Schweser`s Workshop - CFA Society of the UK

Transcription

Derivative Investments
Schweser’s
Workshop
1
Welcome to Schweser’s Workshop

Today we will be discussing Derivatives, a topic
common to all three levels of the CFA® Exam
This lecture is a “sample” of a Schweser Live
Intensive Seminar
2
About Schweser
Schweser is…
an enterprise of full-time finance professors and
CFA® Charterholders, support staff and customer
care representatives
a Division of Kaplan, Inc., a wholly owned
subsidiary of the Washington Post
A participating provider in the CFA Institute® Prep
Provider Guidelines Program
the World’s Trusted Leader in CFA® Review
since 1991
3
Why Schweser?
Why choose Schweser as your CFA® Exam Prep
Provider? Our materials and instruction are…
designed with a proven Learn • Practice • RetainTM
approach
a best-fit educational solution for your individual
learning style
the ESSENTIAL COMPLEMENT to the CFA®
Program assigned curriculum and recommended
readings
4
Register to Win!
If you haven’t already signed up,
please register to WIN:
Schweser Multi-Phase
Online Seminar
(retail value 599 USD)
featuring:
28 hours of instruction over two weekends (March, April)
14 hours of Problem Solving and Exam Strategies (May)
Plus Downloadable Seminar slides,
Full subscription to Schweser Library, and
Archives following each session
*Drawing tonight! Must be present to win.
5
CFA Exam Survival Guide

Identify your Learning Style
Go to schweser.com/learning to take the quiz
Set goals
Select study materials
CAUTION! The curriculum has changed extensively…
make sure you use 2007 materials!
Plan to study a MINIMUM of 250 hours
Create a study plan
Make every minute count
Utilize a variety of tools
Attend a Seminar for clarification and reinforcement
6
Schweser Study Solutions
Schweser Study Solutions…
are packages of study tools with choices to fit
your individual learning needs
are based on a Learn • Practice • RetainTM
approach to the curriculum
may be combined with
a Seminar to provide
the most thorough
CFA® Review
7
Schweser Study Solutions
Essential Solution includes Study
Notes and choice of:

Practice Exam book or Flashcards
SchweserProTM Question Bank or
Audio CDs
Premium Solution includes all of
the above plus choice of:

Online Multi-Phase Seminar or
16-Week Online Seminar or
Video CDs with Workbook
8
UKSIP Review Course
Schweser and Koppel & Wiley will
provide course instruction and study
materials for a Live Review Course
featuring:

4.5 days of intense review and guidance
by topic specialists for each Level
Superior study materials
Convenient location in central London:
The London School of Economics and Political
Science
Perfect timing for curriculum review:
26 – 30 March, 2007
9
Level 1
Derivative Markets and
Instruments
10
LOS 73.a, p. 130
Derivatives

A derivative security derives its value from the
price of another (underlying) asset or an
interest rate
Futures and some options are traded on
organized exchanges
Forward contracts, swaps, and some options
are custom instruments created by dealers
11
LOS 73.b, pp. 130-131
Forward Commitments and Contingent
Claims
Contingent claims are called options
Calls: right to buy
Puts: right to sell
12
LOS 73.b, p. 130
Forward Contracts

Customized: no active secondary market
Long obligated to buy, short obligated to sell
Specified asset (currency, stock, index, bond)
Specified date in the future
Long profits if asset price above forward price
Short profits if asset price below forward price
13
LOS 73.b, p. 130
Futures Contracts

Like forward contracts but standardized

Exchange-traded, active secondary market

Require margin deposit

No default (counterparty) risk
14
LOS 73.b, p. 130
Swaps

Equivalent to a series of forward contracts
Simple interest-rate swap
One party pays a fixed rate of interest
One party pays a variable (floating) rate of
interest
Payments can be based on interest rates or
stock/portfolio/index returns
Can involve two different currencies
15
Level 1
Forward Markets and
Contracts
16
LOS 74.a, p. 135
Forward Contract Positions
Long position (will buy)
The party to the forward contract that agrees to
buy the underlying financial or physical asset
Short position (will sell)
The party to the forward contract that agrees to
sell/deliver the asset
Neither party pays at contract initiation
17
LOS 74.b, p. 136
Forward Contract Settlement

Delivery: short delivers underlying to long for
payment of the forward price
Cash settlement: negative side of contract
pays the positive side
18
LOS 74.b, p. 136
Early Termination of Forward
One party pays the other cash (buys their way
out)
Enter into an offsetting contract
With a different counterparty (default risk
still exists)
With same (original) counterparty (no
default risk)
19
LOS 74.d, p. 137-138
Equity Forward Contract

Can be on a stock, a portfolio, or an equity
index
Example: Forward Contract to buy 10,000
shares of Acme Industries common stock in 90
days for $128,000 (i.e. $12.80 per share)
Index contract based on notional amount and
settlement payment is based on percentage
above or below forward contract index value
20
LOS 74.d, p. 137-138
Equity Index Forward Contract
Example:
90-day S&P 100 forward contract
Forward contract price = 525.2
Notional amount = $10 million
In 90 days index is at 535.7, 2% above 525.2
Long receives 2% × $10 million = $200,000 at
settlement, paid by the short
21
LOS 74.d, p. 138
Forward on Zero-Coupon Bond
Example: 100 day, T-bill forward
Underlying: $10 million T-bill
Forward Price: $9,945,560 (1.96% discount)

