CFA 2018 level 3 schweser practice exam CFA 2018 level 3 question bank CFA 2018 r29 risk management applications of options strategies summary

fair option prices.

“Boxing-in” your profit!

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Strategy

Comment

Covered Call

Profit Calculation

Underlying+ short call
Lowers risk by providing a cushion
Protective Put Underling + long put
Puts a floor on losses
Bull Spread
Long call (X1) + short call (X2)
Make money if underlying goes up
Bear Spread
Long put (X2) + short put (X1)
Make money if underlying goes down
Butterfly
Combine bull and bear, using three exercise prices
Make money if underlying is stable

VT – S0 + c0

Collar

Pay for put by selling call
Limits down side risk… but also cap upside

VT – S0

Straddle

Buy put and call with same X and same expiration
Make money if underlying is volatile

VT – (c0 + p0)

Box

Combine bull and bear, using two exercise prices
Exploit arbitrage opportunity if both BSM and Binomial do
not hold

X2 – X1 – (c1 – c2 + p2 – p1)

VT – S0 – p0
VT – c1 + c2
VT – p2 + p1
VT – c1 + 2c2 – c3

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Interest Rate Options
Underlying is an interest rate and the exercise price is expressed in terms of a rate.
A call option will make money if the option expires with the underlying interest rate above the
exercise rate. Interest rate call option payoff:
NP)max(0, Rate at expiration − Exercise rate)((Days in underlying rate ) 360

A put option will make money if the option expires with the underlying interest rate below the exercise rate.
Interest rate put option payoff:
NP)max(0, Exercise rate − Rate at expiration)((Days in underlying rate ) 360

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Effective Interest Rate Using Call Options
Loan amount
Underlying
Spread
Current Libor
Expiration
Exercise rate
Call premium
Today’s date

$40,000,000
180-day Libor
200 basis points over Libor
5.5 percent
20 August (128 days later)
5 percent
$100,000

14 April

For Libor of 8 percent, the payoff is
$40,000,000𝑚𝑎𝑥(0,0.08 − 0.05)

180
= $600,000
360

The compounded premium is:
$100,000 1 + (0.055 + 0.02)

128
360

= $102,667

The effective loan proceeds are $40,000,000 – $102,667 = $39,897,333
The loan interest is: $40,000,000(0.08 + 0.02)(180/360) = $2,000,000
The effective interest rate is:

$40,000,000 + $2,000,000 − $600,000
$39,897,333

365 180

− 1 = 0.0779

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Effective Interest Rate Using Put Options
Loan amount
Underlying
Spread
Current Libor
Expiration
Exercise rate
Put premium

$50,000,000
90-day Libor
250 basis points over Libor
7.25 percent
1 May (in 47 days)
7 percent
$62,500

For Libor of 6 percent, the payoff is:
$50,000,000max (0, 0.07−0.060) (90/360) =$125,000

The compounded premium is:
$62,500[1+ (0.0725+0.025) (47/360)] = $63,296

Put costs $62,500 on 15 March, which is equivalent to $63,296 on 1 May. The effective amount loaned is
$50,000,000 + $63,296 = $50,063,296.
With Libor at 6 percent, the interest is: $50,000,000[(0.06+0.025)(90/360)] = $1,062,500
The loan interest plus the put payoff is the effective interest on the loan. The effective rate on the loan is:
$50,000,000 + $1,062,500 + $125,000
$50,063,296

365 90

− 1 = 0.0942

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Interest Rate Caps
A floating rate loan requires periodic interest
payments in which the rate is reset on a regularly
scheduled basis. A cap is a combination of interest
rate call options designed to align with rates on a
loan. Each component is called a caplet. It
provides protection against rising interest rates
over the life of the loan.

Date
15 April
15 October
15 April
15 October
15 April
15 October
15 April

Libor
0.0900
0.0850
0.0725

0.0700
0.0690
0.0875

Loan amount
Underlying
Spread
Current Libor
Interest based on
Component caplets
Exercise rate
Cap premium

$10,000,000
180-day Libor
100 basis points over Libor
9 percent
actual days/360
five caplets expiring 15 October, 15 April, …
8 percent
$75,000

Loan Rate
0.1000
0.0950
0.0825
0.0800
0.0790
0.0975

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Interest Rate Caps
A floating rate loan requires periodic interest
payments in which the rate is reset on regularly
scheduled basis. A cap is a combination of interest
rate call options designed to align with rates on a
loan. Each component is called a caplet. It
provides protection against rising interest rates
over the life of the loan.

