50 $ 4.91
$60 $ 0.71
$70 $ 0.00
You own four special put options each with one of the strike prices listed in (iv).
Each of these put options can only be exercised immediately or one year from now.
Determine the lowest strike price for which it is optimal to exercise these special put option(s)
immediately.
(A) $40
(B) $50
(C) $60
(D) $70
(E) It is not optimal to exercise any of these put options.
4.43 (CAS3, 11/07, Q.14) (2.5 points)
Given the following information about a European call option on Stock Z:
• The call price is 5.50.
• The call has a strike price of 47.
• The call expires in two years.
• The current stock price is 45.
• The continuously compounded risk-free rate is 5%.
• Stock Z will pay a dividend of 1.50 in one year.
Calculate the price of a European put option on Stock Z with a strike price of 47 that expires in two
years.
A. Less than 3.00
B. At least 3.00, but less than 3.50
C. At least 3.50, but less than 4.00
D. At least 4.00, but less than 4.50
E. At least 4.50
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 69
4.44 (CAS3, 11/07, Q.16) (2.5 points) An investor has been quoted a price on European
options on the same non-dividend paying stock. The stock is currently valued at 80 and the
continuously compounded risk-free interest rate is 3%. The details of the options are:
Option 1
Option 2
Type
Put
Call
Strike
82
82
Time to expiration 180 days
180 days
Based on his analysis, the investor has decided that the prices of the two options do not present
any arbitrage opportunities. He decides to buy 100 calls and sell 100 puts.
Calculate the net cost of this transaction.
(Hint: A positive net cost means the investor pays money from the transaction. A negative cost
means the investor receives money.)
A. Less than -60
B. At least -60, but less than -20
C. At least -20, but less than 20
D. At least 20, but less than 60
E. At least 60
4.45 (MFE/3F, 5/09, Q.12) (2.5 points) You are given:
(i) C(K, T) denotes the current price of a K-strike T-year European call option on a
nondividend-paying stock.
(ii) P(K, T) denotes the current price of a K-strike T-year European put option on the same stock.
(iii) S denotes the current price of the stock.
(iv) The continuously compounded risk-free interest rate is r.
Which of the following is (are) correct?
(I) 0 ≤ C(50, T) - C(55, T) ≤ 5e-rT
(II) 50e-rT ≤ P(45, T) - C(50, T) + S ≤ 55e-rT
(III) 45e-rT ≤ P(45, T) - C(50, T) + S ≤ 50e-rT
(A) (I) only
(B) (II) only
(C) (III) only
(D) (I) and (II) only
(E) (I) and (III) only
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 70
Section 5, Bounds on Premiums of European Options
Option premiums must stay within certain bounds.
Maximum and Minimum Prices of Calls:89
Since an option does not require its owner to do anything, it never has a negative value.
E[(ST - K)+] ≥ 0. E[(K - ST)+] ≥ 0.
If Joe buys a call option, then he must pay K > 0 if he wants to own a share of stock at time T in the
future.
For S0 Mary can buy a share of the stock now and if she holds on to it, guarantee she owns a share
at time T in the future.
After Mary buys her share of stock and after Joe buys his option, Mary is in a better position than
Joe, even ignoring any dividends Mary may get on the stock. Therefore, the value of the call option
is less than (or equal to) S0 , the current market share of the stock.
Owning a share of stock now is worth at least as much as the option to buy a share of stock in the
future. ⇒ A call is never worth more than the current stock price.
S 0 ≥ C(S0 , K, T) ≥ 0.
P (S ), and will own a share of stock at time T.
In fact, one can pay the prepaid forward price, F0,T
0
This is more valuable than the option to buy a share of stock at time T in the future.
For example, Tim pays Howard the prepaid forward price for a share of ABC stock and in exchange
Howard agrees to give Tim a share of ABC stock two years from now. By paying this amount of
money today, Tim will own a share of ABC stock two years from now. Instead, Tim could buy a two
year $100 strike call on ABC stock from John. Two years from now Tim will have the option to pay
$100 to buy a share of ABC stock from John.
Owning a share of stock two years from now is more valuable than having the option to buy a share
of stock two years from now for $100. The future value of the former is S2 . The future value of the
latter is: (S2 - 100)+ < S2 .
P (S ) ≥ C
90
Therefore, a somewhat more stringent inequality holds: F0,T
0
Eur(S0 , K, T).
89
See page 292 and 293 of Derivative Markets by McDonald.
McDonald does not mention this in Derivative Markets.
In the absence of dividends, the prepaid forward price is just S0 .
90
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 71
Phyllis can buy a forward contract to buy a share of stock at T in the future.
At time T in the future Phyllis will pay F0,T for a share of stock.
Harry can instead buy a call option for this same stock at time T in the future, for a price of
C(S0 , K, T), and can in addition invest PV0,T[K] in a risk-free investment. At time T Harry will then
have the money to buy a share of stock if he chooses to exercise his option.
Phyllis will own a share of stock at time T. Harry can choose to own a share of stock at time T if it is
advantageous to him to exercise his option. Therefore, Harryʼs position is worth more than Phyllisʼs.
C(S0 , K, T) + PV0,T[K] ≥ PV0,T[F0,T]. ⇒ C(S0 , K, T) ≥ PV0,T[F0,T] - PV0,T[K].91
Therefore, recalling that an option can never have a negative value:
S 0 ≥ C(S0 , K, T) ≥ (PV0 , T[F0 , T] - PV0 , T[K])+ .92
However, PV0,T[F0,T] = S0 - PV[Div]. Therefore, C(S0 , K, T) ≥ (S0 - PV[Div] - PV0,T[K])+.
Exercise: K = 100, T = 2, r = 6%, and δ = 2%.
What are the bounds on the call premium as a function of S0 ?
[Solution: PV0,T[F0,T] - PV0,T[K] = S0 e-δT - K e-rT = S0 e-0.04 - 100 e-0.12 = 0.961 S0 - 88.69.
S 0 ≥ C(S0 , K, T) ≥ (0.961 S0 - 88.69)+.
Comment: 0.961 S0 - 88.69 = 0, for S0 = 92.3 = Ke-(r-δ)T.]
For these inputs, here is a graph of the possible call premiums as a function of S0 :
C
140
120
100
80
60
40
20
50
91
92
92.3
From put-call parity: C = P + PV[F0,T] - PV[K]. Also P ≥ 0.
See Equation 9.9 in Derivative Markets by McDonald.
150
S0
⇒ C ≥ PV[F0,T] - PV[K].
Financial Economics,
HCMSA-2012-MFE/3F,
12/14/11,
Page 72
Exercise: S0 = 90, T = 2, r = 6%, and δ = 2%.
What are the bounds on the call premium as a function of K?
[Solution: PV0,T[F0,T] - PV0,T[K] = S0 e-δT - K e-rT = 90 e-0.04 - K e-0.12 = 86.47 - 0.887 K.
90 ≥ C(S0 , K, T) ≥ (86.47 - 0.887 K)+.
Comment: 86.47 - 0.887 K = 0, for K = 97.5 = S0 e(r-δ)T.]
For these inputs, here is a graph of the possible call premiums as a function of K:
C
80
60
40
20
50
97.5
150
K
Maximum and Minimum Prices of Puts:
If one owns a put option, then the most one can gain is K, if the stock price goes to zero.
Therefore, the value of a put is at most its strike price K.93
K ≥ P(S0 , K, T).
In fact, since the possible gain from the put occurs at time T, the value of the put is at most the
present value of K, e-rT K: PV[K] ≥ P(S0 , K, T).
Susan buys a put option on a share of stock with expiration T, and also enters a forward contract to
buy the stock at T, at price F0,T. Then if Susan invests PV0,T[F0,T], she will have the money at time
T to buy the stock as per her forward contract. If the price of the stock is greater than K she will not
exercise her put option and instead she keeps the stock. If the price of the stock at time T is less
than K she will exercise her option, and sell the share of stock for K. In either case, her position at
time T is worth at least K.
93
PEur(S0 , K, T) = E[(K - ST)+] e-Tr ≤ E[(K - ST)+] ≤ K.
Financial Economics,
HCMSA-2012-MFE/3F,
12/14/11,
Page 73
Therefore, P(S0 , K, T) + PV0,T[F0,T] ≥ PV0,T[K]. ⇒ P(S0 , K, T) ≥ PV0,T[K] - PV0,T[F0,T].94
Therefore, recalling that an option can never have a negative value:
K ≥ P(S0 , K, T) ≥ (PV0 , T[K] - PV0 , T[F0 , T])+ .95
However, PV0,T[F0,T] = S0 - PV[Div].
Therefore, P(S0 , K, T) ≥ (PV0,T[K] - S0 + PV[Div])+.
Exercise: K = 80, T = 3, r = 5%, and δ = 1%.
What are the bounds on the put premium as a function of S0 ?
[Solution: PV0,T[K] - PV0,T[F0,T] = K e-rT - S0 e-δT = 80 e-0.15 - S0 e-0.03 = 68.86 - 0.970 S0 .
80 ≥ P(S0 , K, T) ≥ (68.86 - 0.970 S0 )+.
Comment: 68.86 - 0.970 S0 = 0, for S0 = 71.0 = Ke-(r-δ)T.]
For these inputs, here is a graph of the possible put premiums as a function of S0 :
P
80
60
40
20
50
71
150
S0
Exercise: S = 70, T = 3, r = 5%, and δ = 1%.
What are the bounds on the put premium as a function of K?
[Solution: PV0,T[K] - PV0,T[F0,T] = K e-rT - S0 e-δT = K e-0.15 - 70 e-0.03 = 0.861 K - 67.93.
K ≥ P(S0 , K, T) ≥ (0.861 K - 67.93)+.
Comment: 0.861 K - 67.93 = 0, for K = 78.9 = S0 e(r-δ)T.]
94
95
From put-call parity: P = C + PV[K] - PV[F0,T]. Also C ≥ 0.
See Equation 9.10 in McDonald.
⇒ P ≥ PV[K] - PV[F0,T].
Financial Economics,
HCMSA-2012-MFE/3F,
12/14/11,
Page 74
For these inputs, here is a graph of the possible put premiums as a function of K:
P
140
120
100
80
60
40
20
50
78.9
150
K
For a European put, the most you can be paid is K at time T, which has present value Ke-rT.
Therefore, Ke-rT ≥ P(S0 , K, T).96
Relationship to Put-Call Parity:
From Put-Call Parity, we have:
C Eur(K, T) = PEur(K, T) + PV[F0,T] - PV[K].
Since, PEur(K, T) ≥ 0, CEur(K, T) ≥ PV[F0,T] - PV[K], as discussed previously.
Since, CEur(K, T) ≥ 0, PEur(K, T) + PV[F0,T] - PV[K] ≥ 0.
Therefore, PEur(K, T) ≥ PV[K] - PV[F0,T], as discussed previously.
96
McDonald does not mention this in Derivative Markets.
See for example, Options, Futures, and Other Derivatives by John C. Hull, not on the syllabus.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 75
An Increasing Strike Price:97
Rather than being fixed as is usual, the strike price could increase with the time until expiration.
Let the strike price for an option with time until expiration T be: KT = KerT.
In other words, we keep the present value of the strike price constant.98
On a stock that does not pay dividends, when the strike price grows at the risk free rate, then the
premiums on European puts increase with time to expiration.99
Exercise: Demonstrate that an arbitrage opportunity exists in the following situation.
The risk free rate is 6%.
The premium for a one-year 100-strike European put on a non-dividend paying stock is 5.
The premium for the two-year 106.18-strike European put on the same stock is 4.
[Solution: Buy the two-year put and sell the one-year put.
You get 1, which you invest at the risk-free rate. At time 1 you have e0.06 = 1.0618.
If S1 > 100 then you do not pay off on the one-year put you sold.
You have 1.0618 plus the two year put; your position is positive.
If S1 < 100 then you pay off on the one-year put: 100 - S1 .
However, the premium of the 106.18-strike put that you still own is greater than or equal to:
(PV[K] - PV[F0,T])+ = (106.18 e-0.06 - S1 )+ = 100 - S1 .
Thus the put you still own is worth at least as much as the amount you pay off on the put you sold.
Thus you have at least 1.0618; your position is positive. Demonstrating arbitrage.
Comment: Note that the strike for the two year put is: (100) e0.06 = 106.18.
Since the stock pays no dividends, its prepaid forward price is just its current price.
The result did not depend on the particular values other than the fact that 4 < 5.]
If the options start at the money, then the increasing strike price is: S0 erT.
From put-call parity, C - P = Ke-rT - Se-δT.
Thus, in this situation with no dividends and K = S0 erT: C - P = 0.
The call premium is equal to the put premium. They both increase with T.
In general, for a European call on a non-dividend paying stock, when the strike price grows at the risk
free rate, then the premiums increase with time to expiration.100
97
See page 298 of Derivatives Markets by McDonald.
This is analogous to keeping a insurance policy deductible or limit up with inflation.
99
Since the put premium increases with the strike price, if the strike price increases with T at a rate greater than r,
then the put premium increases even faster with T.
100
Since the call premium decreases with the strike price, if the strike price increases with T at a rate less than r, then
the call premium increases even faster with T.
98
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 76
Exercise: For a stock that does not pay dividends, demonstrate that when the strike price grows at
the risk free rate, the premium of a European call increases with time to maturity.
[Solution: KT = KerT. Take similar two calls, with times to expiration T < U.
Assume the premium of the first call with less time to expiration is more than that of the second call.
Buy the U-year call and sell the T-year call
Since we have assumed that the premium of the T-year call is greater than the premium of the
U-year call, we get money in the door, which we invest at the risk-free rate.
If ST ≤ KerT then we do not pay off on the T-year call we sold.
We have money plus the T-year call; our position is positive.
If ST > KerT then we pay off on the T-year call: ST - KerT.
However, the premium of the call we own with strike KerU, that has U - T until expiration, is greater
than or equal to: (PV[Forward contract on the stock] - PV[K])+ = (ST - e-r(U-T) KerT)+
= ST - e-r(U-T) KerU = ST - KerT.
Thus the call we still own is worth at least as much as the amount we pay off on the call we sold.
Thus we have at least the money we invested at time 0, plus the interest we earned on it;
your position is positive. Demonstrating arbitrage.]
Portfolio Insurance for the Long Run:101
If you owned a share of a stock that pays no dividends and purchased a put with strike price S0 erT,
then if at time T: ST < S0 erT, you could use your put to sell the stock for S0 erT. You would
guarantee that over the period time from 0 to T you would earn at least the risk free rate on the stock
plus put, not including the cost to buy the put.
However, as we have seen the cost of such puts increases with T.102 The longer the period of time
we wish to insure, the more the premium. One can apply the same idea to a portfolio of different
stocks.
101
See page 299 of Derivatives Markets by McDonald. See also page 603, to be discussed subsequently.
When there are no dividends. If there are dividends, we would instead have the strike increase at the rate r - δ;
after taking into account the dividends we receive, we would earn at least the risk free rate.
102
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 77
Problems:
5.1 (1 point) A European 140 strike 2 year call is an option on a stock with current price 100 and
forward price of 108. r = 5%.
Which of the following intervals represents the range of possible prices of this option?
A. [0, 100] B. [0, 108] C. [9.05, 100]
D. [9.05, 108]
E. [8, 108]
5.2 (1 point) A European 140 strike 2 year put is an option on a stock with current price 100 and
forward price of 108. r = 5%.