If interest rates rise, P↓, long loses/short gains
If interest rates fall, P↑, long gains/short loses
Coupon bonds: priced at YTM; same principle
Risky bonds: must provide for default possibility
22
LOS 74.e, p. 139
LIBOR-Based Loan Example
Loan value = $1.0 million
Term = 30 days
30-day LIBOR = 6%
Interest payment
= $1,000,000 (0.06) (30/360) = $5,000
Total payment in 30 days
= $1,000,000 + $5,000 = $1,005,000
23
LOS 74.e, p. 139
Forward Rate Agreement (FRA)
Exchange fixed-rate for floating-rate payment
Notional amount
Fixed rate = forward (contract) rate
Floating rate (LIBOR) is underlying rate
Long gains when LIBOR > contract rate
24
LOS 74.e, p. 139
Forward Rate Agreement (FRA)

Long position can be viewed as the obligation
to take a (hypothetical) loan at the contract
rate, i.e., borrow at the fixed rate; gains when
reference rate↑
Short position can be viewed as the obligation
to make a (hypothetical) loan at the contract
rate, i.e., lend at the contract rate; gains when
reference rate↓
25
LOS 74.f, p. 139-140
FRA Example
Term = 30 days
Underlying rate = 90-day LIBOR
Forward rate = 5%
At t = 30 days, 90-day LIBOR = 6%
Underlying floating rate > fixed rate
Long position receives payment
26
LOS 74.f, p. 139-140
FRA Example: Net Payment
27
LOS 74.f, p. 139-140
FRA Settlement Payment to Long
28
Level 2
Forward Markets and
Contracts
LOS 64.a, b / 148,150
Foundation Concepts

No-arbitrage principle: there should be no riskless
profit from combining forward (or futures) contracts
with other instruments
Forward price = price of underlying that would not
permit profitable riskless arbitrage, so value equals
zero
Off-market forward contract price is different than
no-arb price, so value not zero at initiation; side
with positive value pays to enter
Many pricing relationships for forward contracts are
identical for futures contracts
30
LOS 64.a / p. 148
Pricing a Forward Contract

Price for underlying; not price to enter
contract
No-arbitrage price (cost of carry model):
Maturity
Spot price of underlying
FP = S0 × (1+ Rf )
Forward price
T
Risk-free rate
31
LOS 64.a / p. 148
Valuing a Forward Contract

We can determine the value of a forward
contract during its life
As underlying price changes, the forward
contract will accrue value
Long (buy): gains when price of
underlying increases
Short (sell): gains when price of
underlying decreases
32
LOS 64.a / p. 148
Valuing a Forward Contract

The no-arbitrage value of a LONG forward
contract during the life of the contract (Vt) is:
⎡
⎤
FP
⎛ of long position ⎞
⎥
Vt ⎜
⎟ = St − ⎢
T −t
during
life
of
co
ntract
⎝
⎠
⎢⎣ (1 + R f ) ⎥⎦

The value of the short position is simply the
negative of the long position (because it’s a
zero-sum game: I win, you lose)
33
LOS 64.a / p. 148
An Easy(?) Way to Remember Formula
At contract maturity (time T) long position:
1) Receives ST
2) Pays FP
value t = PV ( ST ) − PV (FP )
⎡
⎤
FP
⎥
= St − ⎢
T−t
⎢⎣ (1 + R f ) ⎥⎦
= " spot price − PV of forward price "
34
LOS 64.d / p. 154
Forward Rate Agreements (FRA)

An FRA is an agreement to borrow (long)
or lend (short) money in the future
Usually based on LIBOR with # days/360
basis
In “2×3” FRA the long, in effect, agrees to
borrow money in two months at a fixed
rate for a term that ends in three months
That is, the “underlying” in a 2×3 FRA is a
30-day loan 60 days from now
35
LOS 64.d / p. 154
“2×3” FRA
36
LOS 64.d / p. 154
Pricing an FRA: Overview

The “price” of an FRA is the implied forward
rate for the period beginning when the FRA
expires to the maturity of the underlying
“loan” (e.g., price of 2×3 FRA in previous
slide is implied 30-day forward rate in 60
days, given current 60-day and 90-day rates)
Calculating this forward rate is a L1 topic!
37
LOS 64.d / p. 154
Pricing an FRA: Example

To review the calculation of forward rates
(which gives the FRA price), assume:
1×3 FRA (i.e., a 60-day loan in 30 days)
The 30-day rate is 2.4%
38
LOS 64.d / p. 154
Pricing an FRA: Example

Unannualized 30-day rate is:
R30 =0.024×
30
=0.002
360

Unannualized 90-day rate is:

90
= 0.0075
360 rate 30 days hence is:
Annualized 60-day
R90 =0.03×
⎛ 1.0075
⎞ 360
FR = ⎜
− 1⎟ ×
= 3.3%
⎝ 1.0020
⎠ 60
FRA Price
39
LOS 64.d / p. 154
Valuing an FRA

We can also value an FRA after initiation
The keys to remember are:
1) Value is determined by changes in interest
rates (e.g., you contracted to borrow at 3.3%
when rates increased to 4%)
2) It’s a forward contract on a rate, so the long
“wins” when rates go up
3) The impact of higher (or lower) rates won’t be
realized until the end of the “loan”
40
LOS 64.d / p. 154
Valuing an FRA: Example
Consider our 1×3 FRA with a contracted rate
of 3.3%. Suppose at contract expiration, 60day rates are 4%. Assume a $1,000,000
notional principal. What is the value of the
FRA at settlement?
41
LOS 64.d / p. 154
Valuing an FRA: Example

