WORKBOOK
RAMON
RABINOVITCH
1
DERIVATIVES
ARE
CONTRACTS
Two parties
Agreement
Underlying security
2
DERIVATIVES
FORWARDS
FUTURES
OPTIONS
SWAPS
3
FORWARD MARKET
THE MARKET FOR
DEFERRED DELIVERY
SELLER = SHORT
BUYER = LONG
THE TWO PARTIES MAKE A
CONTRACT THAT
DETERMINES THE
TRANSACTION TO BE
MADE ON A FUTURE DATE
DELIVERY AND PAYMENT WILL
TAKE PLACE IN THE FUTURE AS
SPECIFIED BY THE CONTRACT
BETWEEN THE SHORT AND
LONG.
4
A FORWARD CONTRACT
THE SHORT WILL SELL TO
THE LONG 25,764 AUNCES
OF GOLD OF A CERTAIN
QUALITY. THE PRICE
WILL BE $300/AUNCE.
DELIVERY AND PAYMENT
WILL TAKE PLACE IN A
SPECIFIED PLACE
EXACTLY 95 DAYS FROM
TODAY.
RISKS
5
THE FUTURES MARKET
FUTURES CONTRACTS ARE
STANDARDIZED
FORWARDS TRADED ON
ORGANIZED EXCHANGES
STANDARDIZED ARE THE
COMMODITIES, THE
DELIVERY PROCEEDURES,
THE DELIVERY TIMES
AND PAYMENTS.
CLEARINGHOUSE
6
THE MARKET FOR SWAPS
A SWAPS IS AN
AGREEMENT BETWEEN
TWO PARTIES
ARRANGING CASH FLOWS
DISTRIBUTION BETWEEN
THE TWO PARTIES. THE
CASH FLOWS ARE BASED
ON THE VALUE OF AN
UNDERLYING
COMMODITY
7
THE MARKET FOR SWAPS
FOR THE NEXT 5 YEARS,
PARTY B WILL PAY PARTY
C THE FIXED RATE OF 7%
ON $100,000,000, WHILE
PARTY C WILL PAY PARTY
B THE 6-MONTHS LIBOR.
PAYMENTS WILL TAKE
PLACE TWICE A YEAR.
ONLY THE NET CASH FLOW
WILL EXCHANGE HANDS
8
THE MARKET FOR SWAPS
$3,500,000 =>
B
C
<= 6-month LIBOR
THE UNDERLYING
SECURITY IS $100,000,000
9
OPTIONS
• A contingent claim:
The option’s value is
contingent upon the value
of the underlying asset
Two Types of Options
• Calls
•
Puts
10
Option Buyer, holder or long.
In exchange for making a
payment of money the
premium, the owner of an
option-the long-has a call-the
right, but not the obligation, to
buy or a put-the right to sell- a
specified quantity of the
underlying commodity at the
execise price before the
option’s expiration date.
11
Option Seller, writer or short.
In exchange for receiving the
premium, the option’s writer
has the obligation to sell the
underlying asset (in case of a
call) or purchase the underlying
asset ( in case of a put) at the
predetermined exercise price
upon being served with an
exercise notice during the life
of the option, I.e., before the
option expires.
12
WHY TRADE
DERIVATIVES?
PRICE RISK
UNCERTAINTY
VOLATILITY
IS THE FUNDAMENTAL
REASON FOR TRADING
DERIVATIVES
13
PRICE RISK
At time zero, the
asset’s price at time t
is not known.
Probability
distributio
St
S0
0
t
14
time
OPTIONS NOTATIONS:
S– the underlying
asset’s market price
X- the exercise price
t – the current date
T – the expiration date
T-t - time till expiration
c,p - European call, put
premiums
C,P –American call,put
premiums
15
Buyer of a call option has
the right to buy the
underlying at the strike
price, X, before the call
expires at T.
Thus => expects the
price of the underlying
commodity to increase
during the period of the
option contract.
16
Seller of a call option
must sell the underlying
asset for X, if exercised.
Thus => expects the price
of the underlying asset to
remain below or at the
exercise price during the
option’s life.This way the
writer keeps the premium
17
Buyer of a put option has
the right to sell the
underlying for X before
the put expires at T.
Thus => expects the
market price of the
underlying commodity, S,
to decrease during the
put’s life.
18
Seller of a put must buy
the underlying for X if
exercised.
Thus => expects the S to
remain at or above X
during the put’s life.
This way the put writer
keeps the premium.
19
$
X
X
t
T S
20
Types of Options
• American Options
exercisable any time
before expiration
• European Options
exercisable only on
expiration date
Asian Options
European style on the
underlying average price
during its life
21
At-the-money
S=X
In this case the intrinsic
value for both calls and
puts is zero:
S-X = X-S = 0
and the premium
consists of the extrinsic
value only.
22
In-the-money
Calls
Puts
S>X
S<X
S-X >0 X-S>0
The Intrinsic value
is positive
23
Out-of-the-money
Calls
S<X
S-X<0
Puts
S>X
X-S>0
The intrinsic value is zero
and the premium consists
of the
extrinsic value only
24
Option Price = Premium
Two Components:
Intrinsic Value
The amount by which an
option is in-the-money.
Extrinsic (time) Value
The amount by which
the price of an option
exceeds its intrinsic
value.
25
Option Market Price
Premium
= Intrinsic value
+ extrinsic value
Intrinsic value
Calls
Max{0, S-X)
Puts
Max{0, X-S)
Intrinsic value cannot be
negative
26
Option Parameters
Underlying Price = S
Strike Price = X
Time to Expiration =T-t
Annual Volatility = s
Annual Interest Rate = r
Annual Dividend Rate= q
27
Summary
CALLS
PUTS
S X
50 35
50 40
50 50
50 60
Feb Mar May
18 19 21
12 13.5 16
6.5 8.25 12
3
4
9
Feb
.05
.25
.75
11
Mar
.15
.34
1
12
Apr
.27
.50
1.15
15
