Master of Finance
Faculty of Business and Economics
The University of Hong Kong
Course Syllabus and Outline for Fall 2006
MFIN6003 Derivative Securities
I.
INFORMATION ON INSTRUCTOR
Instructor:
E-mail:
Office Phone:
Fax:
Office:
II.
Prof. Eric C. Chang
[email protected]
(852) 2857-8347
(852) 2858-5614
Room 604, Meng Wah Complex
School of Business
University of Hong Kong
COURSE INFORMATION
Course Description
Derivatives concepts are now almost required for every advanced finance topic. This
course offers a general introduction to derivative products (principally futures, options,
swaps, and basic structured products), their pricing formulas, the markets in which they
trade, and applications. The focus is more on professional and institutional business
strategies and developments than on personal investment and speculation. The concepts
of hedging and arbitrage will be extensively discussed from both the angles of trading
strategies and pricing foundations. The orientation of the course leans toward economic
intuitions and practical investment implications. However, a certain amount of theory is
necessary to understand the methodologies used in generating solutions to the problems.
The course intends to provide a solid foundation for other advanced courses of the
program such as mathematical finance, interest derivatives and financial engineering.
Textbook (Required):
McDonald, Robert L., 2006, Derivatives Markets (Pearson Education, Inc.)
References:
Jarrow, Robert and Stuart Turnbull, 2000, 2nd edition, Derivative Securities
(South-Western College Publishing, Cincinnati, Ohio)
Hull, John, 2005, 5th edition, Fundamentals of Futures and Options Markets (PrenticeHall).
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Cox, John C. and Mark Rubinstein, 1985, Options Markets (Prentice-Hall, Inc.,
Englewood Cliffs, New Jersey).
Course Learning Outcomes
1.
Understand the basic types of derivatives, their developments and the economic
roles they play in the financial markets.
2.
Understand the basic payoff functions and combination of trading strategies of
basic derivatives such as forward, futures and options.
3.
Learn the basic principles of financial product innovations and the economic
functions they serve.
4.
an
Understand no-arbitrage principle and explore its role for pricing derivatives in
efficient financial market.
5.
Learn the relations between the prices of the underlying (spot), forward and
futures without and with market frictions. Understand the relationship between
forward price and the expected future spot price.
6.
Derive the put-call parity and other pricing relations between calls and puts using
no-arbitrage principle.
7.
Understand extensively the binomial approach in pricing European and American
options with a clear discussion of the equivalent martingale probability and the
risk-neutral pricing concept.
8.
Understand the economic determinants of option prices in the framework of
Balck-Scholes option model with an application to innovative product designs.
9.
Understand the mechanism of market-making and process of risk management in
derivative markets with a clear grasp of hedging errors under different hedging
approaches.
10.
Understand the lognormal distribution and establish the argument that it could be
a reasonable approximation for modeling the underlying asset price process.
Course Delivery
1.
Thirty-six lecture hours for the whole course dividing into twelve days with three
hours per session will be delivered by the instructor. Class attendance and active
participation in discussion are expected for all students.
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2.
Three Excel-based team projects will be given to facilitate the group learning
outside the classroom.
3.
The assigned homework will review the concepts delivered in the lectures and
offer an opportunity of self-examining of the strengths and weaknesses of their
class learning.
Measurement of Learning Outcomes
1.
Three Excel-based team projects will be given. Part of the objective is to develop
students’ ability to work as a team to mutually reinforce the classroom learning.
2.
The first project will enable students to experiment the pricing behavior of calls
and puts based on the Black-Scholes model under a variety of scenarios. The
quality of the submitted Excel sheet can be used to evaluate their skills in using
build-in functions in Excel as well as the software in analyzing the pricing
properties of basic derivatives.
3.
The second project devotes to implement the binomial option concepts beginning
from an understanding of the basic return properties of the underlying, applying
the equivalent martingale probability, checking the self-financing condition and
integrating the replication properties of basic derivatives.
4.
The third project intends to evaluate students’ understanding of the general risk
management system utilized by market makers as well as the remaining hedging
risk and other related issues that they face.
5.
