Download Lecture Notes_Chapter 1 - the School of Economics and Finance

Overview and Preparation
Introduction to Derivatives
FINA0301 Derivatives
Faculty of Business and Economics
University of Hong Kong
Dr. Huiyan Qiu
1
Outline
Course Overview and Class Information
Introduction to Derivatives: in general
• What is a derivative?
• Derivatives markets
Technical preparation
• Time value of money
• Basic transaction including short-selling
• No-arbitrage principle
2
Overview of the course
The course is about:
• the concept, the use, the pricing of derivatives.
Objective of the course:
• this course aims to provide you with a general
introduction to the modern financial
engineering: the construction of a financial
product from other products.
3
Topics covered in this course
Chapter 1 Introduction to Derivatives.
Part I: Insurance, hedging, and simple
strategies. Chapter 2, 3 and 4
Part II: Forwards, futures, and swaps. Chapter
5, 6.1, 6.2, 7, and 8
Part III: option pricing. Chapter 9, 10, 11
Part IV: financial engineering and corporate
applications. Chapter 12, 13
4
Assessment
Option 1
Option 2
Tutorial Performance
5%
5%
Assignments
25%
25%
Mid-term Examination
30%
25%
Final Examination
40%
45%
Final grade is curve-based. No more than 25% of
students will be awarded A+, A, or A-. The median
grade is B-.
5
Class Information
Meeting Schedule:
Wednesday 11:30-12:25 (TT 404)
Friday 10:30-12:25 (TT 404)
Number of students: 63
Subclass F and G (19, 51) are offered by other
instructors on Tuesday and Thursday afternoon.
Note: data collection is based on the registration
information as of January 9, 2012.
6
Total
63
%
BEF
40
63.49
BBA (Accounting and Finance)
8
12.69
BBA (Other)
11
17.46
BEng
1
1.59
BSc
1
1.59
BSc (QFin)
1
1.59
BSocSc
1
1.59
Year
63
%
0
9
14.29
2
39
61.9
3
15
23.81
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What are Derivatives?
A derivative security is a financial instrument
whose value derives from that of some other
underlying asset or assets whose price are taken
as given.
Remark: The value of a derivative is in a relative
sense. No need to inquire into the deeper
economic forces that determine the prices of the
underlying assets.
8
Types of Derivatives




Forward contracts and futures contracts are
agreements to buy or sell an asset at a certain
future time T for a certain price K.
Swaps are similar to forwards, except that the
parties commit to multiple exchanges at different
points in time.
A call option gives the holder the right to buy
the underlying asset by a certain date T for a
certain price K .
A put option gives the holder the right to sell
the underlying asset by a certain date T for a
certain price K .
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A Concrete Example
You enter an agreement with a friend that says:
• If the price of a bushel of corn in one year is
greater than $3, you will pay him $1
• If the price is less than $3, he will pay you $1
This agreement is a derivative
Questions:
• What happens one year later? (outcome, carry-out)
• Why do you or your friend want to enter this
agreement at the first place?
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Uses of Derivatives
Risk management
• Hedging: where the cash flows from the
derivative are used to offset or mitigate the cash
flows from a prior market commitment.
Speculation
• Where derivative is used without an underlying
prior exposure; the aim is to profit from
anticipated market movements.
Reduce transaction costs
Regulatory arbitrage
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Three Different Perspectives
End users
• Corporations
• Investment
managers
• Investors
End
user
Intermediaries
• Market-makers
• Traders
Economic
Observers
• Regulators
• Researchers
Observers
Intermediary
End
user
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Derivatives Markets
The over-the-counter or “OTC” market: where two
parties find each other then work directly with
each other to formulate, execute, and enforce a
derivative transaction.
• Forward contracts, most swaps
The exchange market: where buyer and seller
can do a deal without worrying about finding
each other.
• Futures contracts, most options
Risk-sharing is one of the most important
functions of financial markets.
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Measures of Market Size and Activity
Four ways to measure a market
• Open interest: total number of contracts that are
“open” (waiting to be settled). An important
statistic in derivatives markets.
• Trading volume: number of financial claims that
change hands daily or annually.
• Market value: sum of the market value of the
claims that could be traded.
• Notional value: the scale of a position, using with
reference to some underlying asset.
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Exchange Traded Contracts
Contracts proliferated in the last three decades
Examples of futures contracts traded on the three derivatives
market
What were the drivers behind this proliferation?
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Increased Volatility…
Oil prices:
1947–2006
Figure 1.1 Monthly percentage
change in the producer price index
for oil, 1947–2006.