All prices are in $s
Expirations: At the
rd
market close on the 3
Friday of the expiration
month
28
INTEL Thursday, September 21, 2000.
S = $61.48
CALLS - LAST
PUTS - LAST
X
oct
nov jan apr
oct
nov
jan apr
40
22
---
23
---
---
---
0.56
---
50
12
---
---
---
0.63 ---
---
---
55
8.13 ---
11.5 ---
1.25
---
3.63
---
60
4.75 ---
8.75 ---
2.88
4
5.75
---
65
2.50 3.88 5.75 8.63
6.00 6.63 8.38
10
70
0.94 ---
3.88 ---
9.25 ---
11.25 ---
75
0.31 ---
---
13.38 ---
--- 16.79
80
---
---
1.63 ---
---
90
---
---
0.81 ---
---
---
---
95
---
---
0.44
---
---
--- ---
5.13
---
---
---
-----
29
WHY TRADE OPTIONS ON
ORGANIZED EXCHANGES?
IN THE OVER-THE-COUNTER
MARKET (OTC)
INVESTORS ARE EXPOSED
TO
Credit risk
Operational risk
Liquidity risk
30
1.CREDIT RISK
Does the other party
have enough
resources to meet
its obligation?
31
2. Operational risk:
Will the other party
deliver the underlying
if I exercise my call?
Will the other party
take delivery of the
underlying if I exercise
my put ?
32
3.Market liquidity.
In case the long wishes to
get out of the market,
what are the obstacles?
In case the short wishes to
quit its obligation, what to
do?
In other words: how can the
two parties come out of
their respective
obligations?
33
THE GUARANTEE
The exchanges understood
that there will exist no
efficient options markets
until the above problems
are resolved. So they have
created the:
OPTIONS CLEARING
CORPORATION
34
THE OPTION CLEARING
CORPORATION PLACE
IN THE MARKET
EXCHANGE CORPORATION
OPTIONS CLEARING CORPORATION
CLEARING
NONCLEARING
MEMBERS
MEMEBRS
OCC MEMBER
CLIENTES
BROKERS
35
The Options Clearing Corporation
(OCC) gives all the LONGS the
absolute guarantee
of the completion of its side of the
contract:
You will always be able to exercise your
option!!!
The OCC’s absolute guarantee
provides traders with a
default-free market.
Thus, any investor who wishes to engage
in options buying knows that there will
36
be no operational default.
The OCC
also clears all options trading and
maintains the list of all long and short
positions. Every long position must be
MATCHED with a short position.
Hence, the total sum of all options
traders positions must be ZERO at all
times.
The OCC’s absolute guarantee together
with the trading list makes the market
very liquid.
ONE – traders are not afraid to enter the
market
TWO – traders can quit the market at
any point in time by OFFSETTING
their original position.
37
OFFSETTING POSITIONS
A trader with a LONG position who
wishes to get out of the market must
open a SHORT position with equal
number of the same options on the same
underlying asset for the same month of
expiration and for the same exercise
price.
EX: LONG 5, SEP, $125, IBM puts
offsets this position by
SHORT 5, SEP, $125, IBM puts.
The above trades offset each other and
zero out the trader’s position with the
OCC. This trader is out of the market.
The premiums paid and received upon
entering the above positions determine
38
the trader’s profit or loss.
OFFSETTING POSITIONS
A trader with a SHORT position who
wishes to get out of the market must
open a LONG position with equal
number of the same options on the same
underlying asset for the same month of
expiration and for the same exercise
price.
EX: SHORT 25, JAN, $75, BA calls
offsets this position by
LONG
25, JAN, $75, BA calls.
The above trades offset each other and
zero out the trader’s position with the
OCC. This trader is out of the market.
The premiums paid and received upon
entering the above positions determine
39
the trader’s profit or loss.
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.
40
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, is said to
dominate the other project.
41
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.
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 risk-free asset, investors
borrow capita at the risk-free rate.
42
The One-Price Law:
There exists only one risk-free
rate in an efficient economy.
43
Compounded Interest
Any principal amount, P, invested at an
annual interest rate, r, compounded
annually, for n years would grow to:
An = P(1 + r)n.
If compounded Quarterly:
An = P(1 +r/4)4n.
In general, with m compounding periods
every year, the periodic rate becomes
r/m and nm is the total compounding
periods. Thus, P grows to:
An = P(1 +r/m)nm.
44
Monthly compounding becomes:
An = P(1 +r/12)n12
and daily compounding yields:
An = P(1 +r/12)n12.
EXAMPLES:
n =10 years;
r =12%; P = $100
1.
Simple compounding yields:
A10 = $100(1+ .12)10 = $310.58
2.
Monthly compounding yields:
A10 = $100(1 + .12/12)120 = $330.03
3.
Daily compounding yields:
A10 = $100(1 + .12/365)3650 = $331.94.
45
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 in the formula:
mn
r 