A mid-term and a comprehensive final exam will test students’ breadth and depth
of the understanding of the major concepts covered in the course and their ability
to integrate and apply this knowledge.
III.
PROCESS FOR EVALUATION
Basis of Assessment:
Individual class contribution
Mid-term
Group projects
Exam
10%
15%
40%
35%
Total
100%
Attendance:
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To ensure that students gain the maximum benefit from classes, students are required to
attend at least 70% of classes for each course, otherwise they may be treated as having
failed the whole course. Only except unexpected events, should students find themselves
unable to attend a class, they have to inform the teacher concerned or the ME staff
beforehand in writing, for example, by fax or by e-mail. Coming late to class is also not
encouraged. If students are 30 minutes late, they will be regarded as not having attended
the class. The participation in class will be considered in assigning points for individual
class contribution.
Midterm
A closed book mid-term examination will be given on September 14 (Thursday). Please
bring your calculator.
The Comprehensive Final Examination
A closed book final-term examination will be given according to the program schedule,
though students are allowed to bring one-page (A4) formula sheet. No make-up
examination will be given unless the student consults with the professor prior to the
scheduled date of the examination and provides documents to support the reason for
missing the scheduled examination.
Homework Policies
The answers to half of the end-of-chapter questions and problems will be given to
students. No homework assignments will be collected for grading. However, in order to
pass this course, students are strongly urged to practice these questions and problems.
There is virtually no chance that you will do well in this course unless you are diligent in
your completion of the assigned reading materials and practiced problems. For those
spreadsheet project assignments, you are required to practice your Excel skill in solving
those problems. By engaging yourself in this exercise will further develop your
spreadsheet skills as you will need this skill in your future work.
IV.
COURSE POLICIES
Class Conduct
Respect your instructors and your fellow students. Be considerate to others.
Students are required to attend all classes on time. If you have to leave the class early,
please inform the instructor/teaching assistant (TA) before the class begins. Please sit
near the door and exit quietly. If you fail to inform the instructor/TA before you leave,
no credit will be given for your class attendance.
Academic Dishonesty
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The University Regulations on academic dishonesty will be strictly enforced. Please
check the University Statement on plagiarism on the web:
http://www.hku.hk/plagiarism/
Academic dishonesty is behavior in which a deliberately fraudulent misrepresentation is
employed in an attempt to gain undeserved intellectual credit, either for oneself or for
another. It includes, but is not necessarily limited to, the following types of cases:
Plagiarism - The representation of someone else's ideas as if they are one's own. Where
the arguments, data, designs, etc., of someone else are being used in a paper, report, oral
presentation, or similar academic project, this fact must be made explicitly clear by
citing the appropriate references. The references must fully indicate the extent to which
any parts of the project are not one's own work. Paraphrasing of someone else's ideas is
still using someone else's ideas, and must be acknowledged.
Unauthorized Collaboration on Out-of-Class Projects - The representation of work as
solely one's own when in fact it is the result of a joint effort.
Cheating on In-Class Exams - The covert gathering of information from other students,
the use of unauthorized notes, unauthorized aids, etc.
Unauthorized Advance Access to an Exam - The representation of materials prepared at
leisure, as a result of unauthorized advance access (however obtained), as if it were
prepared under the rigors of the exam setting. This misrepresentation is dishonest in
itself even if there are not compounding factors, such as unauthorized uses of books or
notes.
Where a candidate for a degree or other award uses the work of another person or
persons without due acknowledgement:
If you are caught in an act of academic dishonesty or misconduct, you will receive an
“F” grade for the subject.
The relevant Board of Examiners may impose other penalty in relation to the seriousness
of the offence;
The relevant Board of Examiners may report the candidate to the Senate, where there is
prima facie evidence of an intention to deceive and where sanctions beyond those in the
above paragraph might be invoked.
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Syllabus***
Topic 1
A Discussion of Asset Price Dynamics
A.
B.
C.
D.
E.
Examples of Historical Stock Prices
The Normal Distribution
The Lognormal Distribution
A Lognormal Model of Stock Prices
How Are Asset Prices Distributed
Readings: McDonald, Chapter 18
Reference: Jarrow & Turnbull, Chapter 4
Topic 2
Introduction to Derivatives
A.
