CFA 2018 level 3 schweser practice exam CFA 2018 level 3 question bank CFA 2018 CFA 2018 r29 risk management applications of option strategies IFT notes

Risk Management Applications of Option Strategies

IFT Notes

Risk Management Applications of Option Strategies
1. Introduction .............................................................................................................................................. 3
2. Option Strategies for Equity Portfolios ..................................................................................................... 3
2.1. Standard Long and Short Positions .................................................................................................... 3
2.2. Risk Management Strategies with Options and the Underlying ........................................................ 6
2.3. Money Spreads .................................................................................................................................. 8
2.4. Combinations of Calls and Puts........................................................................................................ 11
3. Interest Rate Option Strategies .............................................................................................................. 14
3.1. Using Interest Rate Calls with Borrowing ........................................................................................ 14
3.2. Using Interest Rate Puts with Lending ............................................................................................. 17
3.3. Using an Interest Rate Cap with a Floating-Rate Loan..................................................................... 20
3.4. Using an Interest Rate Floor with a Floating-Rate Loan .................................................................. 21
3.5. Using an Interest Rate Collar with a Floating-Rate Loan ................................................................. 23
4. Option Portfolio Risk Management Strategies ....................................................................................... 25
4.1. Delta Hedging an Option over Time................................................................................................. 26
4.2. Gamma and the Risk of Delta .......................................................................................................... 28
4.3. Vega and Volatility Risk .................................................................................................................... 28
5. Final Comments ...................................................................................................................................... 28
Summary ..................................................................................................................................................... 29
Examples from the Curriculum ................................................................................................................... 34
Example 1 ................................................................................................................................................ 34
Example 2 ................................................................................................................................................ 35
Example 3 ................................................................................................................................................ 37
Example 4 ................................................................................................................................................ 37
Example 5 ................................................................................................................................................ 38
Example 6 ................................................................................................................................................ 39
Example 7 ................................................................................................................................................ 40

Example 8 ................................................................................................................................................ 42
Example 9 ................................................................................................................................................ 43
Example 10 .............................................................................................................................................. 43
Example 11 .............................................................................................................................................. 44
Example 12 .............................................................................................................................................. 45

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Risk Management Applications of Option Strategies

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Example 13 .............................................................................................................................................. 47
Example 14 .............................................................................................................................................. 48
Example 15 .............................................................................................................................................. 48
Example 16 .............................................................................................................................................. 49
This document should be read in conjunction with the corresponding reading in the 2018 Level III CFA®
Program curriculum. Some of the graphs, charts, tables, examples, and figures are copyright
2017, CFA Institute. Reproduced and republished with permission from CFA Institute. All rights reserved.
Required disclaimer: CFA Institute does not endorse, promote, or warrant the accuracy or quality of the
products or services offered by IFT. CFA Institute, CFA®, and Chartered Financial Analyst® are
trademarks owned by CFA Institute.

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1. Introduction
Some basic notations used in options are:
Time 0: It is the time at which the strategy is initiated.
Time T: It is the time the option expires.
c0, cT = price of the call option at time 0 and time T
p0, pT = price of the put option at time 0 and time T
X = exercise price
S0, ST = price of the underlying at time 0 and time T
V0, VT = value of the position at time 0 and time T
Π = profit from the transaction: VT – V0
r = risk-free rate

2. Option Strategies for Equity Portfolios
2.1. Standard Long and Short Positions
Note: Section 2.1 is optional and is a review of concepts learned at earlier levels.
Buy Call (Long Call)
Exhibit 2 shows the profit diagram for a buyer of a call option.

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The important points are:


cT = max(0, ST – X)



Value at expiration = cT



Profit: Π = cT – c0



Maximum profit = ∞



Maximum loss = c0



Breakeven: ST* = X + c0

Sell Call (Short Call)
Exhibit 3 shows the profit diagram for a seller of a call option. Note that it is the mirror image of the long
call.

The important points are:


cT = max(0, ST – X)



Value at expiration = –cT



Profit: Π = –cT + c0



Maximum profit = c0



Maximum loss = ∞



Breakeven: ST* = X + c0

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Refer to Example 1 from the curriculum.
Buy Put
Exhibit 4 shows the profit diagram for a buyer of a put option.

The important points are:


pT = max(0, X – ST)



Value at expiration = pT



Profit: Π = pT – p0



Maximum profit = X – p0



Maximum loss = p0



Breakeven: ST* = X – p0

Sell Put
Exhibit 5 shows the profit diagram for a seller of a put option.

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The important points to note are:


pT = max(0, X – ST)



Value at expiration = –pT



Profit: Π = –pT + p0



Maximum profit = p0



Maximum loss = X – p0



Breakeven: ST* = X – p0

Refer to Example 2 from the curriculum.

2.2. Risk Management Strategies with Options and the Underlying
LO.a: Compare the use of covered calls and protective puts to manage risk exposure to individual
securities
Covered Call
In this strategy we take a long position in the underlying and sell a call option. A covered call provides
some protection against a fall in the price of the underlying. It also generates cash up front, but the
covered call writer could miss out on the upside in a strong bull market.

Exhibit 6 shows the profit diagram for a covered call strategy.

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The important points to note are:


Value at expiration: VT = ST – max(0, ST – X)



Profit: Π = VT – S0 + c0



Maximum profit = X – S0 + c0



Maximum loss = S0 – c0



Breakeven: ST* = S0 – c0

Refer to Example 3 from the curriculum.
Protective Put
In this strategy we are long on the underlying and buy a put. It provides downside protection while
retaining the upside potential. However, we need to pay cash upfront to buy the put option.
Exhibit 7 shows the profit diagram for a protective put.

