Transcript Chapter 1: Introduction

Introduction
Chapter 1
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DERIVATIVES:
FORWARDS
FUTURES
OPTIONS
SWAPS
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DERIVATIVES ARE
CONTRACTS:
Two parties
Agreement, or Contract
Underlying security
Contract termination date
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The Difference between Derivatives:
The contract determines
the rights and/or obligations of the
two parties
in relation to the
sale/purchase
of the
underlying asset.
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The Nature of Derivatives
A derivative is a contract whose value
depends on the values of other more
basic underlying variables
In particular, it depends on the
market price of the so called:
underlying asset.
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Underlying assets:
Stocks
Interest bearing securities: Bonds
Foreign currencies:
Euro, GBP, AUD,…
Metals: Gold, Silver…
Energy commodities: Crude oil, Natural gas,
Gasoline, heating oil…
Agricultural commodities: Wheat, corn, rice,
grain feed, soy beans, pork
bellies…
Stock indexes
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Ways Derivatives are Used
• To hedge risks
• To speculate (take a view on the
future direction of the market)
• To lock in an arbitrage profit
• To change the nature of a liability
• To change the nature of an
investment without incurring the
costs of selling one portfolio and
buying another
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Types of risk:
Price risk
Credit risk
Operational risk
Completion risk
Human risk
Regulatory risk
Tax risk
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IN THIS CLASS WE WILL ONLY
ANALYZE THE
RISK
ASSOCIATED WITH THE
SPOT MARKET PRICE
OF
THE UNDERLYING ASSET
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PRICE RISK: At time t, the
asset’s price at time T is not
known.
Probability
distribution
ST
St
t
T
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time
FORWARDS
A FORWARD
IS A CONTRACT IN WHICH:
ONE PARTY COMMITS TO BUY AND THE
OTHER PARTY COMMITS TO SELL A
SPECIFIED AMOUNT
OF AN AGREED UPON COMMODITY
OF A SPESIFIC QUALITY
FOR A PREDETERMINED PRICE
ON AN AGREED UPON FUTURE DATE
AT A GIVEN LOCATION
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Buy or sell
a forward
t
Delivery
and
payment
T
Time
BUY = OPEN A LONG POSITION
SELL = OPEN A SHORT POSITION
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Forward Price
• The forward price for a contract
is the delivery price that would be
applicable to the contract if were
negotiated today (i.e., it is the
delivery price that would make
the contract worth exactly zero)
• The forward price may be
different for contracts of
different maturities
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Foreign Exchange Quotes for
USD/GBP on DEC 19, 2007
Spot
Bid
1.99480
Ask
1.99720
1-month forward
1.99283
1.99530
3-month forward
1.99127
1.99376
6-month forward
198353
1.98650
12-month forward
1.96990
1.972663
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Profit from a
Long Forward Position
Profit
K
Price of Underlying
at Maturity, ST
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Profit from a
Short Forward Position
Profit
K
Price of Underlying
at Maturity, ST
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A FUTURES
is
A STANDARDIZED FORWARD TRADED ON
AN ORGANIZED EXCHANGE.
STANDARDIZATION
THE COMMODITY
THE QUANTITY
THE QUALITY
PRICE QUOTES
DELIVERY DATES
DELIVERY PROCEDURES
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Futures Contracts
• Agreement to buy or sell an asset
for a certain price at a certain time
• Similar to forward contract
• Whereas a forward contract is
traded OTC, a futures contract is
traded on an exchange
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AN OPTION
IS A CONTRACT IN WHICH ONE PARTY HAS THE RIGHT,
BUT NOT THE OBLIGATION, TO BUY OR SELL A
SPECIFIED AMOUNT OF AN AGREED UPON COMMODITY
FOR A PREDETERMINED PRICE BEFORE OR ON A
SPECIFIC DATE IN THE FUTURE. THE OTHER PARTY HAS
THE OBLIGATION TO DO WHAT THE FIRST PARTY
WISHES TO DO. THE FIRST PARTY, HOWEVER, MAY
CHOOSE NOT TO EXERCISE ITS RIGHT AND LET THE
OPTION EXPIRE WORTHLESS.