B.
C.
D.
What Is a Derivative?
The Role of Financial Markets
Derivatives in Practice
Buying and Short-Selling Financial Assets
Readings: McDonald, Chapter 1
Topic 3
An Introduction to Forwards and Options
A.
B.
C.
D.
E.
Forward Contracts
Call Options
Put Options
Options Are Insurance
Understanding Equity-Linked CDs
Readings: McDonald, Chapter 2
Topic 4
Insurance, Collars, and Other Strategies
A.
B.
C.
D.
E.
Basic Insurance Strategies
Synthetic Forwards
Spreads and Collars
Payoff Diagrams for Elementary Strategies
Speculating on Volatility
Readings: McDonald, Chapter 3
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Topic 5
Introduction to Risk Management
A. Basic Risk Management: The Producer’s Perspective
B. Basic Risk Management: The Buyer’s Perspective
C. Why Do Firms Manage Risk?
Readings: McDonald, Chapter 4
Topic 6
Financial Forwards and Futures
A.
B.
C.
D.
E.
Alternative Ways to Buy Stock
Forward Contracts on Stock
Futures Contracts
Uses of Index Futures
Currency Contracts
Readings: McDonald, Chapter 5
Topic 7
Parity and Other Option Relationships
A. Put-Call Parity
B. Generalized Parity and Exchange Options
Readings: McDonald, Chapter 9
Topic 8
Binomial Option Pricing: I
A.
B.
C.
D.
E.
F.
G.
H.
One-Period Binomial Tree
Arbitraging a Mispriced Option
Two-or More Binomial Periods
Risk-Neutral Pricing
Where does the Tree Come From?
Replicating Option on Spot With Futures
Put Options
American Options
Readings: McDonald, Chapter 10
Reference: Jarrow & Turnbull, Chapter 5
Readings: O'Brien, Thomas J., "The Mechanics of Portfolio
Insurance," Journal of Portfolio Management (Spring
1988), pp. 40-47.
Topic 9:
Binomial Option Pricing: II
A. Understanding Early Exercise
B. Understanding Risk-Neutral Pricing
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C.
D.
E.
F.
The Binomial Tree and Lognormality
Estimating Volatility
Pricing Options with True Probabilities
Why Does Risk-Neutral Pricing Work?
Readings: McDonald, Chapter 11
Topic 10
The Black-Scholes Formula
A.
B.
C.
D.
Introduction to the Black-Scholes Formula
Applying the Formula to Other Assets
Option Greeks
Implied Volatility
Readings: McDonald, Chapter 12
Topic 11
Market-Making and Delta-Hedging
A.
B.
C.
D.
E.
What Do Market-Makers Do?
Market-Maker Risk
Delta-Hedging
The Mathematics of Delta-Hedging
The Black-Scholes Analysis
Readings: McDonald, Chapter 13
Readings: Jarrow & Turnbull, Chapter 10
Topic 12
Financial Engineering and Applications
A. Dynamic Hedging and Portfolio Insurance
B. Pricing and Designing Structured Notes
C. Bonds with Embedded Options
Readings: McDonald, Chapter 15
Topic 13
Martingale Pricing
A.
B.
C.
D.
Relative Prices and Martingales
The Money Market Account
Risk Neutral Valuation
Martingales and No Arbitrage
Readings: Jarrow & Turnbull, Chapter 6
Topic 14
Interest Forwards and Futures
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A.
B.
C.
D.
E.
Bond Basics
Forward Rate Agreements, Eurodollars, and Hedging
Duration
Treasury-Bond and Treasury-Note Futures
Repurchase Agreements
Readings:
Topic 15
McDonald, Chapter 7
Swaps
A. An Example of a Commodity Swap
B. Interest Rate Swaps
C. Currency Swaps
Readings:
***
McDonald, Chapter 8
Additional discussion questions and problems will be announced periodically.
The extent of the material is substantial, and we may not complete it all.
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SPRING 2006
DERIVATIVE SECURITIES
PROJECT 1
[Due September 12 (Tuesday)]
(A) Generate 100 standard normally distributed random number [N(0, 1)] from the
random number generator provided. This series will be the value of z for your
subsequent analysis.