Determine the range of possible prices of this option.
5.3 (3 points) One has a European option on a stock with current market price of 100.
The time until expiration is 2 years. r = 5%.
The two year forward price of the stock is 110.
Which the following statements are true?
A. The premium of a call option with a strike price of 80 could not be 80.
B. The premium of a put option with a strike price of 140 could not be 105.
C. The premium of a call option with a strike price of 80 could not be 25.
D. The premium of a put option with a strike price of 140 could not be 30.
E. The premium of an at-the-money put must be at least the premium of an at-the-money call.
5.4 (1 point) A European 110 strike 2 year put is an option on a stock with current price 100 and
forward price of 108. r = 5%.
Which of the following intervals represents the range of possible prices of this option?
A. [0, 100]
B. [0, 110]
C. [1.81, 108]
D. [1.81, 110]
E. [2, 108]
5.5 (1 point) A European 105 strike 2 year call is an option on a stock with current price 100 and
forward price of 108. r = 5%.
Which of the following intervals represents the range of possible prices of this option?
A. [0, 100]
B. [0, 108]
C. [2.72, 100]
D. [2.72, 108]
E. [8, 108]
Financial Economics,
HCMSA-2012-MFE/3F,
12/14/11,
Page 78
5.6 (2 points) ABC stock pays no dividends. r = 8%.
For a 90 strike 6 month European put on ABC Stock, which of the following graphs represents the
bounds on the put premium as a function of S0 ?
P
P
140
80
120
60
100
40
80
60
40
20
A.
25
50
75
S0
100 125 150
P
20
B.
25
50
75
S0
100 125 150
25
50
75
S0
100 125 150
P
120
80
100
60
80
40
60
40
20
C.
25
50
75
25
50
75
S0
100 125 150
P
80
60
40
20
E.
S0
100 125 150
20
D.
Financial Economics,
HCMSA-2012-MFE/3F,
12/14/11,
Page 79
5.7 (2 points) XYZ stock pays dividends at a continuously compounded rate of 2%.
XYZ stock has a current price of 80. r = 10%.
For a 3 year European call on XYZ Stock, which of the following graphs represents the bounds on
the call premium as a function of K?
C
80
C
140
60
120
100
40
80
20
60
40
A.
25
50
75
K
100 125 150
25
50
75
K
100 125 150
75
K
100 125 150
C
C
C.
20
B.
70
70
60
50
40
60
30
20
10
30
50
40
20
25
50
75
K
100 125 150
25
50
75
K
100 125 150
C
80
60
40
20
E.
10
D.
25
50
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 80
Section 6, Options on Currency
For example, one may be able today to buy one Euro for 1.25 U.S. Dollars.
1€ = $1.25. ⇔ 0.8€ = $1. Instead one could buy one U.S. Dollar for 0.8 Euros.
Let r$ be the interest rate on dollars and r€ be the interest rate on Euros.
Then the forward price in dollars for one Euro in the future is: (current exchange rate) exp[(r$ - r€)T].103
The forward price in Euros for one dollar in the future is: (current exchange rate) exp[(r€ - r$ )T].104
In general, one can exchange one currency for another currency at an exchange rate.105
However, exchange rates change over time. Therefore, it is useful for businesses engaged in
international trade to buy and sell options related to currencies.
Dollar Denominated Options:
For example, one could purchase a call option to buy one euro in 2 years for $1.20.
The owner of this call would have the option in 2 years to use $1.20 to obtain one Euro.
This is an example of a dollar-denominated option, since the strike price and premium (cost) are in
dollars. A dollar-denominated call on Euros would give one the option to obtain Euros at some time
in the future for a specified number of dollars.
If the price of one euro in two years is more than $1.20, then one would exercise the above option.
For example if the exchange rate two years from now is $1.50 per Euro, then one would make
$1.50 - $1.20 = $0.30 by exercising this call.
A dollar-denominated put on Euros would give one the option to sell Euros at some time in the
future for a specified number of dollars. For example, a 2 year $1.20 strike put on Euros would give
its owner the option in 2 years to sell one Euro for $1.20. The owner would exercise this put if
2 years from now the exchange rate is less than $1.20 per Euro.
The payoff of the call option is: (x2 - $1.20)+, where x2 is the exchange rate in two years.
The payoff of the put option is: ($1.20 - x2 )+, where x2 is the exchange rate in two years.
The payoff on a call on currency has the same mathematical expression as for a call on stocks, with
xT substituted for ST: (xT - K)+. Similarly, the payoff on a put on currency is: (K - xT)+.
103
See Equation 5.18 in Derivative Markets by McDonald, on the syllabus of an earlier exam.
Since the current exchange rates are reciprocals of each other, so are the forward prices.
105
For current exchange rates see for example www.xe.com/ucc/
104
Financial Economics,
HCMSA-2012-MFE/3F,
12/14/11,
Page 81
Put-Call Parity for Dollar Denominated Options:106
The same put-call parity relationship holds as was discussed previously:
C(K, T) = P(K, T) + PV[F0,T] - PV[K].
If the option is dollar-denominated, then the strike price K is in dollars. We discount it back to the
present using the interest rate for dollars r$ , usually just written as r in the United States.
The present value of one euro at time T, PV[F0,T], is the amount we would need to invest now in a
euro-denominated risk free bond in order to have 1 euro at time t. A euro-denominated risk free
bond earns interest continuous at a rate of r€.107
If we spend x0 exp[-Tr€ ] dollars to buy exp[-Tr€ ] euros at time 0, and invest in a
euro-denominated risk free bond, we will have 1 euro at time T.
Therefore, PV[F0,T] = x0 exp[-Tr€ ].
Therefore, for dollar-denominated options on euros:
C $ ( x0 , K, T) = P$ ( x0 , K, T) + x0 exp[-Tr€ ] - K e-Tr.108
x0 is the exchange rate for Euros in terms of dollars, $1.25 = 1 Euro, and K is in dollars.
This is analogous to what we had for options on stock: C(K, T) = P(K, T) + S0 e-Tδ - K e-Tr.
x 0 ⇔ S0 .
Dollars act as money:
r$ ⇔ r.
Euros act as the asset: r€ ⇔ δ.
Exercise: x0 = $1.25/€ . T = 2 years. K = $1.20/€ .
The dollar-denominated interest rate is 6% and the euro-denominated interest rate is 4%.
The option premium for a dollar-denominated put on one euro is $0.10.
Determine the premium for a dollar-denominated call on one euro.
[Solution: $0.10 + ($1.25)e-(2)(0.04) - ($1.20)e-(2)(0.06) = $0.190.
Comment: Note how every term in the equation is in dollars. This is one way to check your work.]
106
See page 286 of Derivative Markets by McDonald.
We assume that r$ will usually differ from r€. Similarly, each currency will have its own associated interest rate.
108
See equation 9.4 of Derivative Markets by McDonald.
This is similar to the parity equation for stocks paying continuous dividends, with r€ taking the place of δ.
107
Financial Economics,
HCMSA-2012-MFE/3F,
12/14/11,
Page 82
Foreign Denominated Options:
A Euro-denominated call on dollars would give one the option to obtain dollars at some time in the
future for a specified number of Euros.
For example, a 2 year € 0.70 strike call on dollars would give its owner the option in 2 years to buy
one dollar for € 0.70. The owner would exercise this call if 2 years from now the exchange rate is
more than € 0.70 per dollar.
A Euro-denominated put on dollars would give one the option to sell dollars at some time in the
future for a specified number of Euros.
For example, a 2 year € 0.60 strike put on dollars would give its owner the option in 2 years to sell
one dollar for € 0.60. The owner would exercise this put if 2 years from now the exchange rate is
less than € 0.60 per dollar.
Put-Call Parity for Foreign Denominated Options:
For Euro-denominated options on dollars:
C €( x0 , K, T) = P€( x0 , K, T) + x0 exp[-T r$ ] - K exp[-T r€].
x0 is the exchange rate for dollars in terms of Euros, 0.8 Euro = $1, and K is in Euros.
This is analogous to what we had for options on stock: C(K, T) = P(K, T) + S0 e-Tδ - K e-Tr.
x 0 ⇔ S0 .
Euros act as money:
r€ ⇔ r.
Dollars act as the asset: r$ ⇔ δ.
Exercise: x0 = 0.8€ /$. T = 2 years. K = 0.833€ /$.
The dollar-denominated interest rate is 6% and the euro-denominated interest rate is 4%.
The option premium for a euro-denominated put on one dollar is 0.08 euro.
Determine the premium for a euro-denominated call on one dollar.
[Solution: 0.08 euros + (0.8 euros)e-(2)(0.06) - (0.833 euros)e-(2)(0.04) = 0.0206 euro.
Comment: Since the options are euro-denominated, the roles of the two interest rates are reversed.
Here euros act as the currency, while dollars act as the asset. Therefore, the interest rate on dollars is
analogous to the dividend rate on a stock.
Note how every term in the equation is in euros. This is one way to check your work.]
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 83
Different Points of View:
Assume that currently 1€ = $1.25. ⇔ 0.8€ = $1.
Sam is in the United States. Sam has 1000 dollar denominated calls with strike price $1.30, giving
him the option one year from now to buy 1000 Euros for 1300 dollars.
Exercise: In one year, 1€ = $1.4. What is the payoff on Samʼs calls?
[Solution: $1.4 - $1.30 = $0.10. For 1000 calls: (1000)($0.10) = $100.]
From Samʼs point of view, the future value of each of his calls in dollars is: (x1 - 1.30)+.
Johann is in Germany. Johann has 1300 Euro denominated puts with strike price: 1/1.30 = 0.7692,
giving him the option one year from now to sell 1300 dollars for: (1300)(0.7692) = 1000 Euros.
Exercise: In one year, 1€ = $1.4. What is the payoff on Johannʼs puts?
[Solution: $1 = 1/1.4 = 0.7143 euros. 0.7692 - 0.7143 = 0.0549 euros.
(1300)(0.0549) = 71.4 euros.
Comment: At this future exchange rate, 71.4 euros = (71.4€) ($1.4/€) = $100.]
From Johannʼs point of view, the payoff on each of his puts in euros is: (0.7692 - 1/x1 )+,
where x1 is the exchange rate to buy euros with dollars, and therefore 1/x1 is the exchange rate to
buy dollars with euros. The payoff in dollars on each of Johannʼs puts is:
x1 (0.7692 - 1/x1 )+ = (0.7692x1 - 1)+ = (x1 - 1.30)+ / 1.30.
Samʼs calls have value if x1 > 1.30. Johannʼs puts have value if 0.7682 > 1/x1 ⇔ x1 > 1.30.
Their options have value under the same conditions.
If x1 = 1.4, then as calculated above, Samʼs calls are worth $100 and Johannʼs puts are worth 71.4
euros or $100 at this future exchange rate. Johannʼs puts are worth the same as Samʼs calls.
Both Samʼs and Johannʼs positions give them the option one year from now to use 1300 dollars to
obtain 1000 Euros. Therefore, their positions must be worth the same amount.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 84
Exercise: If each of Johannʼs euro-denominated puts for one dollar with a strike price of 0.7692 euros
has a price of 0.0523 euros, determine the price of Samʼs dollar denominated one year call for one
euro, with a strike price of $1.30.
[Solution: Johannʼs position costs (1300)(0.0523) = 68 euros.
Thus each of Samʼs calls should cost: 68 euros/1000 = 0.068 euros.
At the current exchange rate, this is: (0.068 euros)($1.25/ euro) = $0.085.
Comment: Samʼs position is worth: (1000)($0.085) = $85.]
In general, in order to replicate the dollar denominated call, one needs to buy K euro denominated
puts. Thus the value of the call is K times the value of the put. In order to put the price of the call in
dollars rather than euros, one must multiply by the current exchange rate x0 .
C $ ( x0 , K, T) = x0 K Pf (1/x0 , 1/K, T).109
Exercise: Use the above formula to solve the previous exercise.
[Solution: C$ (1.25, 1.3, T) = (1.25) (1.3) P€ (1/1.25, 1/1.3, T) = (1.25)(1.3)(0.0523) = $0.085.]
Calls from Samʼs point of view was equivalent to puts from Johannʼs point of view.
A call from one point of view is a put from an opposite point of view.
Similarly, P$ ( x0 , K, T) = x0 K Cf (1/x0 , 1/K, T).
For a dollar denominated call, we have the option to use dollars to receive another currency.
For a dollar denominated put, we have the option to receive dollars by using another currency.
For a Euro denominated call, we have the option to use Euros to receive another currency.
For a Euro denominated put, we have the option to receive Euros by using another currency.
109
See equation 9.7 in Derivative Markets by McDonald.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 85
Derivation of the Relationship of Dollar and Foreign Denominated Currency Options:
Let x(t) be the exchange rate at time t; x(t) is the value in dollars of one of the foreign currency at time
t.110
The payoff at expiration of a K-strike dollar denominated call on a foreign currency is in dollars:
(x(T) - K)+.
Now at time T, this payoff would be equal to in the foreign currency:
(x(T) - K)+ / x(T) = (1 - K/x(T))+ = K (1/K - 1/x(T))+.
The payoff at expiration of a 1/K-strike foreign denominated put on dollars is in that foreign currency:
(1/K - 1/x(T))+.
Therefore, both in the foreign currency at expiration:
K (payoff of 1/K-strike foreign denominated put on dollars) =
(payoff of K-strike dollar denominated call on foreign currency).
Therefore, K(premium of 1/K-strike foreign denominated put on dollars) =
(premium in foreign currency of K-strike dollar denominated call on foreign currency) =
(premium in dollars of K-strike dollar denominated call on foreign currency) / x0 .111
In other words, C$ (x0 , K, T) / x0 = K Pf(1/x0 , 1/K, T).
⇒ C$ (x0 , K, T) = x0 K Pf(1/x0 , 1/K, T).
110
For example, the current exchange rate could be $0.8 per Euro.
For example, if one could pay 0.10 Euros to purchase the dollar denominated call on Euros, and the current
exchange rate were $0.8 per Euro, then one could also pay 0.10/0.8 = $0.125 to purchase this call.
111
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 86
Examples of Using Currency Options for Hedging:
Aidan Math is an American consulting actuary. On January 1, Aidan takes a three month assignment
consulting for the Seguros Popular, a Mexican Insurance Company. Aidan will be paid a lump sum
for his work on March 31 at the completion of his assignment. Aidan will be doing all the work at his
office in the United States.
The exchange rate on January 1 is 10 pesos per U.S. dollar, or $0.10 U.S. per Mexican Peso.
If Seguros Popular agrees to pay Aidan 600,000 Pesos on March 31, then Aidan faces a risk of the
exchange rate changing. On January 1, 600,000 Pesos is equivalent to $60,000. If on March 31 the
exchange rate were 12 pesos per U.S. dollar, then Aidan would receive only the equivalent of:
600,000/12 = $50,000.
Aidan would have the choice of just accepting this risk, or buying an option to hedge this risk.
Aidan could buy 60,000 peso denominated 3 month 10 peso European call options to buy
dollars.112 113 Then on March 31, if the exchange rate is more than 10 peso per dollar, he can exercise
his options and use the 600,000 Pesos he is paid by Seguros Popular to buy $60,000.