A CALL = A RIGHT TO BUY THE UNDERLYING ASSET
A PUT = A RIGHT TO SELL THE UNDERLYING ASSET
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Long Call on Microsoft
Profit from buying a European call option
on Microsoft: option price = $5, strike
price = $60
30 Profit ($)
20
10
30
0
-5
40
50
Terminal
stock price ($)
60
70
80
90
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Short Call on Microsoft
Profit from writing a European call
option on Microsoft: option price = $5,
strike price = $60
Profit ($)
5
0
-10
70
30
40
50 60
80
90
Terminal
stock price ($)
-20
-30
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Long Put on IBM
Profit from buying a European put
option on IBM: option price = $7, strike
price = $90
30 Profit ($)
20
10
0
-7
Terminal
stock price ($)
60
70
80
90
100 110 120
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Short Put on IBM
Profit from writing a European put
option on IBM: option price = $7, strike
price = $90
Profit ($)
7
0
60
70
Terminal
stock price ($)
80
90
100 110 120
-10
-20
-30
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A SWAP
IS A CONTRACT IN WHICH THE TWO
PARTIES COMMIT TO EXCHANGE A
SERIES OF CASH FLOWS. THE CASH
FLOWS ARE BASED ON AN AGREED UPON
PRINCIPAL AMOUNT. NORMALLY, ONLY
THE NET FLOW EXCHANGES HANDS.
Principal amount = EUR100,000,000;
semiannual payments. Tenure: three years.
7%
Party B
Party A
6-months LIBOR
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EXAMPLE: A RISK MANAGEMENT SWAP
BONDS MARKET
FL1 = 6-MONTH BANK RATE.
FL2 = 6-MONTH LIBOR.
LOAN
FL1
10%
SWAP DEALER A
BANK
LOAN
FL2
12%
FIRM A
BORROWS AT A
FIXED RATE FOR 5
YEARS
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THE BANK’S CASH FLOW:
12% - FLOATING1 + FLOATING2 – 10% = 2% + SPREAD
Where the
SPREAD = FLOATING2 - FLOATING1
RESULTS
THE BANK EXCHANGES THE RISK
ASSOCIATED WITH THE DIFFERENCE BETWEEN FLOATING1
and 12% WITH THE RISK ASSOCIATED WITH THE
SPREAD = FLOATING2 - FLOATING1.
The bank may decide to swap the SPREAD for
fixed, risk-free cash flows.
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EXAMPLE: A RISK MANAGEMENT SWAP
BOND MARKET
FL1
10%
SWAP DEALER A
BANK
FL2
FL2
12%
FL1
SWAP DEALER B
FIRM A
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THE BANK’S CASH FLOW:
12% - FL1 + FL2 – 10% + (FL1 - FL2 ) = 2%
RESULTS
THE BANK EXCHANGES THE
RISK ASSOCIATED WITH THE
SPREAD = FL2 - FL1
WITH A FIXED RATE OF 2%.
THIS RATE IS A FIXED RATE!
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WHY TRADE DERIVATIVES?
THE FUNDAMENTAL
REASON FOR TRADING
DERIVATIVES IS TO HEDGE:
THE PRICE RISK
Exhibited by the Underlying commodity’s
spot price volatility
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PRICE RISK IS THE
VOLATILITY
ASSOCIATED WITH THE COMMODITY’S
PRICE IN THE
CASH MARKET
REMEMBER THAT THE CASH MARKET IS
WHERE FIRMS DO THEIR BUSINESS. I.E., BUY
AND SELL THE COMMODITY.
ZERO PRICE VOLATILITY
NO DERIVATIVES!!!!
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PRICE RISK: At time t, the
asset’s price at time T is not
known.
Probability
distribution
ST
St
t
T
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time
Derivatives Traders
• Speculators
• Hedgers
•Arbitrageurs
Some of the large trading losses in
derivatives occurred because individuals
who had a mandate to hedge risks
switched to being speculators
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THE ECONOMIC PURPOSES OF
DERIVATIVE MARKETS
HEDGING
PRICE DISCOVERY
SAVING
HEDGING
IS THE ACTIVITY OF MANAGING PRICE RISK
EXPOSURE
PRICE DISCOVERY IS THE REVEALING OF INFORMSTION
ABOUT THE FUTURE CASH MARKET PRICE
OF A PRODUCT.
SAVING
IS THE COST SAVING ASSOCIATED WITH
SWAPING CASH FLOWS
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A Review of Some Financial
Economics Principles
Arbitrage: A market situation whereby an investor
can make a profit with:
no equity and no risk.
Efficiency: A market is said to be efficient if prices
are such that there exist no arbitrage opportunities.
Alternatively, a market is said to be inefficient if prices
present arbitrage opportunities for investors in this
market.
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Valuation: The current market value (price)
of any project or investment is the net
present value of all the future expected
cash flows from the project.
One-Price Law: Any two projects whose
cash flows are equal in every possible state
of the world have the same market value.
Domination: Let two projects have equal
cash flows in all possible states of the world
but one. The project with the higher cash
flow in that particular state of the world has
a higher current market value and thus, is34
said to dominate the other project.
The Holding Period Rate of Return (HPRR):
Buy shares of a stock on date t and sell
them later on date T. While holding the
shares, the stock has paid a cash dividend
In the amount of $D/share.