1
.
252
(B)
Define t =
(C)
Use the following sets of annual parameters to convert the (same) series
generated from (A) to three daily stock continuously compounded return series,
A, B, and C.
Series
µ
σ
A
B
C
50%
50%
50%
30%
50%
80%
1 ⎞
⎛
x = ⎜ µ − σ 2 ⎟t + σ t z
2 ⎠
⎝
Estimate the annualized mean and volatility for return series A, B, and C,
respectively.
(D)
Assume the initial stock price is $100. Convert the three daily return series to
three stock price series A, B, and C whereby
S t +1 = S t e x
(E)
Assume the initial maturity T = 100 days, exercise price K = $100, annual Riskfree interest r = 6.0%, annual dividend yield δ = 0.0%, calculate and plot the three
call option values (according to the Black-Scholes option valuation model offered
in the software) based on the price series A, B and C. (Note that as time goes by,
the remaining maturity of the option also drops until zero.)
(F)
Fix T = 100 days, K = $100, r = 6.0%, δ = 0.0%, and σ = 0.30, 0.50 or 0.80,
respectively, calculate and plot Black-Scholes call values over stock prices range
from $50 to $150 (with $1 increment).
(G)
Fix σ = 0.50, K = $100, r = 6.0%, δ = 0.0%, and T = 50, 100, or 150 days,
respectively, calculate and plot the Black-Scholes call values over stock prices
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range from $50 to $150 (with $1 increment).
(H)
Repeat (E), (F) and (G) with put option.
(I) Fix σ = 0.50, K = $100, r = 6.0%, δ = 0.0%, and T = 50 days, calculate and plot
delta values for the call and the put, respectively, based on the Black and Scholes
option pricing model over stock prices range from $50 to $150 (with $1 increment).
(J) A vertical bull call spread consists of buying a call option and selling an otherwise
identical call option with a higher strike price (Pages. 69 - 76). Fix σ = 0.50, r = 6.0%,
δ = 0.0%, and T = 30 days, K1 = $100, K2 = $110 calculate and plot Black-Scholes
vertical bull call values over stock prices range from $50 to $150 (with $1 increment).
Repeat the above analysis with T = 0 day.
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SPRING 2006
DERIVATIVE SECURITIES
PROJECT 2
[Due September 26 (Tuesday)]
(A)
Create a spreadsheet which can handle a 4-period, 5-state binomial stock price
(underlying portfolio value) dynamics. The input parameters of the spreadsheet
should include:
(a)
(b)
(c)
(d)
(e)
(f)
the volatility of return on the underlying stock, σ;
the continuously compounded annual risk-free rate of interest, r;
the continuously compounded annual dividend yield, δ;
the length of the interval, h, (you can set it to be 0.25 in this exercise);
the initial stock price, S0;
the strike price, K.
[You can set the maturity of the derivatives to be one year.]
[The risk-neutral probability can be determined by the above inputs (P. 312).
(B)
Using your spreadsheet to produce the details of the entire replication (trade by
trade) similar to Figure 5 in O’Brien based on the stock price dynamics and the
risk neutral probability determined by the parameters chosen.
(C)
The basic dynamic trading approach involves replication the insured stock’s
price action with an ever-changing combination of positions in the stock and the
riskless asset. When a futures contract exists on the insured stock, O’Brien points
out that the same replication can be accomplished with either
(a)
(b)
the stock and short stock futures positions or
the riskless asset and long stock futures.
He states that “portfolio insurance in this case is created dynamically by using
the index futures to create a synthetic position in either the riskless security or
the underlying portfolio.
Assuming that (i) underlying stock pays no dividends; (ii) futures are available
with the same delivery date as the horizon date of the insurance program; (iii)
futures prices are correctly determined in each period.
Modify your spreadsheet so that it can perform the details of the entire
replication (similar to Figure 5 of O’Brien’s paper) using the underlying stock
and short futures positions based on your stock price dynamics.