If instead on March 31 the exchange rate were 8 pesos per U.S. dollar, then Aidan would receive
the equivalent of 600,000/8 = $75,000. In this case, Aidan would benefit from the movement in the
exchange rate. In general, if on March 31 the exchange rate were less than 10 pesos per U.S. dollar,
then the 600,000 pesos Aidan receives would be the equivalent of more than $60,000.
Instead of accepting this possibility of being lucky, Aidan could have sold 600,000 peso
denominated 3 month 10 peso European put options to buy dollars.114 In that case, if the exchange
rate were less than 10 pesos per U.S. dollar, then the person who bought these puts from Aidan
would exercise them, and buy the 600,000 Pesos that Aidan receives from Seguros Popular for
$60,000.115 Aidan would no longer benefit if the exchange rate were less than 10 pesos per U.S.
dollar, but in any case he would have the money from selling the puts. This money would help to
offset the cost of the calls he bought.
112
Equivalently, Aidan could buy 600,000 dollar denominated 3 month $0.1 European put options to sell pesos.
Then on March 31, if the exchange rate is more than 10 peso per dollar, in other words less than $0.1 per peso, he
can exercise his option and sell the 600,000 Pesos he is paid by Seguros Popular for $60,000.
113
As will be discussed subsequently, such a call could be priced using the Black-Scholes formula. If σ = 12%, the
interest rate in pesos is 6%, and the interest rate in dollars is 5%, then the premium of each such call would be about
0.25 pesos. Thus 60,000 such calls would cost Aidan about 15,000 pesos on January 1, the equivalent of $1500.
114
Each such put would cost about 0.224 pesos, and Aidan would receive about $1460. Thus if Aidan both buys
the calls and sells the puts, then for a net cost of about $1500 - $1460 = $40, he would be unaffected by changes in
currency exchange rates. In this case, the calls cost more than the puts, since I have assumed the interest rate in
pesos is greater than the interest rate in dollars.
115
For example, if the exchange rate on March 31 were 8 pesos per U.S. dollar, obtaining 600,000 pesos would
normally cost 600,000/8 = $75,000. Using the options bought from Aidan, this person could instead obtain
600,000 pesos for only $60,000.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 87
Since Aidan will be doing all the work at his office in the United States, he would prefer to be paid in
dollars. If Seguros Popular agrees to pay Aidan $60,000 on March 31, then Seguros Popular rather
than Aidan would face a risk of the exchange rate changing. In this case, if on March 31 the exchange
rate were 12 pesos per U.S. dollar, then Seguros Popular would have to use: (12)(60000) =
720,000 pesos in order to pay $60,000 to Aidan.
Similar to the previous situation, Seguros Popular would have the choice of just accepting this risk, or
buying an option to hedge this risk. Seguros Popular could buy 60,000 peso denominated 3 month
10 peso European call options to buy dollars.116 Then on March 31, if the exchange rate is more
than 10 peso per dollar, Seguros Popular can exercise its options and use the 600,000 Pesos to
buy $60,000, with which to pay Aidan. If on March 31 the exchange rate is less than 10 peso per
dollar, then Seguros Popular would benefit from the exchange rate movement; they would need to
use less than 600,000 pesos in order to pay Aidan $60,000.
Rosetta Stone is a Mexican pension actuary. On January 1, Rosetta takes a six month assignment
consulting for the Grate American Cheese Company, which is based in the United States. Rosetta
will be paid a lump sum for her work on June 30 at the completion of her assignment. Rosetta will
be doing all the work at her office in Mexico.
The exchange rate on January 1 is 10 pesos per U.S. dollar, or $0.10 U.S. per Mexican Peso.
Exercise: Grate American Cheese Company has agreed to pay Rosetta $100,000. Briefly
discuss the currency exchange risk faced by Rosetta and how she can hedge it using options.
[Solution: If the exchange rate is less than 10 pesos per dollar, then the $100,000 Rosetta will be
paid is worth less than the 1 million pesos she expected. Rosetta could buy 100,000 peso
denominated 6 month 10 peso-strike European put options to sell dollars. Alternately, Rosetta
could buy 1 million dollar denominated 6 month $0.1-strike European call options to buy pesos.]
Exercise: Grate American Cheese Company has agreed to pay Rosetta 1 million pesos. Briefly
discuss the currency exchange risk faced by the company and how to hedge it using options.
[Solution: If the exchange rate is less than 10 pesos per dollar, then the 1 million pesos Rosetta will
be paid will cost the company more than the $100,000 it expected. The company could buy 1
million dollar denominated 6 month $0.1-strike peso European call options to buy pesos.
Alternately, it could buy 100,000 peso denominated 6 month 10 peso-strike European put options
to sell dollars.
Comment: In this and the previous exercise, Rosetta and the company each face the same risk, and
can hedge that risk the same way. If the company had agreed to pay Rosetta $50,000 and
500,000 pesos, then Rosetta and the company would share the currency exchange risk.]
116
Equivalently, Seguros Popular could buy 600,000 dollar denominated 3 month $0.1 European put options to sell
pesos.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 88
Problems:
6.1 (2 points) The spot exchange rate of dollars per euro is 1.1.
Dollar and euro interest rates are 5.0% and 6.0%, respectively.
The price of a $0.93 strike 18-month call option is $0.28.
What is the price of a similar put?
A. Less than $0.05
B. At least $0.05, but less than $0.10
C. At least $0.10, but less than $0.15
D. At least $0.15, but less than $0.20
E. At least $0.20
6.2 (2 points) Currently one can buy one dollar for 120 yen.
Dollar and yen interest rates are 4.5% and 3.0%, respectively.
The price of a 100 yen strike 2 year put option is 13 yen.
What is the price in yen of a similar call?
A. Less than 15
B. At least 15, but less than 20
C. At least 20, but less than 25
D. At least 25, but less than 30
E. At least 30
6.3 (2 points) Currently one can buy one Swiss Franc for 0.40 British Pounds (£0.4).
The interest rate for Swiss Francs is 5%.
The interest rate for British Pounds is 3%.
The price of a £0.5 strike 3 year call option is £0.05.
Determine the price in Swiss Francs of a 2 Swiss Franc strike 3 year put option.
A. 0.04
B. 0.05
C. 0.10
D. 0.125
E. 0.25
6.4 (2 points) The price of a $0.02 strike 1 year call option on an Indian Rupee is $0.00565.
The price of a $0.02 strike 1 year put option on an Indian Rupee is $0.00342.
Dollar and rupee interest rates are 4.0% and 7.0%, respectively.
How many dollars does it currently take to purchase one rupee?
A. 0.020
B. 0.021
C. 0.022
D. 0.023
E. 0.24
6.5 (2 points) The current exchange rate is $1.90 per British Pound.
A $2 strike 6 month call on the British Pound currently costs $0.0554.
What is the price in pounds of a £0.5 strike 6 month put on United States dollars?
A. £0.010
B. £0.015
C. £0.020
D. £0.025
E. £0.030
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
6.6 (2 points) Currently one can buy 1 Brazilian Real for $0.47.
Dollar and Real interest rates are 4.0% and 7.0%, respectively.
The price of a $0.50 strike 6-month call option to buy 1 Brazilian Real is $0.09.
What is the price of a similar put?
A. Less than $0.05
B. At least $0.05, but less than $0.10
C. At least $0.10, but less than $0.15
D. At least $0.15, but less than $0.20
E. At least $0.20
6.7 (2 points) The current exchange rate is $0.15 per Chinese Yuan Renminbi.
An 8 yuan strike 2 year call on the U.S. Dollar currently costs 0.568 yuan.
What is the price in dollars of a $0.125 strike 2 year put on yuan?
A. Less than $0.006
B. At least $0.006, but less than $0.008
C. At least $0.008, but less than $0.010
D. At least $0.010, but less than $0.012
E. At least $0.012
6.8 (2 points) Currently one can buy 1 U.S. Dollar for 11 Mexican Pesos.
Dollar and Peso interest rates are 4.0% and 6.0%, respectively.
The price of a 12 Peso strike 1-year put option is 1.36 Pesos.
What is the price in Pesos of a similar call?
A. Less than 0.60
B. At least 0.60, but less than 0.70
C. At least 0.70, but less than 0.80
D. At least 0.80, but less than 0.90
E. At least 0.90
6.9 (3 points) The spot exchange rate of dollars per South Korean Won is 0.00097.
Dollar and Won interest rates are 4.0% and 7.0%, respectively.
The price for one million $0.00100 strike 3 year call options on Won is $40.11.
What is the price in Won of a 1000 Won strike 3 year call on dollars?
A. 135
B. 140
C. 145
D. 150
E. 155
Page 89
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 90
6.10 (3 points) The spot exchange rate of yen per Euro is 125.
You are given the following premiums for five year Euro denominated calls on yen.
Strike
Premium
€0.004
€0.003618
€0.006
€0.002132
€0.008
€0.001072
€0.010
€0.000482
€0.012
€0.000203
€0.014
€0.000083
Determine the premium for a five year 100 yen strike put on Euros.
A. 5
B. 6
C. 9
D. 13
E. 17
6.11 (3 points) You are given:
(i) The current exchange rate is 1.16€/£.
(ii) A two-year European call option on Euros with a strike price of £0.80 sells for £0.120.
(iii) The continuously compounded risk-free interest rate on British Pounds is 5%.
(iv) The continuously compounded risk-free interest rate on Euros is 3%.
Calculate the price of 1000 two-year European call options on British Pounds with a strike price of
€1.25.
(A) €44
(B) €46
(C) €48
(D) €50
(E) €52
6.12 (5B, 5/99, Q.14) (1 point) An American manufacturer has contracted to sell a large order of
widgets for one million Canadian dollars, with payment due on delivery in six months.
How could the American company reduce foreign exchange risk?
1. Borrow Canadian currency against its receivables, convert to U.S. dollars at the spot rate,
and invest proceeds in the U.S.
2. Buy an option to sell Canadian dollars in six months at a specific price.
3. Buy Canadian currency forward.
A. 1
B. 1, 2
C. 1, 3
D. 2, 3
E. 1, 2, 3
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 91
6.13 (CAS3, 11/07, Q.15) (2.5 points)
A nine-month dollar-denominated call option on euros with a strike price of $1.30 is valued at $0.06.
A nine-month dollar-denominated put option on euros with the same strike price is valued at 0.18.
The current exchange rate is $1.2/euro and the continuously compounded risk-free rate on dollars is
7%. What is the continuously compounded risk-free rate on euros?
A. Less than 7.5%
B. At least 7.5%, but less than 8.5%
C. At least 8.5%, but less than 9.5%
D. At least 9.5%, but less than 10.5%
E. At least 10.5%
6.14 (MFE/3F, 5/09, Q.9) (2.5 points) You are given:
(i) The current exchange rate is 0.011$/¥.
(ii) A four-year dollar-denominated European put option on yen with a strike price of $0.008
sells for $0.0005.
(iii) The continuously compounded risk-free interest rate on dollars is 3%.
(iv) The continuously compounded risk-free interest rate on yen is 1.5%.
Calculate the price of a four-year yen-denominated European put option on dollars with a strike price
of ¥125.
(A) ¥35
(B) ¥37
(C) ¥39
(D) ¥41
(E) ¥43
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 92
Section 7, Exchange Options117
One can buy an option to exchange one asset for another.
For example, Steve is given the option to exchange one share of a competitorʼs Stock B for one
share of the stock of his employer Stock A. This is a call option with underlying asset Stock A and
strike asset Stock B.
A call exchange option allows one to exchange one share of a Stock B (strike asset) for one share of
the stock of Stock A (underlying asset). So rather than a strike price K in dollars, we have the option
to use an asset, Stock B, in order to obtain Stock A.
Assuming a European call, then at time T, Steve would make the exchange if a share of Stock A is
worth more than a share of Stock B. The future value of the call is (ST - QT)+, where S is price of the
underlying asset, Stock A, and Q is the price of the strike asset, Stock B.
Whether one will exercise the call depends on the relative value of the underlying asset and the
strike asset, both of which are random variables. If at expiration the underlying asset is worth more
than the strike asset, then it would be worthwhile to exercise the call.
For example, if the call expires in 2 years, among the many possible situations:
Price of Stock A
(Underlying Asset)
2 Years from now
Price of Stock B
(Strike Asset)
2 Years from now
Exercise
Exchange
Call?
100
90
Yes
110
130
No
120
100
Yes
70
80
No
A similar put exchange option would instead allow us to sell stock A in exchange for Stock B, in other
words, get B in exchange for A.
117
See Section 9.2 of Derivative Markets by McDonald. Also called an outperformance option.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 93
Put-Call Parity:
Buy one European call, sell a similar put, sell a prepaid forward contract on Stock A, and buy a
prepaid forward contract on Stock B.118
At Time T, you will get a share of Stock B due to the forward contract you bought.
If ST ≥ QT, you will exercise your call and exchange your Share of Stock B for a Share of Stock A.
Then you will provide this share of Stock A to the person who bought a forward contract from you.
If instead ST < QT, then the person who bought your put, will exchange a share of Stock A for your
share of Stock B. You will then provide this share of Stock A to the person who bought a forward
contract from you.
In either case, at time T you end up with nothing after all of the transactions.
Therefore, the correct price for this position is zero.
P (S) be the prepaid forward price of a share of Stock A.
Let F0,T
P (Q) be the prepaid forward price of a share of Stock B.
Let F0,T
P (S) + F P (Q).
We have shown that: 0 = CEur(S0 , Q0 , T) - PEur(S0 , Q0 , T) - F0,T
0,T
P (S) - F P (Q). 119 120
C E u r( S0 , Q0 , T) = PE u r( S0 , Q0 , T) + F0,T
0,T
If the current price of Stock A is $100 and the current price of Stock B is $120, and neither one pays
P (S) = 100 and F P (Q) = 120. Therefore,
dividends, then F0,T
0,T
C Eur(S0 , Q0 , T) = PEur(S0 , Q0 , T) + 100 - 120 = PEur(S0 , Q0 , T) - 20.
Exercise: Golden Goblin stock has a current price of $70, and will pay a dividend of $2 in 1 month.
Wolfram & Hart stock has a current price of $80, and will pay a dividend of $3 in 2 months.
A three month call on Golden Goblin with strike asset of Wolfram & Hart costs $0.89.
If r = 12%, determine the cost of the similar put.
[Solution: Prepaid forward price of Golden Goblin is: 70 - 2/exp[1%] = 68.02.
Prepaid forward price of Wolfram & Hart is: 80 - 3/exp[2%] = 77.06.
P (Q) - F P (S) = 0.89 + 77.06 - 68.02 = $9.93.]
P = C + F0,T
0,T
118
Under a prepaid forward contract, you pay now but receive the asset at a fixed point of time in the future.
See Equation 9.6 in Derivative Markets by McDonald.
120
This put-call parity follows from the fact that: (S(T) - Q(T))+ + Q(T) = (Q(T) - S(T))+ + S(T).
See remark (iii) on MFE Sample Exam Q.55.
119
Financial Economics,
HCMSA-2012-MFE/3F,
Page 94
12/14/11,
We can display the payoffs in this general situation in a table:121
Transaction
Time 0
Buy Call (to use Q to get S)
-C
Sell Put
P
Sell Prepaid Forward on S
Buy Prepaid Forward on Q
Total
At Expiration (TIme T)
If ST ≤ QT
if ST> QT
0
S T - QT
S T - QT
0
P (S)
F0,T
-ST
-ST
P (Q)
- F0,T
QT
QT
0
0
P (S) - F P (Q)
P - C + F0,T
0,T
P (S) - F P (Q) = 0, there would be an
Since the payoff at time T is always zero, unless P - C + F0,T
0,T
opportunity for arbitrage.