Dollar/Pound rate:
1947–2006
Figure 1.2 Monthly percentage
change in the dollar/pound
($/£) exchange rate, 1947–2006.
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…Led to New and Big Markets
Exchange-traded derivatives
Figure 1.3 Millions of futures
contracts traded annually at the
Chicago Board of Trade (CBT),
Chicago Mercantile Exchange
(CME), and the New York
Mercantile Exchange (NYMEX),
1970–2006. The CME and CBT
merged in 2007.
Over-the-counter traded derivatives: even more!
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Derivatives Products in HK
Exchange-traded derivatives products in HKEX
include:
• Equity Index Products (futures and options on
Hang Seng Index, H-shares Index, Mini-Hang Seng
Index, Mini H-shares Index, and Dividend futures)
• Equity Products (stock futures and stock options)
• Interest Rate and Fixed Income Products
(HIBOR futures and Three-year exchange fund
note futures)
• Gold Futures
OTC market products: numerous
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Hong Kong Mercantile Exchange
HKMEX: an electronic commodities exchange
• “… HKMEx seeks to become the preferred platform
where international and mainland market
participants come together to trade commodity
contracts for investment, hedging and arbitrage
opportunities.”
Formally began trading on May 18, 2011
Products
• 32 troy ounce gold futures: May 18, 2011
• 1,000 troy ounce silver futures: July 22, 2011
Website: http://www.hkmerc.com
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Futures Exchanges in China
Dalian Commodity Exchange (1993.02)
Zhengzhou Commodity Exchange (1998.08)
Shanghai Futures Exchange (1999.12)
China Financial Futures Exchange (2006.09)
• CFFEX is the first financial derivatives exchange
in China. http://www.cffex.com.cn/
• 16 April, 2010, the CSI index futures started to
trade
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Technical Preparation
Time value of money, future value, present
value, APR, EAR
Continuous compounding (Appendix B)
Basic transaction: short-selling (§1.4)
No Arbitrage Principle
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Time Value of Money
Time value of money refers to a dollar today is
different from a dollar in the future
Time value of money is measured by the interest
rate for the period concerned.
To compare money flows, we must convert them
to the same time point.
$100
$110
Which one is more valuable?
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Future Value and Present Value
F V  P V  (1  r/m)
n
where FV = future value
PV = present value
r
= the quoted annual interest rate
m = the number of times interest is
compounded per year
n
= the number of compounding periods to
maturity
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A Simple Example
$100 is deposited for a year at quoted annual
percentage rate (APR) of 12% with monthly
compounding.
Given 12% APR, the monthly interest rate is 1%. At
the end of each month, interest is calculated and
added to the principle to earn more interest.
• End of month 1: $100(1+1%)
• End of month 2: $100(1+1%)(1+1%) = 100(1+1%)2
•:
• End of month 12: $100(1+1%)12 = $100(1+12.68%)
12.68% is the effective annual rate (EAR).
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APR and EAR
APR: annual percentage rate
EAR: effective annual rate
APR = 10%
 APR 
1  EAR   1 

n 

Compounding
Frequency
n
Annually
1
10.0000
Quarterly
4
10.3813
Monthly
12
10.4713
Weekly
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10.5065
Daily
365
10.5156
∞
10.5171
Continuously
n
EAR
(% p.a.)
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Continuously Compounding
Continuously compounding: n → ∞ (infinity)
n
 r
lim 1    e r
n 
 n
by definition of e.
Consider a stock paying continuous dividend
with annual rate of δ. Claim: The present
value of 1 share at time T is then S0e-δT.
• How can dividend be continuously paid?
• Example: stock index, mutual fund
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Continuous Dividend Payment
Annual dividend yield is  . Let’s first assume daily
compounding, then daily dividend yield is  / 365.
At day t, per share, there is S t 
365
dividend in cash,
which is equivalent to  / 365 unit of shares.
In stead of keeping cash dividend (varying), we
reinvest to accumulate more shares.
Starting with one share at day 0, at the end of the
year, total number of shares is 1   
 365 
365
.
If continuous compounding  e  shares.
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Continuous Dividend Payment
That it, one share today will result in e  shares
one year later.
To result in one share T years later, number of
shares needed today is thus e T.
Therefore, the present value of 1 share at time T
is S0e-δT.