First, you need to calculate the difference in
interest cost for the loan:
( 0.04 − 0.033 ) ×

60
× $1,000,000 = $1,167
360
Second, you must discount this difference
back to the present (at the current 4% rate):
$1,167
= $1,159
60 ⎞
⎛
1+ ⎜ 0.04 ×
360 ⎟⎠
⎝
42
Level 3
Risk Management
Applications of Forward
and Futures Strategies
43
LOS 49.a
Forward Rate Agreements

A borrower can lock in the interest rate on a
floating-rate loan by buying an FRA
If market rate > FRA rate, the buyer of FRA
receives cash from the seller that offsets the
increase in interest rates
If market rate < FRA rate, the buyer of FRA
pays cash to the seller of FRA that increases
loan interest rate to FRA rate
Continued →
44
LOS 49.a
Forward Rate Agreements cont.
FRA Settlement = S
FRA rate
Market rate
Days
in the
loan
period
(r − f ) ⎛⎜
T ⎞
⎟
360 ⎠
⎝
S = (notional principal) ×
⎧ ⎡
T ⎤⎫
⎨1 + ⎢(r)
⎥⎬
⎩ ⎣ 360 ⎦ ⎭
Continued →
45
LOS 49.a
Forward Rate Agreements: Example

In 2 months a borrower is going to borrow $10
million for 6 months at LIBOR

The FRA rate is 3.05% for 6 months

In 2 months LIBOR= 3.93%
Continued →
46
LOS 49.a
cont.
LIBOR
S = $10m ×
FRA rate
6 months
( 0.0393 − 0.0305 ) ⎛⎜
180 ⎞
⎟
⎝ 360 ⎠ = $43,152
⎧ ⎡
180 ⎤ ⎫
⎨1 + ⎢(0.0393)
⎬
360 ⎥⎦ ⎭
⎩ ⎣
Continued →
47
LOS 49.a
cont.

Since LIBOR > FRA rate, the bank pays the
borrower $43,152 at the inception of the loan
The borrower receives a total of $10,043,152 at
the beginning of the loan period and pays the
bank $10,000,000 plus $196,500 interest (at
3.93%) in six months
BEY on the loan is 3.05% (the same as the FRA
rate)
48
Practice
What You’ve Learned
49
SchweserProTM Question Bank

Thousands of questions at
each level
Browse questions by LOS
Insert personal study notes
and bookmarks
Print custom exams by level of
difficulty, topic, length and
weighting
Download new questions
throughout the season
50
Online Practice Exams

FREE
with UKSIP Review
Course registration
TODAY!

Unique online exams
one-time usage
unlimited review until exam date
Online or printable
Available late-season
51
Practice Exam Book

Three complete,
6-hour exams
Answers and
explanations for
all questions
Scoring breakdown
for all Level 3 essay
answers
52
Level 1
Futures Markets and
Contracts
53
LOS 75.a, p. 146-147
Forwards vs. Futures
Forwards
Private contracts
Unique contracts
Default Risk present
No margin
Little regulation
Futures
Exchange-traded
Standardized
Guaranteed by
clearinghouse
Margin required
Regulated

54
LOS 75.a, p. 146-147
Futures Characteristics
Contract specifies: quality and quantity of
good, delivery time, manner of delivery
Exchange specifies: minimum price
fluctuation (tick), daily price limit
Clearinghouse holds other side of each trade
Margin posted and marked to market daily
Margin is a performance guarantee, not a loan
Long buys and short sells the future
55
LOS 75.c, p. 148
A Futures Trade

July wheat futures call for delivery of 5000/bu. of
wheat in July, futures price is $2 per bushel
Contract value is 5000 × $2 = $10,000

Long obligated to buy 5000 bu. in July at $2

Short obligated to sell 5000 bu. in July at $2

Both the long and short post same margin amount

If future price > $2 long gains, < $2 short gains
56
LOS 75.e, p. 149
Closing a Futures Trade by Offset

Most futures contracts closed prior to the
expiration or delivery date
For example, the long position in July wheat
can be closed out by taking an equal short
position in July wheat
The futures price when the trade is closed out
determines gains or losses on the trade
57
LOS 71.b, p. 147-148
Futures Margins Terms
Initial margin: deposited before trade occurs
Maintenance margin: minimum margin that
must be maintained in a futures account
Variation margin: funds needed to restore
futures account to initial margin amount
Settlement price: average of trades during
closing period, used to calculate margin
58
LOS 75.d, p. 148
Marking to Market

Marking to market is the process of adjusting
margin balance in a futures account each day
for the change in the futures price (add gains,
subtract losses)
The futures exchanges can require a mark to
market more frequently (than daily) under
extraordinary circumstances (increased
volatility)
59
LOS 75.d, p. 149
Margin Calculation Example

Long five July wheat contracts
Size = 5000 bushels
Futures price = $2.00/bu
Initial margin deposit = $150 per contract
Maintenance margin = $100 per contract
Total Initial Margin = 5 × $150 = $750
Total Maintenance margin = 5 × $100 = $500
60
LOS 75.d, p. 149
Margin Calculation Example cont.