A n  P 1   .
m

Observe that continuous time means that
the number of compounding periods, m,
increases without limit, while the periodic
interest rate, r/m, becomes smaller and
smaller.
46
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
 m
This expression may be rewritten as:
m
(
) rn
r


1 

A n  (P)Limit { 1 
}.
m 
m  
r

The solution of this limit yields
the expression for the continuous ly
compounded value of P after n years :
A n  Pe .
rn
47
EXAMPLE, continued:
First, we remind you that the number e
is defined as:
x
1

e  Limit {1   }
x 
x

For example:
x
e
1
2
10
2.59374246
100
2.70481382
1,000
2.71692393
10,000
2.71814592
1,000,000
2.71828046
In the limit e = 2.718281828…..
48
Recall that in our example:
N = 10 years and r = 12% and P=$100.
Thus, P=$100 invested at a 12% annual
rate, continuously compounded for ten
years will grow to:
A n  Pe  $100e
rn
(.12)(10)
 $332.01
Continuous compounding yields the
highest return to the investor:
Compounding
Factor
Simple
3.105848208
Quarterly
3.262037792
Monthly
3.300386895
Daily
3.319462164
continuously
3.320116923
49
Continuous Discounting
From the continuous ly compoundin g
formula, A n  Pe , it is
rn
clear that given A n , r and n,
the continuous ly discounted
This expression may be rewritten as:
value of A n is :
P  A ne .
- rn
More generally, any time period
t cash flow, CFt , can be continuous ly
discounted for the present by
multiplyin g it by :
- rt
e ,
where r is the continuous ly
compounded interest rate.
50
EXAMPLE, continued:
First, we remind you that the number e
is defined as:
x
1

e  Limit {1   }.
x 
x

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:
A n  Pe  $100e
rn
(.12)(10)
 $332.01
Therefore, we can write the continuously
discounted value of $320.01 is:
A0  Ane
-rn
 $332.01e
- (.12)(10)
 $100.
51
PURE ARBITRAGE
PROFIT:
A PROFIT MADE
1. WITHOUT EQUITY
and
2. WITHOUT ANY RISK.
52
Risk-free lending and borrowing
Arbitrage: A market situation in
which 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.
53
Risk-free lending and borrowing
PURE ARBITRAGE PROFIT:
A PROFIT MADE
1.WITHOUT EQUITY INVESTMENT
and
2. WITHOUT ANY RISK
We will assume that
the options market
is efficient.
This assumption implies that one
cannot make arbitrage profits
in the options markets
54
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}
We will only use
continuous discounting
55
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 riskfree rate. I.e., investors may write
IOU notes, promising the risk-free
rate to their buyers, thereby, raising
capital at the risk-free rate.
56
Risk-free lending and borrowing
The One-Price Law:
There exists only one
risk-free rate in an efficient
economy.
Proof: If two risk-free rates exist in
the market concurrently, all investors
will try to borrow at the lower rate
and simultaneously try to invest at
the higher rate for an immediate
arbitrage profit. These activities will
increase the lower rate and decrease
the higher rate until they coincide to
one unique risk-free rate.
57
Risk-free lending and borrowing
By purchasing the risk-free asset,
investors lend capital.
By selling the risk-free asset,
investors borrow capital.
Both activities are at the
risk-free rate.
58
Continuous Discounting:
Recall that continuous compounding and
discounting use the number e, which in
itself is used as the result of
“continualizing” the simple compounding
formula as follows:
mn
r 