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Basic Transactions
Buying and selling a financial asset (cost)
• Brokers: commissions
• Market-makers: bid-ask (offer) spread
Example: Buy and sell 100 shares of XYZ
• XYZ: bid = $49.75, offer = $50, commission = $15
• Buy: (100 x $50) + $15 = $5,015
• Sell: (100 x $49.75) – $15 = $4,960
• Transaction cost: $5,015 – $4,960 = $55
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Short-Selling
When price of an asset is expected to fall
• First: borrow and sell an asset (get $$)
• Then: buy back and return the asset (pay $)
• If price fell in the mean time: Profit $ = $$ – $
What happens if price doesn’t fall as expected?
If the asset pays dividend in between, who gets
the dividend payment?
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Short-Selling
Example: short-sell IBM stock for 90 days
The lender must be compensated for dividends
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received (lease rate of the asset)
Short-Selling (cont’d)
Why short-sell?
• Speculation
• Financing
• Hedging
Credit risk in short-selling
• Collateral and “haircut”
Interest received from lender on collateral
• Scarcity decreases the interest rate
• Repo rate in bond markets
• Short rebate in the stock market
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Example
Assume that you open a 100 share position in
Fanny, Inc. common stock at the bid-ask price of
$32.00 - $32.50. When you close your position the
bid-ask prices are $32.50 - $33.00. If you pay a
commission rate of 0.5% and the effective market
interest rate over your holding period is 2%.
What is your profit or loss if
• Case 1: you purchase the stock then sell;
• Case 2: you short-sell the stock then close the
position.
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Example (cont’d)
You pay ask price when you purchase a stock
and you get bid price when selling a stock.
If the market interest rate is ignored,
• Case 1: loss of $32.50
• Case 2: loss of $132.50
When the market interest rate of 2% is taken
into account,
• Case 1: loss of $97.825
• Case 2: loss of $68.82
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Discussion
Question 1: Why the loss in short-selling is
more than the loss in outright purchase? (In the
case that the market interest rate is zero.)
Question 2: Interest rate seems to have positive
effect on the profit/loss on short-selling but
negative effect on the profit/loss on outright
purchase. Why?
Question 3: at what interest rate, profit/loss
from short-selling or from outright purchase is
the same?
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Pricing Approaches
Much of this course will focus on the pricing of a
derivative security. In general there are two
approaches to price an asset (or a contract or a
portfolio):
Pricing an asset using an equilibrium model:
• Determine cash flows and their risk
• Use some theory of investor’s attitude towards
risk and return (e.g. CAPM) to figure out the
expected rate of return
• Conduct discounted cash flow analysis to find
the present value of future cash flows
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Pricing Approaches
Pricing an asset by analogy (using no-arbitrage):
• Find another asset, whose price you know, that
has the same payoffs of the asset to be priced.
Arbitrage is any trading strategy requiring no cash
input that has some probability of making profits,
without any risk of a loss
• Law of One Price: two equivalent things cannot
sell for different prices.
• Law of No Arbitrage: a portfolio involving zero
risk, zero net investment and positive expected
returns cannot exist.
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Law of No Arbitrage
Can one expect to continually earn arbitrage
profits in well functioning capital markets?
From an economic perspective, the existence of
arbitrage opportunities implies that the economy
is in an economic disequilibrium.
Assumptions:
• No market frictions (transaction costs? bid/ask
spread? restriction on short sales? taxes?)
• No counterparty risk (credit risk? collateral
requirements? margin requirements?)
• Competitive market (liquidity concern?)
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Two Examples
Example 1: the effect of dividend payment on
stock price change
Example 2: how to make arbitrage profit
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Cum-Dividend/Ex-Dividend Prices
A stock that pays a known dividend of dt dollars per
share at date t
Stc = the cum-dividend stock price at date t
Ste = the ex-dividend stock price at date t
Assumptions
• no arbitrage opportunities,
• no differential taxation between capital gains and
dividend income
The following relation can be shown to hold
Stc = Ste + dt
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No Arbitrage Argument
Suppose that Stc < Ste + dt
• buy the stock cum-dividend
• receive the dividend
• sell the stock ex-dividend
• reap the arbitrage profits (Ste + dt) – Stc > 0
Suppose that Stc > Ste + dt
• sell the stock at the cum price
• buy it back immediately after the dividend is paid
• reap the arbitrage profits (Stc – Ste) – dt > 0
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No-Arbitrage Pricing Method
Example:
• Current stock price S0 = $25.00, there is no
dividends payment in the following 6 months
• The continuously compounded risk-free annual
interest rate = 7.00%
• A contract (forward contract): agreement to buy
the stock at time 6 for F0, 6 = $26.00 (forward price)
Is there arbitrage profit to make? (Is the forward
contract fairly priced?)