Each change of $0.01 in the futures price
leads to a change of 5000 × $0.01 = $50 per
contract in the margin account
On the 5 contracts in our example, a $0.01
increase in the July wheat futures price will
increase the long’s margin by $250 and
decrease the margin balance in the short’s
account by $250
61
LOS 75.d, p. 149
Margin Calculation Example cont.
62
LOS 75.e, p. 149-150
Methods to Terminate a Futures
Position at Expiration

Reversal (offsetting trade): common

Delivery of asset (< 1% of trades)

Cash-settlement: may be required

Exchange for physicals: off exchange
63
LOS 75.e,f, p. 149-150
Futures Contracts Delivery Options

T-bond futures
What to deliver: which bonds to deliver
When to deliver: date during expiration
month
Commodities: where to deliver; location
Delivery option is valuable to short
64
Level 2
Futures Markets and
Contracts
Foundation Concepts

Many of the theoretical pricing
relationships expressed for forward
contracts are identical for futures
contracts
No-arbitrage principle: There should be
no riskless profit from combining futures
contracts with other instruments
Futures price = price that would not permit
profitable riskless arbitrage
66
LOS 65.c,d,p. 172-173
Forwards vs. Futures Contracts

Futures are marked to market daily
Any value accrued due to underlying price
changes during the day is immediately
realized by the contracting parties
Like a series of 1-day forward contracts
Hence, the “value” of the contract is reset
to zero each day
No such thing as an “off-market” futures
contract
67
LOS 65.d, p. 173
Core Material and LOS
Pricing a Futures Contract: Quick
Review

Futures are priced just like forwards
(assuming interest rates and underlying
prices are uncorrelated)
Cost of carry model: FP = S0 × (1 + Rf)T
Look familiar? It’s identical to the pricing
relationship reviewed in the last topic
The same applies to equities, fixed incomes,
and currencies!
68
LOS 65.d, p. 173
Pricing a Futures Contract

If interest rates are positively correlated
with the price of the underlying, the futures
will be more valuable than the forward
If interest rates are negatively correlated,
the forward will be more valuable than the
futures
69
LOS 65.e, p. 175
Effect on Futures Price of Benefits and
Costs of Holding Asset

Costs of storing or holding the asset will
increase commodity futures price
Monetary benefits (e.g., dividends) reduce
financial futures price
Non-monetary benefits from holding asset will
reduce futures price (e.g., holding an asset in
short supply with seasonal/highly risky
production process)
Return is called “convenience yield”
70
Forwards and Futures
Level 3
Risk Management
Applications of Forward
and Futures Strategies
71
LOS 49.e
Using Equity Futures
to Obtain a Target Beta

The manager can also attain a target beta to
make the equity portfolio more or less volatile:
⎛ β − βP ⎞ ⎛
⎞
VP
# contracts = ⎜ T
⎟⎜
⎟
⎝ β F ⎠ ⎝ Pf (multiplier) ⎠
Continued →
72
LOS 49.e
to Obtain a Target Beta cont.
Desired change in beta
⎛ β − βP ⎞ ⎛
⎞
VP
# contracts = ⎜ T
⎟⎜
⎟
⎝ β F ⎠ ⎝ Pf (multiplier) ⎠
“Amount” of beta provided
by one futures contract
73
LOS 49.e
to Obtain a Target Beta: Example

A $60 million equity portfolio has a beta of 1.2,
and futures are trading at $2,674 (having a
multiplier of $125 and a beta of 0.96). How
many contracts are needed to reduce the beta
to 0.8?
⎛ 0.8 − 1.2 ⎞ ⎡ $60,000,000 ⎤
# contracts = ⎜
⎟⎢
⎥ = −74.8
⎝ 0.96 ⎠ ⎣ 2,674($125) ⎦
⇒ Sell 75 contracts
74
Retain
What You’ve Learned
75
Flashcards

Color coded by topic
Indexed by study
session, reading, and
LOS
Auto-indexing available
in web-enabled version
76
Audio CDs

Condensed readings of Study Notes
Full LOS coverage
Choice of audio CDs or MP3 CDs
77
Schweser’s Secret SauceTM

Small, convenient text for
summary of critical concepts
Time management hints
Analysis of level-specific
question styles
Essential Exam Strategies
78
Final Review Pack

This late-season package includes:
Secret SauceTM eBook
Online Practice Exams
Improve time management
Track performance by topic
Compare scores against other candidates
Online Workshops
Three level-specific workshops in late May
Highlights of topic areas in each session
Exam strategies and key concepts
79
TYING IT ALL TOGETHER
Learn • Practice • RetainTM
80
Learn • Practice • RetainTM

Learn the curriculum with Study Notes and
online tools
Practice new skills with SchweserProTM and
Practice Exams
Retain information with Flashcards, Audio
CDs, and Secret SauceTM
Attend a seminar for reinforcement,
clarification and identification of “trouble” spots
81
Premium or
Essential Solution
+
UKSIP
Review Course
=
The Ultimate CFA® Review Experience!
SAVE 100USD
when you order both!
82
Level 1
Options Markets and
Contracts
83
Options Basics

Option buyer (owner, long position)
Pays a premium to purchase the right to
exercise an option at a future date and price
Option seller (writer, short position)
Incurs an obligation to perform under the
option contract terms
84
Options Basics

Call option: long has the right to purchase the
underlying asset at the exercise (strike) price;
short has the obligation to sell/deliver the
underlying asset at the exercise price
Put option: long has the right to sell the
underlying asset at the exercise (strike) price;
short has the obligation to purchase the
underlying asset at the exercise price
85
LOS 76.a, p. 157
Options Terminology

American options can be exercised any time
prior to expiration (early exercise) or at
expiration
European options can be exercised only at
expiration
American options are worth at least as much
as otherwise identical European options
86
LOS 76.a, p. 157
Moneyness
Call Options
Put Options
87
LOS 76.a, p. 158-160
Option Value
Option value = intrinsic value + time value