The Holding Period Rate of Return HPRR is:
ST  DT t  St
R Tt 
St
365
R annual 
R Tt
Tt
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Example:
St = $50/share
ST = $51.5/share
DT-t = $1/share
T = t + 73days.
51.5  1 - 50
R 73days 
 .05
50
365
R annual 
[.05]  .25
73
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A proof by contradiction:
is a method of proving that an
assumption, or a set of
assumptions, is incorrect by
showing that the implication of
the assumptions contradicts
these very same assumptions.
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Risk-Free Asset: is a security of investment
whose return carries no risk. Thus, the
return on this security is known and
guaranteed in advance.
Risk-Free Borrowing And Landing: By
purchasing the risk-free asset, investors
lend their capital and by selling the riskfree asset, investors borrow capita at the
risk-free rate.
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The One-Price Law:
There exists only one risk-free rate in an
efficient economy.
Proof: By contradiction. Suppose two risk-free
rates exist in a market and R > r. Since both are
free of risk, ALL investors will try to borrow at r
and invest the money borrowed in R, thus
assuring themselves the difference. BUT, the
excess demand for borrowing at r and excess
supply of lending (investing) at R will change
them. Supply = demand only when R = r.
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Compounded Interest (p. 76)
Any principal amount, P, invested at an
annual interest rate, R, compounded
annually, for n years would grow to:
n
An = P(1 + R) .
If compounded Quarterly:
4n
An = P(1 +R/4) .
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In general:
Invest P dollars in an account which pays
An annual interest rate R with m
Compounding periods every year.
The rate in every period is R/m.
The number of compounding periods is nm.
Thus, P grows to:
An = P(1
mn
+R/m) .
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mn
An = P(1 +R/m)
.
Monthly compounding becomes:
An = P(1 +R/12)
12n
and daily compounding yields:
An = P(1 +R/365)
365n
.
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EXAMPLES:
n =10 years; R =12%; P = $100
1.Simple compounding, m = 1, yields:
10
A10 = $100(1+ .12)
= $310.5848
2.Monthly compounding, m = 12, yields:
120
A10 = $100(1 + .12/12)
= $330.0387
3.Daily compounding, m = 365, yields:
A
3,650
= $100(1 + .12/365)
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= $331.9462.
DISCOUNTING
The Present Value of a future income,
FVT,
on date T hence, is given by
DISCOUNTING:
FVT
PVt 
T
[1  R]
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DISCOUNTING
Let cj, j = 1,2,3,….,; be a sequence of
payments paid out m times a year over the
next n years. Let R be the annual rate
During these years.
DISCOUNTING this cash flow yields the
Present Value:
mn
j
t
j
j1
PV  
c
R
[1  ]
m
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CONTINUOUS COMPOUNDING
In the early 1970s, banks came up with
The following economic reasoning: Since
The bank has depositors money all the
time, this money should be working for
the depositor all the time! This idea, of
course, leads to the concept of continuous
compounding.
We want to apply this idea to the formula:
mn
R

A n  P 1   .
 m
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CONTINUOUS COMPOUNDING
As m increases the time span of every
compounding period diminishes
Compounding
m
Time span
Yearly
1
1 year
Daily
365
1 day
Hourly
8760
1 hour
Every second 3,153,600
One second
Continuously
∞
Infinitesimally
small 47
CONTINUOUS COMPOUNDING
This reasoning implies that in order to
impose the concept of continuous time on
the above compounding expression, we
need to solve:
mn
R

A n  Limit {P 1   }
m
as: m 
This expression may be rewritten
A n  Pe
Rn
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Recall that the number “e” is:
x
 1
e  Limit {1   }
x 
 x
X
e
1
2
100
2.70481382
10,000
2.71814592
1,000,000
2.71828046
In the limit
2.718281828…..
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Recall that in our example:
n = 10 years.
R = 12%
P=$100.
So, P = $100 invested at a 12% annual
rate, continuously compounded for ten
years will grow to:
Rn
A n  Pe
 $100e
 $332.0117
(.12)(10)
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Continuous compounding yields the
highest return:
Compounding
m
Factor
Simple
1
3.105848208
Quarterly
4
3.262037792
Monthly
12
3.300386895
Daily
365
3.319462164
Continuously
∞
3.320116923 51
Continuous Discounting
Pe

A
n
This expression may be rewritten as:
Rn
Thus, given A n , R and n,
P  Ane
- Rn
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Continuous Discounting
In general, CFT ,
can be continuous ly discounted
This expression may be rewritten as:
for the present ti me, t :
PVt  CFT e
- Rt
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Recall that in our example:
P = $100; n = 10 years and R = 12%
Thus, $100 invested at an annual rate of
12% , continuously compounded for ten
years will grow to: $332.0117.