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Example (cont’d)
How to generate a portfolio (synthetic contract)
which duplicates the cash flows and value of the
contract under consideration
Cash flows of the contract:
• Time 0: Zero
• Time 6: Outflow of $26 and inflow of S6 at time 6
(value of the contract: S6 – 26.)
Synthetic contract: borrow $25.00 to buy the
stock
• Time-0 cash flow: Zero
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Example (cont’d)
At time 6,
• Synthetic contract: pay back the borrowed money
and still have the stock. Payment:
25[ e(.07)(6/12) ] = 25.89
• Forward contract: pay $26.00 to have the stock
Conclusion: the contract is over-priced!
Sell it! (Short it!)
At the same time,
buy (long) the synthetic contract!
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Example (cont’d)
At Time 0 (Cash)
• Borrow $25.00 at a 7.00% annual rate for 6 months
• Buy the stock at $25.00
• Write the forward at $26.00
Between 0 and 6 (Carry)
At time 6
• Pay back borrowed money: 25[ e(.07)(6/12) ] = 25.89
• Get $26.00 from the forward (and give up the stock)
• Net payoff: $0.11
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Learn from the Example
Arbitrage-free forward price: F0, T = S0 erT
Forward price is the deferred value of the spot price
The deferred rate is the risk-free rate
Exercise:
• S0= $25.00; F0, 6 = $25.50
• The continuously compounded risk-free annual
interest rate = 7.00%
• What arbitrage would you undertake? How to make
profit?
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Something is worth whatever
it costs to replicate it
Derivatives securities are by definition those for
which a perfect replica can be constructed from
other better-known securities.
The role of models: find the replica.
Buying the replica is the same as buying the
derivative.
Selling the replica is the same as hedging the
derivative.
Absence of arbitrage implies the two have the
same price.
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After-class Exercises
Work on the following problems to check your
knowledge on the time value of money and noarbitrage principle.
1. An interest rate is quoted as 5% per annum with
semiannual compounding. What is the equivalent rate
with (a) annual compounding, (b) monthly
compounding, and (c) continuous compounding?
2. An investor receives $1,100 in one year in return for an
investment of $1,000 now. Calculate the percentage
return per annum with: (a) Annual Compounding, (b)
Semiannual Compounding, (c) Monthly Compounding,
(d) Continuous Compounding
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3. Your grandfather put some money in an account for you
on the day you were born. You are now 18 years old and
are allowed to withdraw the money for the first time. The
account currently has $3,996 in it and pays an 8%
interest rate.
a) How much money would be in the account if you left the
money there until your 25th birthday? 65th birthday?
b) How much money did your grandfather originally put in
the account?
4. You purchased 100 shares of stock which pays share
dividend with continuous compounding rate of 1%. Five
years later, the stock price is HK$50 per share. What is
the worth of your investment?
49
5. If you need €15,000 in three years, how much will you
need to deposit today if you can earn 8 percent per year
compounded continuously?
6. Suppose Bank One offers a risk-free interest rate of 5.5%
on both savings and loans, and Bank Enn offers a riskfree interest rate of 6% on both savings and loans.
a) What arbitrage opportunity is available?
b) Which bank would experience a surge in the demand for
loans? Which bank would receive a surge in deposit?
c) What would you expect to happen to the interest rates the
two banks are offering?
7. Suppose the stock price is $35 and the continuously
compounded interest rate is 5%. Assuming dividends are
zero, what is the 6-month arbitrage-free forward price?
50
8. An Exchange-Traded Fund (ETF) is a security that
represents a portfolio of individual stocks. Consider an
ETF for which each share represents a portfolio of two
shares of Hewlett-Packard (HP), one share of Sears,
Roebuck (S), and three shares of Ford Motor (F). Suppose
the current stock prices of each individual stock are as
shown here:
Stock
HP
S
F
Price
$28
$40
$14
a) What is the price per share of the ETF in a normal market?
b) If the ETF currently trades for $120, what arbitrage
opportunity is available? What trades would you make?
c) If the ETF currently trades for $150, what arbitrage
opportunity is available? What trades would you make?
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9. The table here shows the no-arbitrage prices of securities
A and B.
Cash Flow in One Year
Security
Price Today Weak Economy
Strong Economy
Security A
230.77
0
600
Security B
346.77
600
0
a) What are the payoffs of a portfolio of one share of security
A and one share of security B?
b) What is the market price of this portfolio? What expected
return will you earn from holding this portfolio
c) Suppose security C has a payoff of $600 when the economy
is weak and $1,800 when the economy is strong. What is
the no-arbitrage price of security C?
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