Transcript derivative security - the School of Economics and Finance

MFIN6003 Derivative Securities
Lecture Note One
Faculty of Business and Economics
University of Hong Kong
Dr. Huiyan Qiu
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Outline
Course Overview
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
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Overview of the Course
The course is about:
• the concept, the use, the pricing of derivatives.
1.
Introduction to derivatives in general
2.
Introduction of forwards and options and risk
management using forwards and options
3.
Option spread, collars, and other option
strategies
4.
Pricing of forward and futures and futures
trading
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Overview of the Course (cont’d)
5.
Currency forward / futures, interest rate
forward / futures
6.
Swaps
7.
Parity and other option relationships
8.
Binomial option pricing model
9.
Black-Scholes formula and delta-hedging
10.
Financial engineering and security design,
structured products, exotic options and credit
derivatives
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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.
We examine how to use derivative contracts to
deal with financial risks related to:
– Interest rates
– Commodity prices
– Exchange rates
– Stock prices
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2009 ISDA Derivatives Usage Survey
Types of Risk Managed using Derivatives (%)
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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 $7, you will pay him $1
• If the price is less than $7, 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 including CDS,
structured products
The exchange market: where buyer and seller
can do a deal without worrying about finding
each other.
• Futures contracts, most options
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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 value of a derivative product's
underlying assets at the spot price.
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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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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 = 12%
 APR 
1  EAR   1 

n 

Compounding
Frequency
n
Annually
1
12.0000
Quarterly
4
12.5509
Monthly
12
12.6825
Weekly
52
12.7341
Daily
365
12.7475
∞
12.7497
Continuously
n
EAR
(% p.a.)
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Continuously Compounding
Continuously compounding: n → ∞ (infinity)
n
 r
lim1    er
n
 n
by definition of e.
APR = 12%  EAR = 12.75%
n
 0.12 
0.12
1  EAR  lim1 
  e  1.1275
n
n 

• At 12% continuously compounding annual
interest rate, the future value of $100 is
$112.75.
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Continuous Dividend Payment
Consider a stock (in general, an asset) paying
continuous dividend with annual rate of δ.
Claim: The present value of 1 share at time T
is then S0e-δT.
Reason:
One share at time T
is equivalent to
e-δT shares at time 0 !
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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. Or one share T
T
e
years later is equivalent to
shares today.
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: Cash flows associated with shortselling a share of HSBC for 90 days.
Note that the short-seller must pay the dividend, D,
to the share-lender. In other words, the lender must
be compensated for the dividend.
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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
• The difference between this rate and the market
rate of interest is another cost to your short-sale
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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.
You pay a commission rate of 0.5%.
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
If the effective market interest rate over your
holding period is 2%,
• Case 1: loss of $97.825
• Case 2: loss of $68.82
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Discussion
Question 1: With zero interest rate, why the
loss in short-selling is more than the loss in
outright purchase?
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. Reason?
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 (selling) the replica is the same as buying
(selling) the derivative.
Absence of arbitrage implies the two have the
same price.
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End of the Notes!
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