Intrinsic value also equal to payoff at expiration
Call: max (0, S – X)
Put: max (0, X – S)
Time value
Option premium minus intrinsic value
Also called speculative value
88
LOS 76.a, p. 158-160
Call Option Example

Stock price is $47
Call with exercise price of 50 is trading at $1.50
Out of the money: intrinsic value is zero
Time value is $1.50 – 0 = $1.50
Call with exercise price of $45 is trading at $3.00
In-the-money: intrinsic value is $47– $45 = $2
Time value is $3.00 - $2.00 = $1.00
89
LOS 76.a, p. 158-160
Put Option Example
Stock price is $47
Put with exercise price of 45 is trading at $1.50
Out-of-the-money: intrinsic value is zero
Time value is $1.50 – 0 = $1.50
Put with exercise price of 50 is trading at $4.00
In-the-money: intrinsic value is $50 – $47 =
$3
Time value is $4 – $3 = $1.00
90
LOS 76.b, p. 161
Types of Options: Underlying Assets

Commodity options: e.g., call option on 100 oz.
of gold at $420 per ounce
Stock options: each contract for 100 shares
Bond options: like stock options, payoff based
on bond price and exercise price
Index options: have a multiplier, e.g. payoff is
$250 in cash for every index point the option is
in the money at expiration
91
LOS 76.b,c, p. 161
Types of Options: Underlying Assets
Interest rate options: the payoff is based on the
difference between a floating rate, such as LIBOR,
and the strike rate
Options on futures:
Calls give the option to enter into a futures
contract as the long at a specific futures price
Puts give the option to enter into a futures
contract as the short at the indicated futures
price
92
LOS 76.c,d, p. 162-163
More on Interest Rate Options
Payoff on a LIBOR-based interest rate call is
Max[0,(LIBOR – strike rate)] × notional amount
(long gains when rates rise)
Payoff on a LIBOR-based interest rate put is
Max[0,(strike rate – LIBOR)] × notional amount
(long gains when rates fall)

Payoffs made at the end of the interest
rate term (after option expiration)
93
LOS 76. d, p. 162-163
Interest Rate Option Example

Long 60-day call on 90-day LIBOR
Strike rate = 5%
90-day LIBOR at expiration (in 60 days) = 6%
Payoff (to long):
$1,000,000 × (0.06 – 0.05) × (90/360) = $2,500
Paid by call writer 90-days after expiration
94
LOS 76.c, p. 162
Two Interest Rate Options = One FRA
2%
3%
7%
= 5%
-2%
95
LOS 76.e, p. 163
Interest Rate Caps and Floors
A cap puts a maximum on an issuer’s
payments on a floating rate debt; equivalent to
a series of long interest rate calls; makes
payments to issuer when floating rate > cap
(strike) rate
A floor puts a minimum on an issuer’s
payments on a floating rate debt; equivalent to
a series of short interest rate puts; issuer must
make payments when floating rate < floor
(strike) rate
96
LOS 76.e, p. 163
Cap and Floor Payoffs
97
LOS 76.f, p. 164
Options Notation
St = price of underlying asset at time t
X = exercise price
T = time to expiration
RFR = risk-free rate
ct, pt = European style calls and puts at time = t
Ct, Pt = American style calls and puts at time = t
98
LOS 76.f, p. 165-167
Minimum and Maximum Option Prices
European puts, no early exercise means lower
maximum, PV of X at expiration
99
LOS 76.i, p. 169-170
Deriving Put-Call Parity (European Options)
Protective put = stock + put
If S ≤ X, payoff = S + (X – S) = X
If S ≥ X, payoff = S + 0 = S
Fiduciary call = call + X/(1 +
X
0
RFR)T
X
(bond that pays X at
maturity)
If S ≤ X, payoff = 0 + X = X
If S ≥ X, payoff = (S – X) + X = S
Same payoffs means same values by no-arbitrage
Put-call parity: S + P = C + X/(1 + RFR)T
100
LOS 76.i, p. 170
Parity Conditions
101
LOS 76.i, p. 170
Put-Call Parity Example
Stock XYZ trades at $75
Call premium = $4.50
Expiration = 4 months (T=0.3333)
X = $75
RFR = 5%
What’s the price of the 4-month put on XYZ?
102
LOS 76.i, p. 170
Put-Call Parity Example
103
LOS 76.h,l,p. 169, 171-172
Time, Volatility, RFR, and Strike
Price
Longer time to expiration increases option values
Except for: some far out-of-the-money options and European
style puts
Greater price volatility increases option values
Increase in RFR increases call values and
decreases put values
For X1 < X2:
call at X1 ≥ call at X2
put at X1 ≤ put at X2
104
Level 1
Risk Management
Applications of Option
Strategies
105
LOS 78.a, p. 190-191
Call Intrinsic Value / Payoff at Expiration
106
LOS 78.a, p. 191
Profit and Loss: Call Options
107
LOS 78.a, p. 191-192
Puts Intrinsic Value/ Payoff at Expiration
108
LOS 78.a, p. 192
Profit and Loss: Put Options
109
LOS 78.a, p. 192-193
Example: Call Profit and Loss
Consider a 40 call purchased at $3.00 when the
stock is trading at $39.00
Maximum profit: unlimited
Maximum loss: –$3.00
Breakeven: stock price at expiration = $43.00
If stock price is $42.00 at expiration
Value at expiration is $2.00 but paid $3.00 so,
loss is $1.00 (would be gain to call writer)
110
LOS 78.a, p. 192-193
Example: Put Profit and Loss
Consider a 40 put written/sold at $3.00 when the
stock is trading at $39.00
Maximum profit: $3.00
Maximum loss: –$37.00
Breakeven stock price at expiration: –$37.00
If stock price is $39.00 at expiration
Payoff at expiration is $1.00 but sold for $3.00 so
profit is $2.00 (would be loss to put buyer)
111
LOS 78.b, p. 193-194
Covered Call Strategy (Position)