Therefore, we can write the continuously
discounted value of $320.0117:
A0  Ane
-Rn
 $332.0117e
 $100.
- (.12)(10)
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Equivalent Interest Rates (p.77)
Rm =
The annual rate with m
compounding periods
every year.
R m mn
A n  P[1 
]
m
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Equivalent Interest Rates (p.77)
rc =
The annual rate with
continuous compounding
Bn  Pe
rc n
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Equivalent Interest Rates (p.77)
Rm =
rc =
The annual rate with m
compounding periods every year.
The annual rate with continuous
compounding.
Definition:
if:
Rm and rc are said to be
equivalent
Bn  An
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Equivalent Interest Rates (p.77)
Bn  A n
R m mn
Pe  P[1 
]
m
Rm
rc  mln[1 
]
m
rc m
R m  m[e
 1]
rc n
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Risk-free lending and borrowing
Treasury bills: are zero-coupon bonds, or pure
discount bonds, issued by the Treasury.
A T-bill is a promissory paper which promises its holder
the payment of the bond’s Face Value (Par- Value)
on a specific future maturity date.
The purchase of a T-bill is, therefore, an investment
that pays no cash flow between the purchase date
and the bill’s maturity. Hence, its current market
price is the NPV of the bill’s Face Value:
Pt = NPV{the T-bill Face-Value}
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Risk-free lending and borrowing
Risk-Free Asset: is a security whose return is a
known constant and it carries no risk.
T-bills are risk-free LENDING assets.
Investors lend money to the Government by
purchasing T-bills (and other Treasury notes
and bonds)
We will assume that investors also can
borrow money at the risk-free rate.
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Risk-free lending and borrowing
LENDING:
By purchasing the risk-free asset,
investors lend capital.
BORROWING:
By selling the risk-free asset, investors
borrow capital.
Both activities are at the
risk-free rate.
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We are now ready to calculate the current
value of a T-Bill.
Pt = NPV{the T-bill Face-Value}.
Thus:
the current time, t, T-bill price, Pt , which
pays FV upon its maturity on date T, is:
Pt = [FV]e-r(T-t)
r
is the risk-free rate in the economy.
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EXAMPLE: Consider a T-bill that
promises its holder FV = $1,000 when
it matures in 276 days, with a risk-free
yield of 5%:
Inputs for the formula:
FV = $1,000; r = .05; T-t = 276/365yrs
Pt = [FV]e-r(T-t)
Pt = [$1,000]e-(.05)276/365
Pt = $962.90.
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EXAMPLE: Calculate the yield-to -maturity
of a bond which sells for $965 and
matures in 100 days, with FV = $1,000.
Pt = $965; FV = $1,000; T-t=
100/365yrs.
1
FV
Solving for r: r 
ln[
]
-r(T-t)
Pt = [FV]e
T-t
Pt
1
1,000
r
ln[
]  13%
100
965
365
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SHORT SELLING STOCKS (p. 97)
An Investor may call a broker and ask to “sell a
particular stock short.”
This means that the investor does not own shares of
the stock, but wishes to sell it anyway. The investor
speculates that the stock’s share price will fall and
money will be made upon buying the shares back at
a
lower price. Alas, the investor does not own shares of
the stock. The broker will lend the investor shares
from the broker’s or a client’s account and sell it
in the investor’s name. The investor’s obligation is to
hand over the shares some time in the future, or
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upon the broker’s request.
SHORT SELLING STOCKS
Other conditions:
The proceeds from the short sale cannot be used by
the short seller. Instead, they are deposited in an
escrow account in the investor’s name until the
investor makes
good on the promise to bring the shares back.
Moreover, the investor must deposit an additional
amount of at least 50% of the short sale’s proceeds
in the escrow account. This additional amount
guarantees that there
is enough capital to buy back the borrowed shares and
hand them over back to the broker, in case the
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shares price increases.
SHORT SELLING STOCKS
There are more details associated with short
selling stocks. For example, if the stock pays
dividend, the short seller must pay the dividend
to the broker. Moreover, the short seller does
not gain interest on the amount deposited in
the escrow account, etc.
We will use stock short sales in many of
strategies associated with options trading. In all
of these strategies, we will assume that no
cash flow occurs from the time the strategy is
opened with the stock short sale until the time
the strategy terminates and the stock is
repurchased.
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SHORT SELLING STOCKS
In terms of cash flows per share:
St
is the cash flow/share from
selling the stock short thereby,
opening a SHORT POSITION on
date t.
-ST
is the cash flow from purchasing
the stock back on date T (and
delivering it to the lender
thereby, closing the SHORT
POSITION.)
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