If instead the underlying asset pays dividends at a continuous rate δS and the strike asset pays
P (S) = S exp[-Tδ ] and F P (Q) = Q exp[-Tδ ].122
dividends at a continuous rate δQ, then F0,T
0
S
0
Q
0,T
Therefore: CE u r( S0 , Q0 , T) = PE u r( S0 , Q0 , T) + S0 exp[-TδS ] - Q0 exp[-TδQ ].
Exercise: The current price of Stock A is $100 and the current price of Stock B is $120.
Stock A pays dividends at a continuous annual rate of 1%, while Stock B pays dividends at a
continuous annual rate of 2%. What is the difference in price between a 3 year European call option
to exchange Stock B for Stock A, and the similar put?
[Solution: CEur(S0 , Q0 , 3) - PEur(S0 , Q0 , 3) = (100)exp[-(3)(1%)] - (120)exp[-(3)(2%)] = -15.97.
Comment: In this case, the put is worth more than the similar call. Exchange Call: can use B to get A.
Exchange Put: can use A to get B. Since Stock B is currently worth much more than Stock A, the put
is more likely to be worth something at expiration than is the call.]
P (S ) - F P (Q ).
C Eur(S0 , Q0 , T) - PEur(S0 , Q0 , T) = F0,T
0
0
0,T
Therefore, the premium of the put is equal to the premium of the similar call, if and only if the two
stocks have the same prepaid forward price.
121
See Table 9.2 in Derivative Markets by McDonald.
While you pay for the prepaid forward contract now, you do not get the dividends that are paid on the stock
between now and time T, since you do not own the stock until time T.
122
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 95
Note that this form of put-call parity includes the situations previously discussed as special cases.
For example, if the strike asset is money, specifically K dollars, then the “dividend” on money is
interest at rate r, and
C Eur(S0 , K, T) = PEur(S0 , K, T) + S0 exp[-TδS] - K exp[-Tr].
“Parity is the observation that buying a European call and selling a European put with
the same strike price and time to expiration is equivalent ot making a leveraged
investment in the underlying asset, less the value of cash payments to the underlying
asset over the life of the option.”123
Points of View:
As discussed previously, a call from one point of view is a put from another point of view.
Neither Stock A nor Stock B pay dividends.
Adam has a 2 year European call with underlying asset Stock A and strike asset Stock B.
Adam has the option 2 years from now to buy a share of Stock A in exchange for a share of Stock
B. He will do so if two years from now Stock A is worth more than Stock B.
Eve has a 2 year European put with underlying asset Stock B and strike asset Stock A.
Eve has the option 2 years from now to sell a share of Stock B in exchange for a share of Stock A .
She will do so if two years from now Stock A is worth more than Stock B.
Two years from now they will each own whichever of the two stocks is worth more.
Adam and Eve have options with the same outcome and therefore the same value.
In the absence of dividends, C(S0 , Q0 , T) = P(Q0 , S0 , T).
Call to buy Stock A for price Stock B: A is the underlying asset and B is the strike asset.
Put to sell Stock B for price Stock A: B is the underlying asset and A is the strike asset.
Despite the different language, both give you the option to use Stock B to obtain Stock A.
123
See page 305 of Derivatives Markets by McDonald.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 96
Problems:
7.1 (2 points) Consider the case of a European exchange option in which the underlying stock is
Widget Co. with a current price of $65 per share. The strike asset is Gadget Inc., with a per share
price of $62. Neither stock will pay dividends within the next 3 months. Interest rates are 5% and the
3 month call option is trading for $8. What is the price of the similar put?
A. 3 B. 5 C. 7 D. 9 E. 11
7.2 (2 points) A 2 year European exchange call option has underlying asset one share of MGH
Shipping with a current price of $100 per share. The strike asset is 2 shares of Galactus Transport,
with a per share price of $60. One will have the option to use 2 shares of Galactus in order to obtain
one share of MGH. The price of this call option is $11.
MGH Shipping pays dividends at a continuous rate of 2% per year.
Galactus Transport pays dividends at a continuous rate of 1% per year.
What is the price of the similar put?
A. Less than $20
B. At least $20, but less than $25
C. At least $25, but less than $30
D. At least $30, but less than $35
E. At least $35
7.3 (2 points) A 3 year European exchange option has underlying asset one share of Peach
Computer Company with a current price of $200 per share. The strike asset is the Silicon Valley
Stock Index, with a current price of $210. The price of this call option is $13.
Peach Computer Company pays dividends at a continuous rate of 3% per year.
Silicon Valley Stock Index pays dividends at a continuous rate of 1% per year.
What is the price of the similar put?
A. Less than $20
B. At least $20, but less than $25
C. At least $25, but less than $30
D. At least $30, but less than $35
E. At least $35
7.4 (2 points) Consider four-year European exchange options involving the stocks of Global
Dynamics and Deon International. The price of an option to exchange a share of Global Dynamics
for a share of Deon International is the same as the price of an option to exchange a share of Deon
International for a share of Global Dynamics.
Global Dynamics has a current price of 92 and pays dividends at a continuous rate of 2% per year.
Deon International pays dividends at a continuous rate of 0.9% per year.
What is the current price of a share of Deon International?
A. 85
B. 86
C. 87
D. 88
E. 89
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 97
7.5 (2 points) Daedalus stock has a current price of $65, and will pay a dividends of $1 in 2 months
and 5 months.
Hooper stock has a current price of $60, and will pay dividends of $1 in 1 month and 4 months.
A six month put on Daedalus stock with strike asset of Hooper stock costs $2.79.
If r = 6%, determine the cost of the similar call.
A. 7.00
B. 7.20
C. 7.40
D. 7.60
E. 7.80
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 98
Section 8, Futures Contracts124
As discussed previously, a forward contract is an agreement that sets the terms today,
including price and quantity, at which one will buy or sell an asset or commodity at a
specific time in the future. A futures contract is similar, except that the buyer and seller
post margin, the contract is marked to market, and the contract is typically traded on an
exchange.125
An Example of a Futures Contract:
Frank is a farmer. He expects to harvest 10,000 bushels of wheat in July. On April 15, Frank enters
into a futures contract for $4 per bushel agreeing to deliver 10,000 bushels of wheat on July 15 to
Momʼs Bakery.126 Frank has sold a futures contract.127 Frankʼs position would be referred to as the
short position; he has agreed to deliver wheat. Momʼs Bakery has bought a futures contract. Momʼs
Bakery has a long position; it has agreed to accept delivery of wheat at a predetermined price.128
Frank has guarded against the risk of the price of wheat falling between April and July. Note that
Frank has not guarded against for example his crop being destroyed by hail prior to harvest.
Momʼs Bakery has guarded against the risk of the price of wheat rising between April and July.
There is no investment required to enter a futures contract; however, Frank and Momʼs Bakery must
each post margin with the broker who arranged the contract. The total value is $40,000, so for
example, 10% margin would require that they each post a $4000 margin with the broker.
Let us assume that on April 16, the futures price on July 15 delivery of wheat is $3.99.
Then Frank has made money on his contract, since he would in essence get $4.00 per bushel for his
wheat, instead of the $3.99 he would expect to get in the absence of his futures contract.
The contract is marked to market daily.129 Thus Frankʼs margin account will be increased by:
($0.01)(10,000) = $100, while Momʼs margin account will be decreased by $100.
Each of them earns one day of interest. Assuming r = 5%, they earn $4000(e0.05/365 - 1) = $0.55.
124
See Section 5.4 of Derivative Markets by McDonald, on the syllabus of a previous exam.
One can think of mark to market as follows, as the for ward price moves up or down the owner of the futures
contract either makes or loses money right away.
In contrast, for a forward contract the profit or loss is realized at expiration.
126
Futures contracts can be on other assets like gold, stock indices such as the S&P 500, currency such as yen, etc.
127
A contract would be for 5000 bushels, so Frank has actually sold two contracts.
128
Some futures contracts are settled by delivery, but most are closed out prior to the delivery date.
When you enter into a futures contract you are obligated to make payments if the forward price moves in the wrong
direction for you. Thus the effect of the movement in the forward price is captured monetarily on an ongoing basis.
129
Therefore, Frank will take his profit now, and the new futures price is $3.99.
125
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 99
Thus Frankʼs margin account is now: $4000 + $100 + $0.55 = $4100.55.
Momʼs margin account is now: $4000 - $100 + $0.55 = $3900.55.
Exercise: On April 17 the futures price on July 15 delivery of wheat is $4.02.
What happens to Frankʼs and Momʼs margin accounts?
[Solution: The one day move in price is an increase of $0.03.
Frankʼs margin account has ($0.03)(10,000) = $300 subtracted.
He earns one day of interest of: $4100.55(e0.05/365 - 1) = $0.56.
Frankʼs margin account is now: $4100.55 - $300 + $0.56 = $3801.11.
Similarly, Momʼs margin account is now: $3900.55 + $300 + $0.54 = $4201.09.]
Options on Futures Contracts:
One can buy or sell calls and puts on futures contracts.
If a call is exercised, then the owner of the call acquires a long position in a futures contract with futures
price equal to the strike price of the call. The owner of the call if and when he exercises it, will enter
into an agreement to accept delivery of the commodity such as corn.
If a call is exercised, then the writer (seller) of the call acquires a short position in a futures contract with
futures price equal to the strike price of the call. The writer of the call if and when it is exercised, will
enter into an agreement to deliver the commodity such as gold.
If a put is exercised, then the owner of the put acquires a short position in a futures contract with
futures price equal to the strike price of the put. The owner of the put if and when he exercises it, will
enter into an agreement to deliver the commodity such as oil.
If a put is exercised, then the writer (seller) of the put acquires a long position in a futures contract with
futures price equal to the strike price of the put. The writer of the put if and when it is exercised, will
enter into an agreement to accept delivery of the commodity such as cattle.
An Example of Using Options on Futures:
We have seen how Frank can use a futures contract to guard against the risk of the price of wheat
falling between April and July. Frank could instead buy an option on such a futures contract.130
For example, on March 15 Frank could purchase a 3 month put to sell a futures contract on 10,000
bushels of wheat for July delivery. This put would give Frank the option to sell in June such a futures
contract at a given strike price.131
130
By buying an option on a futures contract Frank would get price protection without limiting his profit potential.
There are two important dates: when the option expires, May, and the delivery date in the futures contract, July.
One could instead of a European option have an American option, to be discussed subsequently. In that case, one
could exercise the option at any date up to expiration. Options on futures contracts are often American, with
expiration date one month before the delivery date of the commodity.
131
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 100
Assume for example the strike price were $4.00 per bushel. Then in June, Frank would have the
option to exercise his put and enter into an agreement to deliver wheat in July at $4.00 per bushel.
If the futures price in June were more than $4.00 per bushel, then Frank would not exercise his put. If
instead the futures price in June were less than $4.00 per bushel, then Frank would exercise his put.
For example, if the futures price in June were $3.80 per bushel, then Frank would exercise his put.132
He would now have locked in a price of $4.00 per bushel for his wheat, which is better than the
$3.80 he could get by entering into a futures contract in June.133
The payoff to Frank from the put would be: $4.00 - $3.80 = $0.20 per bushel. The payoff on his
put is: (Strike Price - Futures Price)+. This is the same payoff formula as for a put on stock, with the
futures price taking the place of the stock price. Thus even though there is no money required to
invest in a futures contract, one can apply the same mathematics to options on futures contracts as to
options on stocks, with appropriate modifications.134
Such a put would protect Frank against the price of wheat declining between March and June. If such
a price decline occurs, Frankʼs wheat is worth less, but his put is worth more.
Similarly, on March 15, Momʼs Bakery could purchase a 3 month $4 per bushel strike call to buy a
futures contract on 10,000 bushels of wheat for July delivery. This call would give Momʼs Bakery the
option to buy in June such a futures contract. In June, Momʼs would have the option to exercise its
call and enter into an agreement to accept delivery of wheat in July at $4.00 per bushel.
If the futures price in June were less than $4.00 per bushel, then Momʼs Bakery would not exercise
this call. If the futures price in June were more than $4.00 per bushel, then Momʼs Bakery would
exercise this call.
If for example the futures price in June were $4.10 per bushel, then Momʼs would exercise its call
and now have entered into a futures contract to receive wheat in July for $4.00 a bushel. Momʼs
would now have locked in a price of $4.00 per bushel for buying wheat, which is better than the
$4.10 it could get via a futures contract. The payoff to Momʼs from the call would be: 4.10 - 4.00 =
$0.10 per bushel. The payoff on this call is: (Futures Price - Strike Price)+. This is the same payoff
formula as for a call on stock, with the futures price taking the place of the stock price.
Such a call would protect Momʼs Bakery against the price of wheat increasing between March and
June. If such a price increase occurs, Momʼs Bakery would have to pay more for wheat, but the call
is worth more.
132
Alternately, Frank could make money by selling his put which would have increased in value.
In June, Frank could enter into a futures contract for July delivery of wheat with a $3.80 per bushel price. I
Instead using his put option on a futures contract, Frank can enter into a futures contract in June for July delivery of
wheat with a $4.00 per bushel price.
134
A form of put-call parity for options on future contracts will be discussed subsequently. In subsequent sections,
will be discussed how to price options on futures contracts, using either Binomial Trees or the Black-Scholes
formula, in a manner parallel to pricing options on stocks.
133
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 101
Advantages of Using Options on Futures rather than Options on the Underlying Asset:135
Options on futures contracts are preferable to options on the underlying asset when it is cheaper
and/or more convenient to deliver/receive the futures contract on the asset rather than the asset itself.
It is easier to deliver or receive a futures contract on cattle than it is to deliver or receive the cattle.
The futures contract will usually be closed out prior to delivery, and therefore, options on futures
contracts are usually settled in cash.
In many situations options on futures contracts tend to have lower transaction costs than options on
the underlying asset.
Trading on futures and options on futures are usually conducted side by side on the same
exchange, which makes it easier to hedge, etc., and makes the markets more efficient.
Put-Call Parity for Options on Futures Contracts:
PV[F] = F0,T e-rT.
Therefore, the put-call parity relationship for options on futures contracts is:
C = P + F0 , T e-rT - Ke-rT.
The put-call parity relationship for options on futures contracts can be obtained from
that for options on stocks by: S0 ⇔ F0 , T, and δ ⇔ r.
Exercise: On March 15 a 3 month $4 per bushel strike put on a futures contract on 10,000 bushels
of wheat for July 15 delivery has a premium of $815. (The buyer of this put would on June 15
have the option to sell, to the person who wrote the put, a futures contract on 10,000 bushels of
wheat for July 15 delivery at $4 per bushel.)
Currently, the futures price for July 15 delivery of wheat is $4.20 per bushel.
If r = 5%, what is the premium for the corresponding call?
[Solution: C = P + F0,T e-rT - Ke-rT =
$815 + (10,000)($4.20) e-(0.05)(1/4) - (10,000)($4.00) e-(0.05)(1/4) = $2790.]