Writer owns the stock and sells a call
Any loss will be reduced by premium
received
Writer trades the stock’s upside potential
for the option premium
112
LOS 78.b, p. 193-194
Example: Covered Call
Buy stock at $39
Sell a 40 Call at $3
Net cost of covered call position: 39 – 3 = $36
Breakeven stock price at expiration: $36
Maximum gain: 40 – 36 = $4
Maximum loss: $36 (if stock goes to zero)
At stock price of $39 → profit = $3
At stock price of $32 → loss = –$4
113
LOS 78.b, p. 194
Payoff and Profits: Covered Call
114
LOS 78.b, 194-195
Protective Put Strategy (Position)

Put buyer owns the stock and buys a put

Any gain will be reduced by premium paid

Put buyer pays for protection against any
stock price below the strike price of the put
115
LOS 78.b, p. 194-195
Example: Protective Put
Buy stock at $41
Buy a 40 Put for $3
Cost of Protective Put strategy = 41+ 3 = $44
Breakeven: stock price at expiration $44
Maximum gain: unlimited
Maximum loss: $4 (if stock price ≤ 40)
At stock price of 47 → profit = $3
At stock price of 38 → loss = $4
116
LOS 78.b, p. 195
Payoff and Profit: Protective Put
117
Level 2
Option Markets and
Contracts
LOS 66.d, p. 204
The Black-Scholes-Merton Model

As the # of time periods increases (interval
length decreases), the binomial model
“converges” to a continuous time model called
Black-Scholes-Merton (BSM):
c
C 0 = ⎡⎣ S 0 × N ( d1 ) ⎤⎦ – ⎡ X × e –R f ×T × N ( d 2 ) ⎤
⎣⎢
⎦⎥
ln ⎛⎜
d1 = ⎝
S0
(
(
)
⎞ + ⎡R c + 0.5 × σ 2 ⎤ × T
X ⎟⎠ ⎣ f
⎦
d 2 = d1 – σ ×
σ×
T
)
T
119
LOS 66.d, p. 204
The BSM Model and “Greek Risk”

The BSM formula has five inputs:
S, the asset price
σ, volatility
Rf, the interest rate
T, time to expiration
X, the exercise price
Changing an input will change the value of
the option (call or put)
Each sensitivity (except X) is a “Greek”
120
LOS 66.d, p. 204
The BSM Model and “Greek Risk”
121
LOS 66.e, p. 205
Delta and Dynamic Hedging

Delta (1st derivative of BSM relative to S)
is the change in the price of an option for
a one-unit change in the price of the
underlying stock
C – C0 ΔC
Discrete time : Deltacall = 1
=
S1 – S0 ΔS
Continuous time : Deltacall = N ( d1 )
122
LOS 66.e, p. 205
Delta Summary

Call deltas range from 0 to 1:
Far out-of-the-money: delta approaches 0
Far in-the-money: delta approaches 1
Put deltas range from –1 to 0:
Far out-of-the-money: delta approaches 0
Far in-the-money: delta approaches –1
Put delta = Call delta –1
123
LOS 66.e, p. 205
A Graphical Depiction of Delta
124
LOS 66.e, p. 205
Delta and Dynamic Hedging

For small changes in the stock price (∆S):
ΔC ≈ N( d1) ×ΔS
Delta of the comparable put option is call
delta minus one, so for small changes in S:
ΔP ≈ ⎡⎣1− N( d1) ⎤⎦ ×ΔS
125
Level 3
Options and Swaps
Strategies and
Applications
LOS 50.f
Delta Hedging

If a dealer sells a naked call, the most commonly
available hedge is to buy the stock
Delta tells us how the call and stock returns are
related and how much stock to buy:
ΔCALL = ΔC / ΔS
0 ≤ ΔCALL ≤ 1
127
LOS 50.f
Delta Hedging Example

Delta = 0.5 and the dealer has sold calls on 100
shares

Dealer should buy 50 shares of stock to hedge
changes in the value of the calls
But delta will change as time passes and as the
stock price changes

To maintain the hedge, the number of shares
must be rebalanced on a periodic basis
Continued →
128
LOS 50.f
Delta Hedging Example cont.

Over time, even with no change in stock price, the
call price and delta will decrease. The manager
will sell stock and earn the risk-free rate on the
proceeds.
If the stock price falls by a small amount, the delta
will fall. The manager will sell stock (e.g., the delta
changes from 0.50 to 0.42, the dealer will sell eight
shares [100 × (0.50 – 0.42)].
Gamma is to delta as convexity is to duration.
129
→
LOS 50.f
Delta Hedging Example cont.