A n  Limit {P 1   }
m
m

The solution of this limit
yields the expression for the
continuous ly compounded
value of P after n years :
A n  Pe .
rn
59
EXAMPLE:
First, we remind the reader that the
number e is defined as:
x
1

e  Limit {1   }
x 
x

On your own calculator you may try:
x
e
1
2
10
2.59374246
100
2.70481382
1,000
2.71692393
10,000
2.71814592
1,000,000
2.71828046.
In the limit e = 2.718281828…..
60
From the continuous ly
compoundin g formula,
A t  Pe ,
rt
it is clear that given P,
r and t, we can calculate
A t for may
any time
t.
This expression
be rewritten
as:
But first,
QUESTION:
Given P and r, how long it takes to double
our money? - “the 72 rule”
Ans.:
2P = Pert ;
t = [ln2]/r
t = 69.31/r.
r = 10% ==> t = 6.931yrs.
61
Continuous Discounting
From the continuous ly compoundin g
formula,
A n  Pe , it is
rn
clear tha t given A n , r and n,
the continuous ly discounted
This
expression
value
of A ismay
: be rewritten as:
n
P  A e- rn .
n
More generally, any time period
t cash flow, CFt , can be continuous ly
discounted for the present by
multiplyin g it by :
rt
e ,
where r is the continuous ly
compounded interest rate.
62
Again,
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:
A n  Pe  $100e
rn
(.12)(10)
 $332.01
The continuously discounted value of
$332.01 is:
A 0  A n e  $332.01e
-rn
- (.12)(10)
 $100.
63
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)
Clearly, r is the risk-free rate in the
economy.
64
EXAMPLE: Consider a T-bill that
promises its holder FV = $1,000
when it matures in 276 days, with
a yield-to-maturity 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.
65
EXAMPLE:
The yield-to -maturity of
a bond which sells for $945 and
matures in 100 days, promising the
FV = $1,000 is:
r=?
Pt = $945; FV = $1,000; T-t= 100 days.
Inputs for the formula:
FV = $1,000; Pt= $945; T-t = 100/365.
Solving Pt = [FV]e-r(T-t) for r:
1
FV
r
ln[ ]
T-t
Pt
r = [365/100]ln[$1,000/$945]
r = 10.324%.
66
SHORT SELLING STOCKS
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 upon the
broker’s request.
67
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 shares price increases.
68
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.
In terms of cash flows:
St
is the cash flow from selling the stock
short on date t, and
-ST is the cash flow from purchasing the
back on date T.
69
Options Risk-Return
Tradeoffs
PROFIT PROFILE OF A STRATEGY
A graph of the profit/loss as a
function of all possible
market values of the
underlying asset
We will begin with profit
profiles at the option’s
expiration; I.e., an instant
before the option expires.
70
Options Risk-Return Tradeoffs At
Expiration
1. Only at expiry; T-t = 0
2. No time value; T-t = 0
3. At maturity
CALL is exercised If S>X
expires worthless If S  X
Cash Flow = Max{0, S – X}
PUT is exercised If S<X
expires worthless If SX
Cash Flow = Max{0, X – S}
71
4. All legs of the strategy
remain open till expiry.
5. A Table Format
Every row is one leg of the
strategy. Every row is
analyzed separately.The
total strategy is the vertical
sum of the rows.The profit
is the cash flow at expiry
plus the initial cash flows
at the of the strategy,
disregarding the time value
of money
72
6.A Graph of the
profit/loss profile
The profit/loss from the
strategy as a function of
all possible prices of the
underlying asset at
expiration.
73
The algebraic expressions
Of profit/loss at expiration:
Cash Flows:
Long stock: ST – S0
Short stock:S0 - ST
Long call:-c + Max{0, ST -X}
Short call:c + Min{0,X- ST }
Long put:-p + Max{0,X- ST}
Short put:p + Min{0, ST -X}
74