135
See Options, Futures, and Other Derivatives by Hull, not on the syllabus.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 102
Problems:
8.1 (2 points) A futures contract provides for the delivery of 1000 barrels of Light Sweet Crude Oil
in September.
In January, a 4 month $100 per barrel strike call on this futures contract has a premium of $3400.
In January, the futures price for delivery of oil in September is $95 per barrel.
If r = 6%, determine the premium for the corresponding put.
A. $7900
B. $8000
C. $8100
D. $8200
E. $8300
8.2 (2 points) The premium for a 6 month 1500 strike put on a futures contract is 53.
The current futures price is 1600.
If r = 4%, determine the premium for the corresponding call.
A. 130
B. 140
C. 150
D. 160
E. 170
8.3 (3 points) On January 1, Saul enters into a futures contract on 5000 bushels of soybeans at
$12.40 per bushel for delivery in August. Saul has a long position; he has agreed to accept delivery
of soybeans in August. Saul post a margin of 10%. The interest rate is 6%.
The futures prices for August delivery are on the following dates:
January 1
$12.40
January 2
$12.60
January 3
$12.55
January 4
$12.30
January 5
$12.40
Determine the amount in Saulʼs margin account on each of these days.
8.4 (1 point) The premium for an at-the money call on a futures contract is $5. What is the premium
for an at-money put on the same futures contract with the same time until maturity?
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 103
Section 9, Synthetic Positions136
Put-call parity followed from creating a synthetic position which matched the cashflows of another
investment. We made use of the “law of one price” which states that two positions that
generate the exact same cashflows should have the same cost. Creating synthetic
positions can be useful to derive results, help ones understanding, and may also have practical
applications in investing.
Synthetic Share of Stock:
We can create a synthetic share of stock by buying a call, selling a put, lending the present value of
the strike price K, and lending the present value of any dividends.
Exercise: Show that this position is equivalent to buying the stock now and holding it until time T.
[Solution: If we buy and hold the stock, then we will receive the dividend payments and own the
stock at time T. In the above position, we will receive the dividend payments as that loan is repaid.
Also we will be paid back K at time T. If ST ≥ K, then we use our call to buy a share for K.
If instead ST < K, then we buy a share of Stock for K from the person to whom we sold the put. In
either case, we own a share of stock at time T, as well as having received the dividend payments.]
This shows that S0 = CEur(K, T) - PEur(K, T) + K e-rT + PV[Div], one of the put-call parity
relationships discussed previously.
One way to remember how to create a given synthetic position is to write the put-call
parity relationship with the desired item alone on one side of the equation.
If the stocks pays dividends continuously, then S0 = eδT {CEur(K, T) - PEur(K, T) + K e-rT}.
We can create a synthetic share of stock by: buying eδT calls, selling eδT puts, lending K e-(r-δ)T.
136
See pages 285-286 of Derivative Markets by McDonald.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 104
Synthetic Treasury Bill:
K e-rT + PV[Div] = S0 - CEur(K, T) + PEur(K, T).
Therefore, we can create a synthetic T-Bill by buying the stock, selling a call, and buying the put.
This is called a conversion. One is lending money.
If ST > K, then we sell our share for K to the person who bought the call. If instead ST ≤ K, then we
sell a share of stock for K using our put. In either case, we have K at time T.
Thus for an investment now, we get K at time T. This is equivalent to a risk free Treasury Bill that
returns K at time T, plus we get the present value of any dividends on the stock.
Alternately, one could also borrow the present value of dividends.
This is equivalent to a risk free Treasury Bill that returns K at time T, since:
Ke-rT = S0 - CEur(K, T) + PEur(K, T) - PV[Div].
Alternately, if dividends are paid continuously, then one could instead buy exp[-δT] shares of stock,
and reinvest the dividends. Then we would end up with one share of stock at time T. This is
equivalent to a risk free Treasury Bill that returns K at time T, since
Ke-rT= S0 e-δT - CEur(K, T) + PEur(K, T).
One can create a synthetic short T-Bill position by shorting the stock, buying a call, and selling the
put. This is called a reverse conversion. One is borrowing money.
Synthetic Options:
We can create a synthetic call by: buying the stock, buying a put, borrowing the present value of the
strike price, and borrowing the present value of any dividends.
Exercise: Show that this position is equivalent to buying the call.
[Solution: We repay one loan as we receive the dividend payments.
If ST ≥ K, then we sell our share for ST, repay the loan by paying K, and are left with ST - K.
If instead ST < K, then we use our put to sell our share for K, repay the loan by paying K, and are
left with 0. These are the same payoffs as if we had purchased the call.]
This shows that CEur(K, T) = S0 + PEur(K, T) - K e-rT - PV[Div], one of the put-call parity
relationships discussed previously.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 105
We can create a synthetic put by buying a call, shorting the stock, lending the present value of the
strike price, and lending the present value of any dividends.
Exercise: Show that this position is equivalent to buying the put.
[Solution: We borrow a share of stock, sell it for S0 , and will give this person a share of stock when
the option expires. We also must pay this person the stock dividends they would have gotten on
the stock, when they would have gotten them. We pay the dividend payments from one of the
loans. If ST ≥ K, then we buy a share for K, using our call and the money from the other loan.
After returning the share to the person we borrowed it from, we are left with nothing.
If instead ST < K, then we buy a share for ST. After returning the share to person we borrowed it
from, we are left with K - ST. These are the same payoffs as if we had purchased the put.]
This shows that PEur(K, T) = CEur(K, T) - S0 + K e-rT + PV[Div], one of the put-call parity
relationships discussed previously.
In the case of continuous dividends, PEur(K, T) = CEur(K, T) - e-δTS 0 + K e-rT.
Thus we can create a synthetic put by buying a call, shorting e-δT shares of stock,
and lending the present value of the strike price.
C Eur(K, T) = PEur(K, T) + e-δTS 0 - K e-rT.
Thus we can create a synthetic call by buying a put, buying e-δT shares of stock,
and borrowing the present value of the strike price.
In the case of discrete dividends, CEur(K, T) = PEur(K, T) + S0 - K e-rT - PV[Div].
Thus we can create a synthetic call by buying a put, buying a share of stock, borrowing the present
value of the strike price, and borrowing the present value of any dividends.
Synthetic Forward Contract:137
We can create a synthetic forward contract by buying a call and selling a put.
If ST ≥ K, then we buy a share for K, using our call.
If instead ST < K, then we buy a share for K, from the person to whom we sold the put.
Thus we have a forward contract to buy the stock at price K. This has present value PV0,T[F0,T - K].
Therefore, PV0,T[F0,T - K] = CEur(K, T) - PEur(K, T), as discussed previously.
137
See page 282 of Derivative Markets by McDonald.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 106
Summary of Synthetic Positions:
Synthetic Stock (discrete dividends): Buy Call, Sell Put, Lend PV of Strike, Lend PV of Dividends.
Synthetic Stock (continuous dividends): Buy eδT Calls, Sell eδT Puts, Lend K e-(r-δ)T.
Synthetic T-Bill (conversion): Buy Stock, Sell Call, Buy Put.138
Synthetic Short T-Bill (reverse conversion): Sell Stock, Buy Call, Sell Put.139
Synthetic Call (discrete dividends):
Buy Stock, Buy Put, Borrow PV of Strike, Borrow PV of Dividends.
Synthetic Call (continuous dividends): Buy e-δT shares of Stock, Buy Put, Borrow PV of Strike.
Synthetic Put (discrete dividends): Sell Stock, Buy Call, Lend PV of Strike, Lend PV of Dividends.
Synthetic Put (continuous dividends): Sell e-δT shares of Stock, Buy Call, Lend PV of Strike.
In each case, one way to remember how to create a given synthetic position is to write the put-call
parity relationship with the desired item alone on one side of the equation.
Synthetic Forward Contract on a Stock: Buy Call, Sell Put.
138
139
One could in addition borrow the present value of any dividends.
One could in addition lend the present value of any dividends.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 107
Problems:
9.1 (2 points) The price of a stock is $135. The stock pays dividends at the continuous rate of 2%.
The price of a European call with a strike price of $140 is $17.39 and the price of a European put
with a strike price of $140 is $19.14. Both options expire in eight months.
Calculate the annual continuously compounded risk-free rate on a synthetic T-Bill created using these
options.
A. Less than 3%
B. At least 3%, but less than 3%
C. At least 4%, but less than 4%
D. At least 5%, but less than 6%
E. At least 6%
9.2 (2 points) Consider options on a non-dividend paying stock.
The price of a 3 year European call with a strike price of $100 is $39.70.
The price of a 3 year European put with a strike price of $100 is $8.23.
r = 6%.
Calculate the price of a synthetic share of stock created using these options.
A. 100
B. 105
C. 110
D. 115
E. 120
9.3 (3 points) The stock of Rich Resorts has a current price of 150, and pays no dividends.
The stock of Cereal Growers has a current price of 80, and pays no dividends.
This coming summer will either be dry or wet.
If the summer is dry, then 4 months from now Rich Resorts stock price will be 180, and
Cereal Growers stock price will be 70.
If the summer is wet, then 4 months from now Rich Resorts stock price will be 130, and
Cereal Growers stock price will be 90.
What is the continuously compounded risk free rate?
A. 3.25%
B. 3.50%
C. 3.75%
D. 4.00%
E. 4.25%
9.4 (2 points) Consider options on a dividend paying stock.
The price of a 9-month European call with a strike price of $60 is $5.67.
The price of a 9-month European put with a strike price of $60 is $7.52.
r = 7%.
Using these options you create a synthetic 9-month forward contract on a share of stock.
Calculate the price, with payment on delivery, of this forward contract.
A. 55
B. 56
C. 57
D. 58
E. 59
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 108
9.5 (CAS3, 5/07, Q.13) (2.5 points) The price of a non-dividend paying stock is $85.
The price of a European call with a strike price of $80 is $6.70 and the price of a European put with a
strike price of $80 is $1.60.
Both options expire in three months.
Calculate the annual continuously compounded risk-free rate on a synthetic T-Bill created using these
options.
A. Less than 1%
B. At least 1%, but less than 2%
C. At least 2%, but less than 3%
D. At least 3%, but less than 4%
E. At least 4%
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 109
Section 10, American Options
An American style option may be exercised at any time up to the expiration date.
For example, Jim buys a 3 year American put option on ABC Stock with a strike price of $100.
This gives Jim the right to sell a share of ABC Stock for $100 at any time during the next 3 years.140
American Versus European Options:141
Since Jim may choose to exercise his American option prior to its expiration, an American option
may be worth more than a European option. Since Jim may choose to never exercise early and
instead only exercise his American option on its expiration date, an American option is worth at
least as much as the similar European option.
The put-call parity relationships discussed previously for the European options do not hold for an
American option.142
Maximum and Minimum Prices:
As discussed previously, S0 ≥ CEur ≥ (PV0,T[F0,T] - PV0,T[K])+ ≥ 0.
In fact, since a European call can only be exercised at expiration: PV0,T[F0,T] ≥ CEur.
Using the same reasoning as used previously with respect to a European option, an American call
option can not be worth more than the current stock price. Therefore, since CAmer ≥ CEur we have:
S 0 ≥ CAmer( S0 , K, T) ≥ CE u r( S0 , K, T) ≥ (PV0 , T[F0 , T] - PV0 , T[K])+ .143
In fact, since an American call can be exercised right away: CAmer ≥ (S0 - K)+.144
140
Recall that for a European style option, Jim would get the option to sell a share of stock on only a single day.
See Section 9.3 of Derivatives Markets by McDonald.
142
Appendix 9.A of McDonald, not on the syllabus, discusses parity bounds for American options.
CAmer(S0 , K, T) + K - PV0,T[F0,T] ≥ PAmer(S0 , K, T) ≥ CAmer(S0 , K, T) + PV0,T[K] - S.
P Amer(S0 , K, T) + S - PV0,T[K] ≥ CAmer(S0 , K, T) ≥ PAmer(S0 , K, T) + PV0,T[F0,T] - K.
Note that for European Options, CEur(S0 , K, T) + K - PV0,T[F0,T] ≥ CEur(S0 , K, T) + PV0,T[K] - PV0,T[F0,T]
= PEur(S0 , K, T) ≥ CEur(S0 , K, T) + PV0,T[K] - S.
143
See Equation 9.9 in McDonald.
144
Not discussed in Derivatives Markets by McDonald. If S0 > K, in other words if the call is in the money, then one
could exercise the American call immediately, getting a payoff of S0 - K.
If the dividend rate is small, then PV0,T[F0,T] - PV0,T[K] > S0 - K.
If the “dividend rate” were large, such as when dealing with options on currency, the reverse could be true.
141
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 110
As discussed previously, K ≥ PEur ≥ (PV0,T[K] - PV0,T[F0,T])+ ≥ 0.
In fact, since a European put can only be exercised at expiration: PV0,T[K] ≥ PEur.145
Using the same reasoning as used previously with respect to a European option, an American put
option can not be worth more than the strike price. Therefore, since PAmer ≥ PEur we have:
K ≥ PAmer( S0 , K, T) ≥ PE u r( S0 , K, T) ≥ (PV0 , T[K] - PV0 , T[F0 , T])+ .146
In fact, since an American put can be exercised right away: PAmer ≥ (K - S0 )+.147
Time Until Expiration:
Owen buys an American option with one year to expiration.
Tom buys an otherwise similar American option with two years to expiration.
Tom decides to exercise his option whenever Owen exercises his option within the first year, and
Tom throws his option away if he still holds it at the end of one year and it is not in the money.148
If Tom mirrors Owen actions, then Tomʼs option is worth the same as Owenʼs option.
In general, an American option is worth at least as much as a similar option with less
time to expiration.
If instead Tom employs a better strategy, then his option would be worth more than Owenʼs.
Tomʼs option provides more opportunities for action than Owenʼs option with a shorter time until
expiration. This is not true for a European option.
Olivia buys a one year European option and Tracy buys an otherwise similar two year European
option. Olivia can only exercise her option at time = 1, while Tracy can only exercise her option at
time = 2. While Tracyʼs option is usually worth more, this is not always the case.
145
See MFE Sample Exam Q. 26.
See Equation 9.10 in McDonald.
147
Not discussed in Derivatives Markets by McDonald. See MFE Sample Exam Q. 26.
If S0 < K, in other words if the put is in the money, then one could exercise the American put immediately, getting a
payoff of K - S0 .
148
Mirroring Owen is not an optimal strategy for Tom.
146
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 111
Exercise: Olivia and Tracy have purchased puts with strike prices of $100.
At time = 1 the stock is worth $0.01. Whose option is worth more?
[Solution: Olivia exercises her option at time = 1 and makes $99.99.
The present value is $99.99e-r.
The most Tracy can make by exercising her option at time = 2 is $100.
The present value is $100e-2r, less than $99.99e-r for any reasonable value of r.
Comment: We have shown that it is possible that a European put could be worth less than a similar
option with less time until expiration.]
Exercise: Olivia and Tracy have purchased calls with strike prices of $100.
At time = 1 the stock is worth $125. At time equal 1.1 the stock will pay a dividend of $120.
Whose option is worth more?
[Solution: Olivia exercises her option at time = 1 and makes $25.
After paying the very large dividend, the price of the stock will decline to something in the range of
$5. It is extremely unlikely that the price of the stock by time = 2 will exceed 100. At time = 2, the
only time Tracy can exercise her option, it is very unlikely that her call option is in the money.