If the stock price increases by a small amount, the
delta will increase. The manager will buy stock by
borrowing {e.g., the delta changes from 0.50 to
0.55, the dealer will buy 5 shares [100 × (0.55 –
0.50)]}.
130
LOS 50.g
The Gamma Effect

Delta is only an approximation because the
relationship between call and stock prices is
nonlinear

Delta will underestimate the increase in the call
price given a stock price increase
Delta will overestimate the decrease in the call
price given a stock price decrease
The 2nd order effect, gamma, increases the
precision (rate of change in delta)
Continued →
131
LOS 50.g
The Gamma Effect cont.
Gamma =

Δ delta
ΔS
Gamma becomes more important when:

The option is at the money, and/or

The option is near expiration
132
Level 1
Swaps Markets and
Contracts
133
LOS 77.b, p. 180-183
Swap Contracts: Overview
If A loans money to B for a fixed rate of interest
and B loans the same amount to A for floating
rate of interest, it’s an interest rate swap
If one of the returns streams is based on a
stock portfolio or index return, it’s an equity
swap
If the loans are in two different currencies, it’s a
currency swap
134
LOS 77.a, p. 179
Characteristics of Swap Contracts

Custom instruments
Not traded in any organized secondary market
Largely unregulated
Default risk is a concern
Most participants are large institutions
Private agreements
Difficult to alter or terminate
135
LOS 77.a, p. 179
Swap Contract Terminology

Notional principal: amount used to calculate
periodic payments

Floating rate: usually US LIBOR

Tenor: time period covered by swap

Settlement dates: payment due dates
136
LOS 77.b, p. 180
Currency Swap

Assume current exchange rate is $1.20/euro
Company A lends $1.2 million to Company B
at 5%/yr
Company B lends 1 million euros to Company
A at 4%/yr
Loans are for two years and interest is paid
semiannually
137
LOS 77.b, p. 181
Currency Swap Cash Flows
At time 0:
Company A lends $1.2 million to Company B
Company B lends € 1 million to Company A
At each semi-annual settlement date (t = 1,2,3,4):
A pays 0.04/2 × €1 million = € 20,000 to B
B pays 0.05/2 × $1.2 million = $30,000 to A
138
LOS 77.b, p. 181
Currency Swap Cash Flows
At t = 4 settlement date, in addition to the interest
payments:
Company B repays $1.2 million to Company A
Company A repays €1 million to Company B
This is a fixed-for-fixed currency swap because
both loans carried a fixed rate of interest
Could be fixed-for-floating or floating-for-floating
139
LOS 77.b, p. 182
Plain Vanilla Interest Rate Swap

Fixed interest-rate payments are exchanged
for floating-rate payments
Notional Amount is not exchanged at the
beginning or end of the swap; both loans are
in same currency and amount
140
LOS 77.b, p. 182
Interest payments are netted
On settlement dates, both interest payments
are calculated and only the difference is paid
by the party owing the greater amount
Floating rate payments are typically made in
arrears, payment is made at end of period
based on beginning-of-period LIBOR
141
LOS 77.b, p. 182
FR = fixed rate
T = # days in settlement period
NP = notional principal
142
LOS 77.b, p. 182-183
Fixed-for-Floating Swap Example
Example: 2-year, semiannual-pay, LIBOR, plain
vanilla interest rate swap for $10 million with a fixed
rate of 6%
Semiannual fixed payments are:
(.06/2) × $10 million = $300,000
LIBOR t0 = 5%, t1= 5.8%, t2= 6.2%, t3= 6.6%
1st payment: Fixed-rate payer pays $50,000 net
(.06 - .05 )(180/360)(10 million) = $50,000
First net payment is known at swap initiation!
143
LOS 77.b, p. 182-183
LIBOR 5% at t0, 5.8% at t1, 6.2% at t2, 6.6% at t3
***********************************************************
2nd payment: Fixed-rate payer pays $10,000 net
(.06 – .058)(180/360)(10 million) = $10,000
3rd payment: Floating rate payer pays $10,000 net
(.06 – .062)(180/360)(10 million) = –$10,000
4th payment: Floating rate payer pays $30,000 net
(.06 – .066)(180/360)(10 million) = –$30,000
144
LOS 77.b, p. 183-184
Equity Swaps
Payments based on equity returns are exchanged
for fixed rate or floating rate payments
Equity Return based on
Individual stock or
Stock portfolio or
Stock index
Can be capital appreciation
or total return including dividends
145
LOS 77.b, p. 183-184
Equity Swaps
Equity return payer:
Receives interest payment
Pays any positive equity return
Receives any negative equity return
Interest payer:
Pays interest payment
Receives positive equity return
Pays any negative equity return
146
LOS 77.b, p. 183-184
Equity Swap Example
2-year $10 million quarterly-pay equity swap
Equity return = S&P 500 Index
Fixed rate = 8 percent
Current index level = 986

Q1, S&P 500 = 1030
Q2, S&P 500 = 968
Q3, S&P 500 = 989
Return = 4.46%
Return = –6.02%
Return = 2.17%
Holder of index portfolio + swap gets 2% per
quarter (plus dividends) for any index value!!
147
LOS 77.b, p. 184
Equity Swap Example cont.
Index return payer pays (+) receives (–):
Q1: 4.46% – 2.00% = 2.46%
$246,000 net payment
Q2: –6.02% – 2.00% = –8.02%
–$802,000 net payment
Q3: 2.17% – 2.00% = 0.17%
$17,000 net payment
148
Level 2
Swap Markets and
Contracts
LOS 67.a, p. 221
Swaps: Pricing vs. Valuation