Date
15 April
15 October
15 April
15 October
15 April
15 October
15 April

Libor
0.0900
0.0850
0.0725
0.0700
0.0690
0.0875

Loan Rate

0.1000
0.0950
0.0825
0.0800
0.0790
0.0975

Loan amount
Underlying
Spread
Current Libor
Interest based on
Component caplets
Exercise rate
Cap premium

$10,000,000
180-day Libor
100 basis points over Libor
9 percent
actual days/360
five caplets expiring 15 October, 15 April, …
8 percent
$75,000

Days in Period Interest Due Caplet Payoffs

Effective Interest

183

182
183
182
183
182

$508,333
455,000
419,375
404,444
401,583
455,000

$508,333
480,278
419,375
404,444
401,583
492,917

$25,278
0
0
0
37,917

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Interest Rate Floor

An interest rate floor is a series of interest
rate put options that expire on various
interest rate reset dates. Each component is
called a floorlet. It provides protection to
the lender against falling interest rates.

Date
15 April
15 October
15 April
15 October
15 April
15 October
15 April

Libor
0.0900
0.0850
0.0725
0.0700
0.0690
0.0875

Loan Rate
0.1000
0.0950
0.0825
0.0800
0.0790
0.0975

Loan amount
Underlying
Spread
Current Libor
Interest based on
Component floorlets
Exercise rate
Floor premium

$10,000,000
180-day Libor
100 basis points over Libor
9 percent
actual days/360
five floorlets expiring 15 October, 15 April, etc.
8 percent
$72,500

Days in Period

Interest Due

183
182
183
182
183
182

$508,333
480,278
419,375
404,444
401,583
492,917

Floorlet Payoffs

Effective Interest

$0
38,125
50,556
55,917
0

$508,333
480,278
457,500
455,000
457,500
492,917
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Interest Rate Collar
A collar combines a long position in a cap with a short
position in a floor. The sale of a floor provides a
premium that can be used to offset the purchase of a

cap. In this strategy, the borrower pays for the cap by
giving away some of the gains from the possibility of
falling interest rates. A collar establishes a range. Any
rate increases above the cap exercise rate will have no
net effect, and any rate decreases below the floor
exercise rate will have no net effect.
Date

Libor

Loan Rate

15 April

0.0900

0.1000

15 October

0.0850

15 April

Loan amount
Underlying
Spread
Current Libor
Interest based on
Component options

Exercise rate
Premium

$10,000,000
180-day Libor
100 basis points over Libor
9 percent
actual days/360
five caplets and floorlets expiring
15 October, 15 April, etc.
8.625 percent on cap, 7.5 percent on floor
no net premium

Days in Period

Interest Due

0.0950

183

$508,333

0.0725

0.0825

182

480,278

$0

$0

480,278

15 October

0.0700

0.0800

183

419,375

0

–12,708

432,083

15 April

0.0690

0.0790

182

404,444

0

–25,278

429,722

15 October

0.0875

0.0975

183

401,583

0

–30,500

432,083

182

492,917

6,319

0

486,598

15 April

Caplet Payoffs

Floorlet Payoffs

Effective Interest
$508,333

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Strategy

Comment

Interest Rate Calls with Borrowing

Used by a borrower. Establishes a maximum rate
for a loan to be taken out in future.

Interest Rate Puts with Lending

Used by a lender. Establishes a minimum rate for a
loan to be given out in future.

Interest Rate Cap with Floating Rate Loan

Used by a borrower. It provides protection against
rising interest rates over the life of the loan.

Interest Rate Floor with Floating Rate Loan

Used by a lender. It provides protection against
falling interest rates.

Interest Rate Collar with Floating Rate Loan

Used by a borrower. It establishes a range, if
interest rates move beyond this range it will have
no net effect on the borrower.

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Delta Hedging
Delta =

Change in option price
Change in underlying price

To delta hedge a call option position, the number of shares to purchase can be found using the
following equation:
𝑁𝑐
1

=−
𝑁𝑆
∆𝑐 ∆𝑆

Delta changes with price. Hence we need to change position when underlying price changes.
Delta also changes with the passage of time. Hence we need to adjust the delta hedge as expiration
approaches.

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Gamma
Gamma measures sensitivity of delta to change in underlying.
Gamma =

Change in delta
Change in underlying price

High gamma means that the delta changes a lot for a given change in underlying. This creates a problem
for delta hedgers.
Gamma is relatively high for at the money options.
Gamma is highest near expiration for at the money options.

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