Tracyʼs option will almost always turn out to be worthless.
Comment: We have shown that it is possible when dividends are paid that a European call could
be worth less than a similar option with less time until expiration.]
For European calls on stocks that do not pay dividends, the premium is the same as that of the
corresponding American call. Therefore, for a European call on a given non-dividend paying
stock, the call is worth at least as much as a similar call with less time to expiration.
However, the same relationship does not necessarily hold for either European calls on
stocks that pay dividends, or for European puts.149
Early Exercise:
It turns out that for an American option, depending on the current price of the stock, the strike price,
and the time until expiration, etc., sometimes it is worthwhile to exercise the option prior to the
expiration date, and sometimes it is not. Here we are talking about the present value of each
strategy, rather than a 20-20 hindsight view of what actually turned out to happen to the stock price
after one had made a decision.
As will be shown subsequently, if the stock pays no dividends, then it is never worthwhile
to exercise an American call option early.150
149
See page 297 of Derivative Markets by McDonald.
If the stock does pay dividends, then we are only concerned about any dividends paid between the time the
option is bought and the time it expires.
150
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 112
Thus for a stock that pays no dividends, an American call option is
worth the same as a European call option.
As discussed previously, an American option is worth at least as much as a similar American option
with less time to expiration.
Therefore, for a stock that pays no dividends, a European option is worth at least as much as a
similar option with less time to expiration.
Difficulty of Deciding Whether to Exercise Early:
There are various simple strategies one could use to decide when to exercise an American Option.
For example, one could choose to always exercise the option right away. Jim buys a 3 year
American put option on ABC Stock with a strike price of $100. If the current market price of ABC
Stock is $120, this is a stupid strategy, since Jim would be better off selling the stock for the $120
market price rather than the $100 strike price.
Assume that in one year, it turns out that ABC Stock has a market price of $90. If Jim buys a share
of ABC Stock for $90 and then exercises his option to sell at $100, he makes $10. However, this
may or may not be the best decision. If over the next two years, the price of ABC Stock never falls
below $90, then Jim should have exercised his option early. If over the next two years, the price of
ABC Stock falls to $80, then Jim could have made $20 if he had waited to exercise his option.
Out of the many possible paths, let us look at four, all of which have a price of $90 at time 1:151
Time
Price Path 1 Price Path 2 Price Path 3 Price Path 4
1
$90
$90
$90
$90
2
$95
$90
$85
$85
3
$100
$90
$80
$100
In path #1, Jim should exercise his option at time 1. If he instead waits until time 2 to exercise his
option, then he makes only $5 rather than $10, and also loses the time value of money. If he does
not exercise his option until expiration, he is out of luck and makes nothing!
In path #2, Jim should exercise his option at time 1. If he waits to exercise his option, he still makes
$10, but loses the time value of money.
In path #3, Jim should wait to exercise his option. If he waits until time 2 he makes $15 instead of
$10. If he waits until time 3 he makes $20!
In path #4, Jim should also wait to exercise his option. If he waits until time 2 he makes $15 instead
of $10. However, if he then waits until time 3 he makes nothing!
151
Jim can exercise his option on any day through expiration. I have only looked at three points in time for simplicity.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 113
Assuming Jim can not see the future, it can be a tough decision whether or not to exercise his
American option early. Ideally, Jim should exercise his option whenever the value of doing so is
greater than the present value of not doing so.
Thus a key tool would be some way for Jim to estimate the present value of his put option based
on the current stock price, time to expiration, strike price, etc. In subsequent sections, we will discuss
ways to tackle this problem.
Early Exercise of a Call:152
The continuation value of an option is the value of the option at a given point in time and
at a given stock price, if we do not exercise the option immediately.
In other words, the continuation value is the actuarial present vale of waiting.
One should exercise early if the continuation value of the option is less than the value of exercising
the option. In other words, one should exercise a call at time t < T if CAmer(St, K, T- t) < St - K.
If one exercises an American call before expiration, one will own the stock, and benefit from
dividends. If there is a particularly large stock dividend, this would make it worthwhile to exercise the
call just before the payment of the dividend.153
By exercising an American call before expiration, one loses interest one can earn on the strike price.
Also by exercising early, one loses the “implicit protection against the stock price moving below the
strike price.” If one exercises the call early and buys the stock, we take on the risk that the stock price
might go down a lot. We can understand this by examining the value of a similar European option.
Put-call parity states that for a stock that pays dividends continuously, at time t:
C Amer(St, K, T - t) ≥ CEur(St, K, T - t) = PEur(St, K, T - t) + Ste-δ(T-t) - K e-r(T-t)
= (St - K) + PEur(St, K, T - t) + K(1 - e-r(T-t)) - St {1 -e-δ(T-t)}
= (Exercise Value of the Call) + (Insurance against ST < K) + (Time value of money on K)
- (Present Value of future Dividends).
For a stock that does not pay dividends, put-call parity states that at time t:
C Amer(St, K, T - t) = CEur(St, K, T - t) = (St - K) + PEur(St, K, T - t) + K{1 - e-r(T-t)}
= (Exercise Value of the call) + (Insurance against ST < K) + (Time value of money on K).154
152
See pages 294 to 296 and Section 11.1 of Derivatives Markets by McDonald.
A dividend is paid to the person who owns the stock on a specific day. If for example that day is March 30, and if
one buys the stock on March 31, then the price is ex-dividend. If Dave buys the stock from Judy on March 31, Judy
will receive the dividend, even if the dividend is actually paid in early April.
154
See equation 9.11 in Derivatives Markets by McDonald.
153
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 114
PEur(St, K, T - t) ≥ 0 and K(1 - e-r(T-t)) ≥ 0.
Therefore, for a stock that does not pay dividends: CAmer(St, K, T - t) ≥ St - K.
Therefore, there is never a reason to exercise an American call early in the absence of dividends.
For American calls there are two reasons to wait and one reason to exercise early:
1. The time value of the strike price is lost if one exercises early.
2. The implicit insurance protection against the stock price moving below the strike price
is lost if one exercises early.
3. The value of dividends is gained if one exercises early.
If we exercise early, then we spend the strike price. If instead we were to wait to exercise, then we
do not spend the strike price and we can earn interest on that money.
If we exercise early, then we own the stock, and risk the stock price going down. If instead we were
to wait to exercise, then we retain this implicit insurance protection against the stock price moving
below the strike price.155 156 Put another way, if we wait to exercise the call, we avoid the possibility
of capital losses on the stock.
If we exercise the call early, then we own the stock and get the dividends.
C Amer(St, K, T - t) ≥ CEur(St, K, T - t) = PEur(St, K, T - t) + PVt,T[Ft,T] - PVt,T[K]
≥ PVt,T[Ft,T] - PVt,T[K]. Therefore, CAmer ≥ PV[Ft,T] - PV[K], as discussed previously.
The payoff for exercising early an American call at time t is:
S t - K = PV[Ft,T] + PV[Div] - PV[K] - (K - PV[K])
= PV[Ft,T] - PV[K] + PV[Div] - (K - PV[K]) ≤ CAmer(St, K, T - t) + PV[Div] - (K - PV[K]).
Therefore, if PVt,T[Div] - (K - PVt,T[K]) < 0, then St - K ≤ CAmer,
and it would not be worthwhile to exercise early.
If K - PVt , T[K] > PVt , T[Div], then one should not exercise an American call early
at time t.157 158 In other words, if K{1 - exp[-r(T- t)]} is greater than the present value of the future
dividends that will be paid until the call expires, then it is better to wait rather than exercise early.
For example, if K = $100, r = 5%, and there are 6 months to expiration of an American call, then if
PV[Div] < 100(1 - e-0.025) = $2.47, it would not be worthwhile to exercise at this time. For example,
if there is a single $2 dividend tomorrow it would not be worthwhile to exercise at this time.
155
Personally, I do not find McDonaldʼs use of the term “insurance implicit in the call” to be helpful.
This implicit protection can be thought of as the implicit put from put-call parity.
157
See Equation 9.12 in Derivative Markets by McDonald. K - PVt,T[K] = K(1 - exp[-r(T- t)])
Specifically, if there are no dividends, then one should not exercise an American call early.
158
If K - PV[K] < PV[Div], then it may or may not make sense to exercise an American call early.
Another necessary condition to exercise early an American call is that St > K.
156
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 115
In this case, the benefit of getting interest on the strike price exceeds the benefit from exercising
early and thus receiving the stock dividends; therefore, it is optimal to wait.
If instead, PV[Div] > $2.47, then it might be worthwhile to exercise at this time. For example, if there
is a single $3 dividend two months from now, then PV[Div] = 3e-0.05/6 = 2.98 > 2.47, and it might
be worthwhile to exercise at this time.
In this case, the benefit from exercising early and thus receiving the stock dividends exceeds the
benefit of getting interest on the strike price. However, if we exercise early, besides losing the
interest on the strike, we would also lose the implicit insurance protection. Therefore, it may or may
not be optimal to exercise early.
In fact, CAmer(St, K, T - t) ≥ PEur(St, K, T - t) + PVt,T[Ft,T] - PVt,T[K].
Therefore, the payoff for exercising early an American call at time t is:
St - K = PV[Ft,T] + PV[Div] - PV[K] - (K - PV[K]) = PV[Ft,T] - PV[K] + PV[Div] - (K - PV[K])
≤ CAmer (St, K, T - t) - PEur(St, K, T - t) + PV[Div] - (K - PV[K]).
Therefore, if PVt,T[Div] - (K - PVt,T[K]) - PEur(St, K, T - t) < 0, then St - K ≤ CAmer,
and it would not be worthwhile to exercise early.
If PEur(St, K, T - t) + K - PVt,T[K] > PVt,T[Div], then one should not exercise an American call
early at time t.159 If were to exercise an American Call early, then we would own the stock and
collect its dividends, but we would lose the interest on the strike price and also lose the implicit
insurance protection represented by the put premium.
Let us assume there is a single $3 dividend two months from now.
Jubal and Anderson each have identical 6 month $100 strike American calls.
Jubal exercises his call now, pays $100 and owns the stock.
Anderson instead invests $100 e-0.05/6 = $99.17 at the risk free rate for two months.
Two months from now, just before the dividend is paid, Anderson has $100 and uses it exercises
his call and own the stock. At that point in time both Jubal and Anderson own the stock and get the
dividend payment. However, Jubal invested $100 today, while Anderson only invested $99.17.
Thus Andersonʼs strategy was better.160
In general, if it is optimal to exercise an American call early, and dividends are paid at
discrete times, then it is best to exercise right before the payment of a (large) dividend,
in other words at the last moment before the ex-dividend date.161
159
Not discussed in Derivative Markets by McDonald.
While Anderson does better than Jubal, it may still not be optimal to exercise the call early.
161
Note that this statement is conditional on it being optimal to exercise the call early.
160
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 116
Early Exercise of a Put:162
The situation for early exercise of puts is somewhat reversed from calls.
For American puts there are two reasons to wait and one reason to exercise early:
1. The time value of the strike price is gained if one exercises early.
2. The implicit insurance protection against the stock price moving above the strike price
is lost if one exercises early.
3. The value of dividends is lost if one exercises early.
If we exercise early, then we receive the strike price and we can earn interest on that money.
If we exercise the put early and use it to sell the stock, then we do not own the stock, and fail to gain
if the stock price goes up. If instead we were to wait to exercise, then we retain this implicit insurance
protection against the stock price moving above the strike price.163 Saying it another way, if we wait
to exercise the put, we retain the possibility of additional capital gains on the stock.
If we exercise the put early, then we do not own the stock and do not get the dividends.
It can make sense to exercise an American put early, whether or not the stock pays dividends.
By Put-Call Parity: PEur = Ke-rT - ST + PV[Div] + CEur.
PAmer ≥ PEur. ⇒
PAmer ≥ K - ST + PV[Div] + CEur - K (1 - e-rT) =
(Exercise Value of the Put) + (Present Value of Future Dividends)
+ (Insurance against ST > K) - (Time Value of Money on Strike).
Moneyness:164
If at a point in time it is worthwhile to exercise an American call with a strike price of K, it also makes
sense to exercise an otherwise similar call with a strike price lower than K. For example, if one should
exercise an option to buy at 100, then one should also exercise a similar option to buy at 80.
If at a point in time it is worthwhile to exercise an American put with a strike price of K, it also makes
sense to exercise an otherwise similar put with a strike price higher than K. For example, if one
should exercise an option to sell at 60, then one should also exercise a similar option to sell at 70.
If it makes sense to exercise an option that is in the money, then it also makes sense to exercise an
option that is more in the money.
162
See pages 296 to 297 and Section 11.1 of Derivatives Markets by McDonald.
This implicit protection can be thought of as the implicit call from put-call parity.
164
See page 304 of Derivatives Markets by McDonald.
163
Financial Economics,
HCMSA-2012-MFE/3F,
12/14/11,
Page 117
Early Exercise Boundaries:
For an American option, using for example Binomial Trees to be discussed subsequently, one can
determine the stock price at which one should exercise early.
In the case of a call, one would exercise early if the stock price were greater than a given value. For
example, at time 1.5 one might exercise a call early if the stock price were more than 200. In the
case of a put, one would exercise early if the stock price were less than a given value. For example,
at time 3 one might exercise a put early if the stock price were less than 55.
For a given option, as a function time, the set of stock prices at which the continuation
value is equal to the value of immediate exercise is called the early exercise boundary.
The following graph shows early exercise boundaries for a 5-year 100 strike American call, for the
volatility of the future price of the stock, σ, either 30% or 50%:165 166
Stock Price
350
sigma= 0.5
300
250
sigma= 0.3
200
150
1
2
3
4
5
Time
We note two features. First, as σ increases, the exercise boundary is higher. For larger σ, the implicit
insurance protection is larger. Therefore, the larger σ, the larger the continuation value. Therefore, a
higher stock price, and thus a higher value of exercising the call immediately, is required in order to
make it worthwhile to exercise the option early.
Second, as the option gets closer to expiration, the exercise boundary approaches the strike price.
As the option approaches expiration, we lose less insurance protection by exercising early.
165
Based on Figure 11.1 of Derivatives Markets by McDonald, which was computed using Binomial Trees, to be
discussed subsequently. The volatility will be discussed in subsequent sections.
166
While one needs to know how to work with Binomial Trees, the huge amount of work required to determine the
exercise boundary is well beyond what you could be required to do on the exam, even if one were supplied with a
computer!
Financial Economics,
HCMSA-2012-MFE/3F,
12/14/11,
Page 118
The following graph shows early exercise boundaries for a 5-year 100 strike American put, for σ
either 30% or 50%:167
Stock Price
100
90
80
70
60
sigma= 0.3
50
40
sigma= 0.5
30
1
2
3
4
5
Time
We note two features. First, as σ increases, the exercise boundary is lower. For larger σ, the implicit
insurance protection is larger. Therefore, the larger σ, the larger the continuation value. Therefore, a
lower stock price, and thus a higher value of exercising the put immediately, is required in order to
make it worthwhile to exercise the option early.
Second, as the option gets closer to expiration, the exercise boundary approaches the strike price.
As the option approaches expiration, we lose less insurance protection by exercising early.
For both puts and calls, the early-exercise criterion becomes less stringent as the option
has less time until expiration.
For both puts and calls, the early-exercise criterion becomes more stringent as the
volatility of the stock increases.