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SPRING 2006
DERIVATIVE SECURITIES
PROJECT 3
[Due October 10 (Tuesday)]
(Useful reference can be found in Chapter 10 of Jarrow and Turnbull
and Chapter 13 of McDonald )
(A)
Generate a set of random numbers from a standard normal distribution with mean
zero and variance 1. The sample size will depend on the need of the simulation.
This series will be the value of z for your subsequent analysis.
(B)
Define t = 1/252.
(C)
For your simulation, let’s set the annual expected value of the continuously
compounded rate of return to be 40% (µ) and its volatility 60% (σ). The return is
normally distributed. The current (day 0) stock price is $100.
Use the following equation
⎛ S ⎞ ⎛
1 ⎞
ln⎜⎜ t ⎟⎟ = ⎜ µ − σ 2 ⎟t + σ t z
2 ⎠
⎝ S t −1 ⎠ ⎝
to generate a random sequence of stock prices.
(D)
Use the software provided to figure out the prices and the four major risk
parameters (delta, gamma, theta and vega) for European Call Option A over
these 50 days. European Call Option A has the following properties at date 0:
Initial Stock Price (date 0)
Stock Price Interval
Strike Price
Initial Maturity
Volatility
Risk-free Rate
Dividend yield
(E)
100
1 day
100
50 days
60%
6.00%
0.0%
Assuming now that you have issued 1 million units of call option A on date 0.
The outstanding calls will be hedged by constructing a replicating portfolio with
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the underlying stocks and an investment in the money market account. The
replicating portfolio is rebalanced at the end of each trading day. Since you are
not trading in continuous time, the replicating portfolio may not be self-financing
each day. Let’s denote the daily insufficient (or surplus) amount as the hedging
error of the day. Therefore, you perform additional borrowing (lending) each day
to cover the deficit (surplus) so that you still can rebalance the portfolio by the
end of the day. Prepare a spreadsheet showing the composition of your daily
replicating portfolio, the daily hedging error and the cumulative hedging error
(with interests) over the entire hedging period. Express the cumulative hedging
error as a percentage of the original call premium.
(F)
Suppose now Call Option B is also available on date 0 with unlimited supply.
Use the software provided to figure out the prices and the four major risk
parameters (delta, gamma, theta and vega) for European Call Option B over the
next 50 days. European Call Option B has the following properties at date 0:
Initial Stock Price (date 0)
Stock Price Interval
Strike Price
Initial Maturity
Volatility*
Risk-free Rate**
Dividend yield
100
1 day
100
100 days
60%
6.00%
0.0%
(G)
Knowing that hedging by our replicating portfolio in (D) can only make your
overall position delta neutral yet not gamma neutral, you can incorporate call
option B into your replicating portfolio to reduce the hedging error. A naïve
strategy would be to first use option B to neutralize the gamma of the overall
position and then to use some of the underlying stocks to make the position delta
neutral. Of course, some amount of money market investment is still needed to
ensure a nearly perfect replication of the outstanding option. Again, you
rebalance the replicating portfolio at the end of each trading day. Since you are
still not trading in continuous time, the replicating portfolio may not be selffinancing each day. Again, let’s denote the daily insufficient (or surplus) amount
as the hedging error of the day. Prepare a spreadsheet showing the composition of
your daily replicating portfolio (unit of option B, number of the underlying shares,
and the amount in the money market account), the daily hedging error and the
cumulative hedging error (with interests) over the entire hedging period. Express
the cumulative hedging error as a percentage of the original call premium.
(H)
Essentially, you have used a long-term option (that is B) to hedge the mediumterm option (that is A) over the 50 days. In comparison with a replicating
portfolio consisting of only the underlying stock and the money market account,
what is the percentage of cumulative hedging error reduction contributed by
option B?
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(I)
Keeping the same µ and σ values and repeating steps (A) to (H) 40 times, you
have obtained a distribution of cumulative hedging errors (in percentage form)
from (E), (G) and (H). Compile a list of these numbers and a table of summary
statistics (mean, standard deviation, maximum, etc.). There is no need to show
me the spreadsheet for this repetition job (other than the first set). Discuss what
you have learned (pricing and risk management implications) from the simulation.
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