“Pricing” swaps requires the determination
of the swap fixed rate (or swap rate)
The swap fixed rate is the rate paid by the
pay-fixed side. The swap rate is set so that:
PVfixed-payments = PVfloating-payments
Valuing a swap requires finding the
difference in the value of the fixed payments
and floating payments after initiation
150
LOS 67.c, p. 223
Swaps as Combinations of Other
Securities
Swaps can be mimicked or replicated by
combinations of other capital market
instruments such as bonds, forward
contracts, options, and FRAs
“Replicated” means that two positions
generate the same pattern of cash flows

151
LOS 67.c, p. 223
Replicating Swaps With Bonds

Fixed-rate payer side (a payer swap)
could be replicated by:
1) Issuing fixed rate bonds (match
maturity and payment dates)
2) Using proceeds to purchase floating
rate notes at LIBOR
This is a key insight that helps us price
and value swaps
152
LOS 67.c, p. 223
The Swap Fixed Rate

Guiding principle: swap fixed rate must be
set so swap value at initiation (for the payer
and receiver) is zero
Method: value the swap as a combination of
fixed-rate bond and floating rate bond
Value of payer swap = value of
“replicating” floating rate bond – value of
“replicating” fixed rate bond
153
LOS 67.c, p. 223
The Swap Fixed Rate
Important points:
1) Bonds have principal payments;
swaps do not. However, we replicate
swaps with bonds and therefore
include the bond’s principal payments
in the valuation procedure.
2) On each settlement date, the value of
a floating rate note (FRN) will always
reset to par (interest rates “adjust” to
market rates).
154
LOS 67.c, p. 223
The Swap Fixed Rate

Using some magic hand waving (and a
little elementary algebra), we get the
formula for the swap rate:
C=
1 − Z4
Z1 + Z2 + Z3 + Z 4
Price of n-period $1 zero coupon bond
= n-period discount factor
155
LOS 67.c, p. 223
The Swap Fixed Rate: Example

Calculate the swap rate and the fixed
payment on a 1-year, quarterly settlement
swap with a notional principal of $10MM
156
LOS 67.c, p. 223

Applying the formula for C, we get the
quarterly swap rate:
1 − 0.94340
0.98888 + 0.97561 + 0.96038 + 0.94340
= 0.0146 = 1.46%
157
LOS 67.c, p. 223

Quarterly fixed-rate payment is:
$10M × 0.0146 = $146,000
Annualizing the quarterly rate gives us the
annual swap rate of 5.84% (= 1.46% × 4)
158
Level 3
Options and Swaps
Strategies and
Applications
LOS 51.a
Interest Rate Swaps

Can be used to convert floating-rate debt to
fixed-rate debt (e.g., a firm has floating rate debt
outstanding at LIBOR + 2%)
They enter into a swap where they receive
LIBOR flat and pay 6% fixed
Paying
LIBOR + 2
–LIBOR
6 fixed
= 8 fixed
In net, they will pay 8% fixed
handy solution
160
LOS 51.a
Interest Rate Swaps cont.
0
200

→
Qtn
24
Can be also used to convert fixed-rate debt into
floating-rate debt (e.g., a firm has fixed-rate debt
at 5%)
They enter into a swap where they receive 4%
fixed and pay LIBOR +1%
In net, they will pay LIBOR + 2%
handy solution
Paying
5 fixed
– 4 fixed
LIBOR + 1
= LIBOR + 2
161
LOS 51.b
Duration of an Interest Rate Swap
D pay floating = D fixed – D floating > 0
75% of
Duration
→ maturity
Guidelines
50% of
payment
interval
For example, if the duration of the fixed payments is
2.3 and the swap is reset semiannually, the
duration of the swap is 2.3 – 0.25 = 2.05 at its
inception.
Continued →
162
LOS 51.b
Duration of an Interest Rate Swap cont.
D receive floating = D floating – D fixed < 0

If a floating-rate borrower swaps for a fixed rate
by receiving floating in a swap, the duration
changes from ≈ 0 to negative
Thus, the risk of payment variability is swapped
for interest rate risk
Continued →
163
LOS 51.b

To help you remember the sign on the
duration of a swap, think of the swap as a
portfolio, and remember that the duration of a
portfolio is the duration of the assets minus
duration of the liabilities:
DP = DA − DL
Continued →
164
LOS 51.b

Taking the pay-fixed arm of a swap is like
having a fixed-rate liability and a variable-rate
asset:
If pay fixed, DL > DA so
DP = DA − DL < 0
Continued →
165
LOS 51.b

Taking the receive-fixed arm of a swap is like
having a fixed-rate asset and a variable-rate
liability:
If receive fixed, DA > DL so
DP = DA − DL > 0
166
LOS 51.d
Changing Duration Using an
Interest Rate Swap
Value of the position
⎛ MD target – MD V
NP = VP ⎜
⎜
MD swap
⎝
⎞
⎟⎟
⎠
We can determine the required
notional principal
Continued →
167
LOS 51.d
Interest Rate Swap cont.

A manager has a $100m bond portfolio with a
MD = 5.0 and would like to decrease it to 3.8.
A swap with a net MD of 2.7 is available
Calculate the required notional principal for the
swap
To reduce duration, we know the manager will
have to take the pay-fixed arm of the swap.
168
→
LOS 51.d
Interest Rate Swap cont.
⎛ 3.8 − 5.0 ⎞
NP = $100 ⎜
⎟ = $44.44 million
⎝ −2.7 ⎠

The sign of the swap’s duration depends upon
whether you pay fixed or floating
Negative in this case because we know the
manager must be pay fixed to reduce his
portfolio duration
169