Put-Call Parity:168
The put-call parity relationships discussed previously for the European options do not hold for an
American option. However, it is the case that for American Options:
S 0 - PV[Div] - K ≤ CAmer - PAmer ≤ S0 - Ke-rT.
167
Based on Figure 11.2 of Derivatives Markets by McDonald, which was computed using Binomial Trees, to be
discussed subsequently.
168
See Appendix 9.A of McDonald, not on the syllabus, discusses parity bounds for American options.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 119
Properties of Premiums of American Options:169
The same three properties hold for American Options as for European Options.
Strike Prices: K1 < K2 < K3 . C is the premium for a Call, while P is the premium for a put.
Condition
Arbitrage if the Condition is Violated
C(K1 ) ≥ C(K2 ).
Buy the K1 Call and Sell the K2 Call (Call Bull Spread)
C(K1 ) - C(K2 ) ≤ K2 - K1 .
Sell the K1 Call and Buy the K2 Call (Call Bear Spread)
C(K1) - C(K2 )
K2 - K1
Buy λ = (K3 - K2 )/(K3 - K1 ) of K1 , Sell 1 of K2 , Buy 1 - λ of K3 .170
≥
C(K2) - C(K3 )
.
K3 - K2
(Asymmetric butterfly spread)
P(K2 ) ≥ P(K1 ).
Sell the K1 Put and Buy the K2 Put (Put Bear Spread)
P(K2 ) - P(K1 ) ≤ K2 - K1 .
Buy the K1 Put and Sell the K2 Put (Put Bull Spread)
P(K2 ) - P(K 1)
K2 - K1
Buy λ = (K3 - K2 )/(K3 - K1 ) of K1 , Sell 1 of K2 , Buy 1 - λ of K3 .171
≤
169
P(K3 ) - P(K 2)
.
K3 - K2
(Asymmetric butterfly spread)
See page 300 of Derivative Markets by McDonald.
There are other sets of amounts of each call that would also demonstrate arbitrage.
171
There are other sets of amounts of each put that would also demonstrate arbitrage.
170
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 120
Proving the Three Properties of Call Premiums:172
Let us assume we have three otherwise similar calls with different strikes.
Strike Prices: K1 < K2 < K3 . C is the premium for a call.
If C(K2 ) > C(K1 ), then we can demonstrate arbitrage by buying a call bull spread:
buy the K1 call and sell the K2 call.173
We lend: C(K2 ) - C(K1 ) > 0.
If we are dealing with European options, then if at expiration ST ≤ K1 , both options expire worthless.
So we have: {C(K2 ) - C(K1 )} erT > 0.
If at expiration K2 ≥ ST > K1 , then the option we sold expires worthless.
So we have: (ST - K1 ) + {C(K2 ) - C(K1 )} erT > 0.
If at expiration ST > K2 , then both options expire in the money.
So we have: (ST - K1 ) - (ST - K2 ) + {C(K2 ) - C(K1 )} erT = (K2 - K1 ) + {C(K2 ) - C(K1 )} erT > 0.
Thus we have demonstrated arbitrage for European calls.
If instead we have American calls, we can deal with the same three cases.
If the call we sold is not exercised early, then the same logic applies as for European calls.174
If instead the option we sold is exercised early at time t, then it must be that St > K2 .
We can then also exercise at time t the call that we bought.175
We then have: (St - K1 ) - (St - K2 ) + {C(K2 ) - C(K1 )} ert = (K2 - K1 ) + {C(K2 ) - C(K1 )} ert > 0.
Thus we have demonstrated arbitrage for American calls.
172
See Appendix 9.B of McDonald, not on the syllabus.
The low-strike call is underpriced so we buy it; the high-strike call is overpriced so we sell it.
174
We can wait to expiration to exercise the call we bought and be guaranteed ending with a positive amount.
175
If at time t we can sell the call we bought for more than its exercise value, so much the better for us.
173
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 121
If C(K1 ) - C(K2 ) > K2 - K1 , then we can demonstrate arbitrage by buying a call bear spread:
sell the K1 call and buy the K2 call.176
We lend: C(K1 ) - C(K2 ) > K2 - K1 > 0.
If we are dealing with European options, then if at expiration ST ≤ K1 , both options expire worthless.
So we have: {C(K1 ) - C(K2 )} erT > 0.
If at expiration K2 > ST ≥ K1 , then the option we bought expires worthless.
So we have: {C(K1 ) - C(K2 )} erT - (ST - K1 ) > {K2 - K1 } erT - (ST - K1 ) >
{K2 - K1 } erT - (K2 - K1 ) > 0.
If at expiration K2 < ST, then both options expire in the money.
So we have: {C(K1 ) - C(K2 )} erT + (ST - K2 ) - (ST - K1 ) = {C(K1 ) - C(K2 )} erT - (K2 - K1 ) >
{K2 - K1 } erT - (K2 - K1 ) > 0.
Thus we have demonstrated arbitrage for European calls.
If instead we have American calls, we can deal with the same three cases.
If the call we sold is not exercised early, then the same logic applies as for European calls.177
If instead the option we sold is exercised early at time t, then it must be that St > K1 .
If K2 > St ≥ K1 , then the option we bought expires worthless.
So we have: {C(K1 ) - C(K2 )} ert - (St - K1 ) > {K2 - K1 } ert - (St - K1 ) >
{K2 - K1 } ert - (K2 - K1 ) > 0.
If St > K2 , then we can also exercise at time t the call that we bought.178
We then have: {C(K1 ) - C(K2 )} ert + (St - K2 ) - (St - K1 ) = {C(K1 ) - C(K2 )} ert - (K2 - K1 ) >
{K2 - K1 } ert - (K2 - K1 ) > 0.
Thus we have demonstrated arbitrage for American calls.
176
This the reverse of what we did to demonstrate arbitrage if the previous property was violated.
We can wait to expiration to exercise the call we bought and be guaranteed ending with a positive amount.
178
If at time t we can sell the call we bought for more than its exercise value, so much the better for us.
177
HCMSA-2012-MFE/3F,
If
Financial Economics,
12/14/11,
Page 122
C(K1) - C(K2 ) C(K2) - C(K3 )
<
⇔ (K2 - K1 ) {C(K2 ) - C(K3 )} - (K3 - K2 ) {C(K1 ) - C(K2 )} > 0
K2 - K1
K3 - K2
⇔ (K3 - K1 ) C(K2 ) - (K3 - K2 ) C(K1 ) - (K1 - K2 ) C(K3 ) > 0,
then we can demonstrate arbitrage by buying an asymmetric butterfly call spread:
Buy λ = (K3 - K2 )/(K3 - K1 ) of low strike, sell 1 of medium strike, and, buy 1 - λ of the high strike call.
We lend:
C(K2 ) - λ C(K1 ) - (1 - λ) C(K3 ) =
(K3 - K1 )C(K2 ) - (K3 - K 2)C(K1) - (K 1 - K2)C(K3 )
> 0.
K3 - K1
If we are dealing with European options, then if at expiration ST ≤ K1 , all options expire worthless.
So we have: {C(K2 ) - λ C(K1 ) - (1 - λ) C(K3 )} erT > 0.
If at expiration K2 ≥ ST > K1 , then the low strike option we bought expires in the money.
So we have: {C(K2 ) - λ C(K1 ) - (1 - λ) C(K3 )} erT + λ(ST - K1 ) > 0.
If at expiration K3 ≥ ST > K2 , then the low and medium strike options expire in the money.
So we have: {C(K2 ) - λ C(K1 ) - (1 - λ) C(K3 )} erT + λ(ST - K1 ) - (ST - K2 ) >
C(K2 ) - λ C(K1 ) - (1 - λ) C(K3 ) - (1 - λ)ST - λK1 + K2 >
C(K2 ) - λ C(K1 ) - (1 - λ) C(K3 ) - (1 - λ)K3 - λK1 + K2 =
C(K2 ) - λ C(K1 ) - (1 - λ) C(K3 ) - K3
K2 - K 1
K - K2
- K1 3
+ K2 =
K3 - K 1
K3 - K 1
C(K2 ) - λ C(K1 ) - (1 - λ) C(K3 ) > 0.179
If at expiration K3 < ST, then all three options expire in the money.
So we have: {C(K2 ) - λ C(K1 ) - (1 - λ) C(K3 )} erT + λ(ST - K1 ) - (ST - K2 ) + (1 - λ)(ST - K3 ) >
C(K2 ) - λ C(K1 ) - (1 - λ) C(K3 ) - (1 - λ)K3 - λK1 + K2 =
C(K2 ) - λ C(K1 ) - (1 - λ) C(K3 ) - K3
K2 - K 1
K - K2
- K1 3
+ K2 =
K3 - K 1
K3 - K 1
C(K2 ) - λ C(K1 ) - (1 - λ) C(K3 ) > 0.
Thus we have demonstrated arbitrage for European calls.
179
Note that from the definition of λ = (K3 - K2 )/(K3 - K1 ), it follows that: K2 - λK1 - (1 - λ)K3 = 0.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 123
If instead we have American calls, we can deal with the same four cases.
If the call we sold is not exercised early, then the same logic applies as for European calls.180
If instead the option we sold is exercised early at time t, then it must be that St > K2 .
If K3 > St ≥ K2 , then the high strike option we bought has an exercise value of zero and we can
exercise the low strike call we bought.181
So we have: {C(K2 ) - λ C(K1 ) - (1 - λ) C(K3 )} ert + λ(St - K1 ) - (St - K2 ) >
C(K2 ) - λ C(K1 ) - (1 - λ) C(K3 ) - (1 - λ)St - λK1 + K2 >
C(K2 ) - λ C(K1 ) - (1 - λ) C(K3 ) - (1 - λ)K3 - λK1 + K2 =
C(K2 ) - λ C(K1 ) - (1 - λ) C(K3 ) - K3
K2 - K 1
K - K2
- K1 3
+ K2 =
K3 - K 1
K3 - K 1
C(K2 ) - λ C(K1 ) - (1 - λ) C(K3 ) > 0.
If St > K3 , then we can also exercise at time t both calls that we bought.182
So we have: {C(K2 ) - λ C(K1 ) - (1 - λ) C(K3 )} ert + λ(St - K1 ) - (St - K2 ) + (1 - λ)(St - K3 ) >
C(K2 ) - λ C(K1 ) - (1 - λ) C(K3 ) - (1 - λ)K3 - λK1 + K2 =
C(K2 ) - λ C(K1 ) - (1 - λ) C(K3 ) - K3
K2 - K 1
K - K2
- K1 3
+ K2 =
K3 - K 1
K3 - K 1
C(K2 ) - λ C(K1 ) - (1 - λ) C(K3 ) > 0.
Thus we have demonstrated arbitrage for American calls.
180
We can wait to expiration to exercise the calls we bought and be guaranteed ending with a positive amount.
If at time t we can sell the calls we bought for more than their exercise value, so much the better for us.
182
If at time t we can sell the calls we bought for more than their exercise value, so much the better for us.
181
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 124
Proving the Three Properties of Put Premiums:183
Let us assume we have three otherwise similar puts with different strikes.
Strike Prices: K1 < K2 < K3 . P is the premium for a put.
If P(K2 ) < P(K1 ), then we can demonstrate arbitrage by buying a put bear spread:
sell the K1 put and buy the K2 put.184
We lend: P(K1 ) - P(K2 ) > 0.
If we are dealing with European options, then if at expiration ST ≥ K2 , both options expire worthless.
So we have: {P(K1 ) - P(K2 )} erT > 0.
If at expiration K2 > ST ≥ K1 , then the option we sold expires worthless.
So we have: (K2 - ST) + {P(K1 ) - P(K2 )} erT > 0.
If at expiration K1 > ST, then both options expire in the money.
So we have: (K2 - ST) - (K1 - ST) + {P(K1 ) - P(K2 )} erT = (K2 - K1 ) + {P(K1 ) - P(K2 )} erT > 0.
Thus we have demonstrated arbitrage for European puts.
If instead we have American puts, we can deal with the same three cases.
If the put we sold is not exercised early, then the same logic applies as for European puts.185
If instead the option we sold is exercised early at time t, then it must be that St < K1 .
We can then also exercise at time t the put that we bought.186
We then have: (K2 - St) - (K1 - St) + {P(K1 ) - P(K2 )} ert = (K2 - K1 ) + {P(K1 ) - P(K2 )} ert > 0.
Thus we have demonstrated arbitrage for American puts.
183
See Appendix 9.B of McDonald, not on the syllabus.
The high-strike put is underpriced so we buy it; the low-strike put is overpriced so we sell it.
185
We can wait to expiration to exercise the put we bought and be guaranteed ending with a positive amount.
186
If at time t we can sell the put we bought for more than its exercise value, so much the better for us.
184
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 125
If P(K2 ) - P(K1 ) > K2 - K1 , then we can demonstrate arbitrage by buying a put bull spread:
buy the K1 put and sell the K2 put.187
We lend: P(K2 ) - P(K1 ) > K2 - K1 > 0.
If we are dealing with European options, then if at expiration ST ≥ K2 , both options expire worthless.
So we have: {P(K2 ) - P(K1 )} erT > 0.
If at expiration K2 > ST ≥ K1 , then the option we bought expires worthless.
So we have: {P(K2 ) - P(K1 )} erT - (K2 - ST) > {K2 - K1 } erT - (K2 - ST) >
{K2 - K1 } erT - (K2 - K1 ) > 0.
If at expiration K1 > ST, then both options expire in the money.
So we have: {P(K2 ) - P(K1 )} erT + (K1 - ST) - (K2 - ST) = {P(K2 ) - P(K1 )} erT - (K2 - K1 ) >
{K2 - K1 } erT - (K2 - K1 ) > 0.
Thus we have demonstrated arbitrage for European puts.
If instead we have American puts, we can deal with the same three cases.
If the put we sold is not exercised early, then the same logic applies as for European puts.188
If instead the option we sold is exercised early at time t, then it must be that St < K2 .
If K2 > St ≥ K1 , then the option we bought expires worthless.
So we have: {P(K2 ) - P(K1 )} ert - (K2 - St) > {K2 - K1 } ert - (K2 - St) >
{K2 - K1 } ert - (K2 - K1 ) > 0.
If St < K1 , then we can also exercise at time t the put that we bought.189
We then have: {P(K2 ) - P(K1 )} ert - (K2 - St) + (K1 - St) = {P(K2 ) - P(K1 )} ert - (K2 - K1 ) >
{K2 - K1 } ert - (K2 - K1 ) > 0.
Thus we have demonstrated arbitrage for American puts.
187
This the reverse of what we did to demonstrate arbitrage if the previous property was violated.
We can wait to expiration to exercise the put we bought and be guaranteed ending with a positive amount.
189
If at time t we can sell the put we bought for more than its exercise value, so much the better for us.
188
Financial Economics,
HCMSA-2012-MFE/3F,
If
12/14/11,
Page 126
P(K2 ) - P(K 1) P(K3 ) - P(K 2)
>
⇔ (K2 - K1 ) {P(K2 ) - P(K3 )} - (K3 - K2 ) {P(K1 ) - P(K2 )} > 0
K2 - K1
K3 - K2
⇔ (K3 - K1 ) P(K2 ) - (K3 - K2 ) P(K1 ) - (K1 - K2 ) P(K3 ) > 0,
then we can demonstrate arbitrage by buying an asymmetric butterfly put spread:
Buy λ = (K3 - K2 )/(K3 - K1 ) of low strike, sell 1 of medium strike, and, buy 1 - λ of the high strike put.
We lend:
P(K2) - λ P(K1 ) - (1 - λ) P(K3 ) =
(K3 - K1 )P(K2) - (K 3 - K2)P(K1 ) - (K1 - K 2)P(K3 )
> 0.
K3 - K1
If we are dealing with European options, then if at expiration ST ≥ K3 , all options expire worthless.
So we have: {P(K2 ) - λ P(K1 ) - (1 - λ) P(K3 )} erT > 0.
If at expiration K3 > ST ≥ K2 , then the high strike option we bought expires in the money.
So we have: {P(K2 ) - λ P(K1 ) - (1 - λ) P(K3 )} erT + (1 - λ)(K3 - ST) > 0.
If at expiration K2 > ST ≥ K1 , then the high and medium strike options expire in the money.
So we have: {P(K2 ) - λ P(K1 ) - (1 - λ) P(K3 )} erT + (1 - λ)(K3 - ST) - (K2 - ST) >
P(K2 ) - λ P(K1 ) - (1 - λ) P(K3 ) + λS T + (1 - λ)K3 - K2 >
P(K2 ) - λ P(K1 ) - (1 - λ) P(K3 ) + λK1 + (1 - λ)K3 - K2 =
P(K2 ) - λ P(K1 ) - (1 - λ) P(K3 ) + K1
K3 - K 2
K - K1
+ K3 2
- K2 =
K3 - K 1
K3 - K 1
P(K2 ) - λ P(K1 ) - (1 - λ) P(K3 ) > 0.190
If at expiration K1 > ST, then all three options expire in the money.
So we have: {P(K2 ) - λ P(K1 ) - (1 - λ) P(K3 )} erT + λ(K1 - ST) + (1 - λ)(K3 - ST) - (K2 - ST) >
P(K2 ) - λ P(K1 ) - (1 - λ) P(K3 ) + (1 - λ)K3 + λK1 - K2 =
P(K2 ) - λ P(K1 ) - (1 - λ) P(K3 ) + K3
K2 - K 1
K - K2
+ K1 3
- K2 =
K3 - K 1
K3 - K 1
P(K2 ) - λ P(K1 ) - (1 - λ) P(K3 ) > 0.
Thus we have demonstrated arbitrage for European puts.
190
Note that from the definition of λ = (K3 - K2 )/(K3 - K1 ), it follows that: K2 - λK1 - (1 - λ)K3 = 0.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 127
If instead we have American puts, we can deal with the same four cases.
If the put we sold is not exercised early, then the same logic applies as for European puts.191
If instead the option we sold is exercised early at time t, then it must be that St < K2 .
If K2 > St ≥ K1 , then the low strike option we bought has an exercise value of zero and we can
exercise the high strike put we bought.192
So we have: {P(K2 ) - λ P(K1 ) - (1 - λ) P(K3 )} ert + (1 - λ)(St - K3 ) - (St - K2 ) >
P(K2 ) - λ P(K1 ) - (1 - λ) P(K3 ) + λS t + (1 - λ)K3 - K2 >
P(K2 ) - λ P(K1 ) - (1 - λ) P(K3 ) + λK1 + (1 - λ)K3 - K2 =
P(K2 ) - λ P(K1 ) - (1 - λ) P(K3 ) + K1
K3 - K 2
K - K1
+ K3 2
- K2 =
K3 - K 1
K3 - K 1
P(K2 ) - λ P(K1 ) - (1 - λ) P(K3 ) > 0.
If St < K1 , then we can also exercise at time t both puts that we bought.193
So we have: {P(K2 ) - λ P(K1 ) - (1 - λ) P(K3 )} ert + λ(K1 - St) + (1 - λ)(K3 - St) - (K2 - St) >
P(K2 ) - λ P(K1 ) - (1 - λ) P(K3 ) + (1 - λ)K3 + λK1 - K2 =
P(K2 ) - λ P(K1 ) - (1 - λ) P(K3 ) + K3
K2 - K 1
K - K2
+ K1 3
- K2 =
K3 - K 1
K3 - K 1
P(K2 ) - λ P(K1 ) - (1 - λ) P(K3 ) > 0.
Thus we have demonstrated arbitrage for American puts.
Bermudan Options:194
In the case of a Bermudan option, the buyer has the right to exercise only at a set of discretely
spaced times. This is intermediate between a European option, which allows exercise at a single
time, and an American option, which allows exercise at any time up to expiration. For example, a
Bermudan interest rate swap might allow exercise at six month intervals.
In a subsequent section, we will discuss pricing American options via Binomial Trees. If we used a
tree with nodes spaced 3 months apart, then we would really be pricing a Bermudan option where
one can only exercise at 3 month intervals. As the time between the nodes approaches zero, the
premium of the American option is a limit of those of the Bermudan options.
191
We can wait to expiration to exercise the puts we bought and be guaranteed ending with a positive amount.
If at time t we can sell the puts we bought for more than their exercise value, so much the better for us.
193
If at time t we can sell the puts we bought for more than their exercise value, so much the better for us.
194
Not on the syllabus.
192
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 128
Problems:
10.1 (2 points) r = 0%. Stock ABC pays dividends. You own an American call on ABC.
Is it ever optimal to exercise this option before expiration? Briefly explain why.
10.2 (1 point) You have an American put option to exchange one share of Stock A for one share of
Stock B. Neither stock pays dividends. In what circumstances might you exercise this option early?
10.3 (1 point) Which of the following statements about American style options is not true?
A. The option may be exercised at any time during its life.
B. If the underlying stock does not pay a dividend, one should not exercise a call option early.
C. The option is worth at least as much as a similar European style option.
D. The option is worth at least as much as a similar option with more time to expiration.
E. All of A, B, C, and D are true.
10.4 (2 points) On March 1, ABC company stock is worth 20.
However, ABC has announced that on April 1 it will pay a liquidating dividend; ABC will pay its
entire value to its stockholders.
On March 1, compare the values of a European 10-strike call on ABC stock that expires
on March 15 and a similar option that expires on April 15.
10.5 (2 points) r = 0%. Stock ABC pays dividends. You own an American put on ABC.
Is it ever optimal to exercise this option before expiration? Briefly explain why.
10.6 (1 point) An American 85 strike 18 month put is an option on a stock with current price 81 and
forward price of 84. r = 3%.
Which of the following intervals represents the range of possible premiums for this option?
A. [0, 81]
B. [0, 84]
C. [0, 85]
D. [0.96, 84]
E. [0.96, 85]
10.7 (1 point) Which of the following statements is true?
1. A European put is worth at least as much as a similar put with more time until expiration.
2. A European call is worth at least as much as a similar call with more time until expiration.
3. A European option is worth at most as much as a similar American style option.
A. 1
B. 2
C. 3
D. 1, 2, 3
E. None of A, B, C, or D
10.8 (1 point) You have an American call option to exchange one share of Stock A for one share of
Stock B. Neither stock pays dividends. In what circumstances might you exercise this option early?
10.9 (3 points) On January 1 you purchase a 1 year $90 strike American call. r = 6%.
The stock will pay dividends of $1 each on: March 1, June 1, September 1, and December 1.
At what points in time might it be optimal to exercise your call?
Financial Economics,
HCMSA-2012-MFE/3F,
12/14/11,
Page 129
10.10 (2 points) XYZ company goes bankrupt on January 1 and its stock becomes worthless.
On January 1, compare the values of a European 100-strike put on XYZ stock that expires April 1
and a similar option that expires July 1.
10.11 (1 point) You have an American call option with strike price of $100. r = 5% and δ = 2%.
If the stock price had zero volatility, under what circumstances should you exercise this option early?
10.12 (2 points) r = 6%.
Consider otherwise similar American and European options on a stock that does not pay dividends.
Which of the following could be a graph of the option premiums as a function of the initial stock price,
holding everything else constant?
In each case, the American option premium is shown as dashed.
C
25
C
20
20
15
15
10
10
5
A.
5
B.
90
100
110
S0
120
P
90
100
110
S0
120
90
100
110
S0
120
P
20
15
15
10
10
5
5
C.
90
100
110
S0
120
D.
E. None of A, B, C, or D
10.13 (2 points) Briefly discuss early exercise of American options if the stock has a volatility of
zero, σ = 0, in other words if the future price of the stock were deterministic rather than random.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 130
10.14 (1 point) An American 95 strike 3 year call is an option on a stock with current price 90 and
forward price of 100. r = 5%. Determine the width of the range of possible prices of this option.
A. 84
B. 86
C. 88
D. 90
E. 92
10.15 (4 points) Use the following information:
• A 2-year American put option on Euros has K = $1.30.
• r$ = 4%.
• r€ = 6%.
Draw and label a diagram to show bounds (ignoring transaction costs) on the possible values of the
American put as a function of the exchange rate when the option is bought.
The bounds illustrated should be independent of any model of the movement of exchange rates.
You should explain the rationale behind the bounds shown.
10.16 (2 points) You own a six-month 50-strike American call option on a stock.
The stock pays dividends at the continuously compounded annual rate of 2%.
The stock currently sells at 54, and is expected to sell at either 50 or 62 six months from now.
What annual continuously compounded risk-free interest rate would equate the value of exercising
the call immediately with the present value of waiting until the end of the term?
A. 1%
B. 2%
C. 3%
D. 4%
E. 5%
10.17 (2 points) Which of the following statements is true?
i. For an American option, the early exercise criterion becomes less stringent closer to expiration.
ii. For American calls, the early exercise boundary is higher for larger volatility.
iii. For American puts, the early exercise boundary is higher for larger volatility.
(A) Only (i) is true
(B) Only (ii) is true
(C) Only (i) and (ii) are true
(D) Only (i) and (iii) are true
(E) (i), (ii) and (iii) are true
10.18 (1 point) An American 70-strike six-month put is an option on a stock with current price of 74.
δ = 1%. r = 4%.
Determine the range of possible put premiums.
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 131
10.19 (2 points) An 80-strike American call has a premium of 21.
A similar 110-strike American call has a premium of 9.
Based on each of the three properties of American option premiums, what is the range of possible
prices for a similar 100-strike American call?
10.20 (2 points) Consider an American 70-strike six-month call on a stock.
The stock will pay a dividend of $1 in one month and another dividend of $1 in four months.
If r = 8%, when might it be optimal to exercise this call? Briefly explain why.
10.21 (4 points) Use the following information:
• A 5-year American call on a stock has K = 100.
• r = 4%.
• δ = 3%.
Draw and label a diagram to show bounds (ignoring transaction costs) on the possible values of the
American call as a function of the stock price when the option is bought.
The bounds illustrated should be independent of any model of the movement of stock prices.
You should explain the rationale behind the bounds shown.
10.22 (2 points) A 50-strike American put has a premium of 7.
A similar 80-strike American put has a premium of 28.
Based on each of the three properties of American option premiums, what is the range of possible
prices for a similar 60-strike American put?
10.23 (3 points) Consider an American 100-strike one-year call on a stock.
The stock will pay a dividends in two months, five months, eight months, and eleven months.
Each of these dividends will be of the same size D.
r = 7%.
What is the smallest value of D such that is possible that this American Call will be worth more than
the similar European call?
A. Less than 0.50
B. At least 0.50, but less than 0.75
C. At least 0.75, but less than 1.00
D. At least 1.00, but less than 1.25
E. At least 1.25
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 132
10.24 (8 points) Consider European and American options on a stock. You are given:
(i) All options have the same strike price of 50.
(ii) All options expire in 2 years.
(iii) The continuously compounded risk-free interest rate is 7%.
(iv) Dividends are paid at the continuously compounded rate of 3%.
For each option, graph the area of possible premiums as a function of the current stock price.
Put the possible stock prices on the horizontal axis, and put the possible option premiums on the
vertical axis.
(a) American Call
(b) European Call
(c) American Put
(d) European Put
10.25 (5B, 11/98, Q.35) (2 points) You have purchased a two-year American call option on a
stock that pays an annual dividend. The dividend will always equal 10% of the share price and the
next dividend is payable one year from today. The exercise price is equal to the current share price.
Each year the stock may increase in value by 20% or decrease in value by 20%. The value of the
stock will drop by the dividend amount immediately after it is paid. The continuously compounded
risk-free rate is 5%.
Assuming that investors are indifferent to risk, under what circumstances would you exercise the
option prior to expiration? Explain.
10.26 (5B, 5/99, Q.34) (1 point) You own a one-year American put option on a non-dividend
paying stock with an exercise price of $42. The stock currently sells at $40, and is expected to sell at
either $38 or $48 one year from now. What annual continuously compounded risk-free interest rate
would equate the value of exercising the put immediately with the present value of waiting until the
end of the term? Show all work.
10.27 (IOA 109 Specimen Exam 5/99, Q.5) (3.75 points) Draw a diagram to show bounds
(ignoring transaction costs) on the possible values of an American call option as a function of the
price of the underlying security. The bounds illustrated should be independent of any model for the
price process followed by the underlying security. You should explain the rationale behind the
bounds shown.
10.28 (5B, 11/99, Q.15) (1 point)
For which of the following options might it be rational to exercise before maturity?
1. American put on a dividend-paying stock
2. American put on a non-dividend-paying stock
3. American call on a dividend-paying stock
A. 1
B. 3
C. 1, 2
D. 2, 3
E. 1, 2, 3
Financial Economics,
HCMSA-2012-MFE/3F,
12/14/11,
Page 133
10.29 (MFE Sample Exam, Q.26)
Consider European and American options on a nondividend-paying stock. You are given:
(i) All options have the same strike price of 100.
(ii) All options expire in six months.
(iii) The continuously compounded risk-free interest rate is 10%.
You are interested in the graph for the price of an option as a function of the current stock
price. In each of the following four charts I - IV, the horizontal axis, S, represents the
current stock price, and the vertical axis, π, represents the price of an option.
π
π
I.
II.
π=S
π=S
100
95.12
π = S - 95.12
π = S - 100
S
100
π
100
S
95.12
π
95.12
III.
IV.
π = 100 - S
π = 95.12 - S
100
S
95.12
Match the option with the shaded region in which its graph lies.
If there are two or more possibilities, choose the chart with the smallest shaded region.
European Call
American Call
European Put
American Put
(A)
(B)
(C)
(D)
(E)
I
II
II
II
II
I
I
I
II
II
III
IV
III
IV
IV
III
III
III
III
IV
S
HCMSA-2012-MFE/3F,
Financial Economics,
12/14/11,
Page 134
10.30 (CAS3, 5/07, Q.12) (2.5 points)
Which of the following effects are correct on the price of a stock option?
I. The premiums would not decrease if the options were American rather than European.
II. For European put, the premiums increase when the stock price increases.
III. For American call, the premiums increase when the strike price increases.
A. I only
B. I and II only
C. I and III only
D. II and III only
E. I, II and III
10.31 (CAS3, 11/07, Q.13) (2.5 points) Given the following chart about call options on a particular
dividend paying stock, which option has the highest value?
Option Option Style Time Until Expiration Strike Price Stock Price
A
European
1 year
50
42
B
American
1 year
50
42
C
European
2 years
50
42
D
American
2 years
50
42
E
American
2 years
55
42
A. Option A B. Option B C. Option C D. Option D E. Option E