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The important points to note are:


Value at expiration: VT = ST + max(0,X – ST)



Profit: Π = VT – S0 – p0



Maximum profit = ∞



Maximum loss = S0 + p0 – X



Breakeven: ST* = S0 + p0

Refer to Example 4 from the curriculum.
LO.b: Calculate and interpret the value at expiration, profit, maximum profit, maximum loss,
breakeven underlying price at expiration, and general shape of the graph for the following option
strategies: bull spread, bear spread, butterfly spread, collar, straddle, box spread
This LO is covered in sections 2.3 and 2.4

2.3. Money Spreads
A spread is a strategy in which we buy one option and sell another option which is identical except for
exercise price or time to expiration.
If the expiration time is different, then the spread is called a time spread (not covered in this reading).
If the exercise price is different, then the spread is called a money spread. This strategy is called a spread
because the payoff is based on the difference, or spread, between option exercise prices.
We will cover the following money spread strategies:
 Bull spread
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Risk Management Applications of Option Strategies



IFT Notes

Bear spread
Butterfly spread

Bull Spread
In this strategy we combine a long position in a call with exercise price X1 and a short position in a call
with a higher exercise price X2. This strategy is designed to make money when the market goes up.
Exhibit 8 shows the profit diagram for a bull spread.

The important points to note are:


Value at expiration: VT = max(0, ST – X1) – max(0, ST – X2)



Profit: Π = VT – c1 + c2



Maximum profit = X2 – X1 – c1 + c2



Maximum loss = c1 – c2



Breakeven: ST* = X1 + c1 – c2

Refer to Example 5 from the curriculum.
Bear Spread
This strategy is exactly opposite to a bull spread. Here we sell a call with a lower exercise price X1 and
buy a call with a higher exercise price X2. Alternatively we can also execute this strategy by buying puts,
we would buy a put with a higher exercise price X2 and sell a put with a lower exercise price X1. This
strategy is designed to make money when the market goes down.

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Exhibit 9 shows the profit diagram for a bear spread. Note that it is a mirror image of the bull spread.

The important points to note are:


Value at expiration: VT = max(0, X2 – ST) – max(0, X1 – ST)



Profit: Π = VT – p2 + p1



Maximum profit = X2 – X1 – p2 + p1



Maximum loss = p2 – p1



Breakeven: ST* = X2 – p2 + p1

Refer to Example 6 from the curriculum.
Butterfly spreads
This strategy combines a bull spread and a bear spread. It is designed to make money when the market
has low volatility and stays within a range.
Consider three exercise prices X1, X2, and X3. As shown above, we can construct a bull spread by buying
the call with exercise price of X1 and selling the call with exercise price of X2. Similarly we can construct a
bear spread by buying the call with exercise price X3 and selling the call with exercise price X2. The end
result is that we own the calls with exercise price X1 and X3 and have sold two calls with exercise price X2.
Exhibit 10 shows the profit diagram for a butterfly spread.

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The important points to note are:


Value at expiration: VT = max(0, ST – X1) – 2max(0, ST – X2) + max(0, ST – X3)



Profit: Π = VT – c1 + 2c2 – c3



Maximum profit = X2 – X1 – c1 + 2c2 – c3



Maximum loss = c1 – 2c2 + c3



Breakeven: ST* = X1 + c1 – 2c2 + c3 and ST* = 2X2 – X1 – c1 + 2c2 – c3

Refer to Example 7 from the curriculum.
Note: We can also construct a butterfly spread using puts. We could buy the puts with exercise prices X1
and X3 and sell two puts with exercise price of X2 to obtain the same result. The formulae are identical
and can be obtained by replacing ‘c’ with ‘p’.

2.4. Combinations of Calls and Puts
Collar
In this strategy we buy a put and sell a call.
Recall that in a protective put strategy, the holder of the asset buys a protective put to provide
downside protection. However there is a cost involved in purchasing the put. To reduce this cost he can
sell a call option and collect a premium. This strategy is called a collar. When the premiums offset each
other, it is called a ‘zero cost collar’.
Exhibit 11 shows the profit diagram of a zero-cost collar.

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The important points to note are:


Value at expiration: VT = ST + max(0, X1 – ST) – max(0, ST – X2)



Profit: Π = VT – S0



Maximum profit = X2 – S0



Maximum loss = S0 – X1



Breakeven: ST* = S0

Refer to Example 8 from the curriculum.
Straddle
In this strategy we buy a call and a put on the underlying with the same exercise price and same
expiration. It can be used when we expect the markets to be volatile, but are not sure which direction
the market will move.
Exhibit 12 shows the profit diagram for a straddle.

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The important points to note are:


Value at expiration: VT = max(0, ST – X) + max(0, X – ST)



Profit: Π = VT – (c0 + p0)



Maximum profit = ∞



Maximum loss = c0 + p0



Breakeven: ST* = X ± (c0 + p0)

Refer to Example 9 from the curriculum.
Note: Some variations of this strategy are: Strap (A call is added to the straddle), Strip (A put is added to
the straddle) and Strangle (The call and put have different exercise prices).
Box spreads

Box spread can be used to exploit an arbitrage opportunity, when neither binomial nor BSM model
holds. It is a combination of a bull spread and a bear spread.
The important points to note are:


Value at expiration: VT = X2 – X1



Profit: Π = X2 – X1 – (c1 – c2 + p2 – p1)



Maximum profit = (same as profit)



Maximum loss = (no loss is possible, given fair option prices)



Breakeven: no breakeven; the transaction always earns the risk-free rate, given fair option

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Risk Management Applications of Option Strategies

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prices.
Refer to Example 10 from the curriculum.

3. Interest Rate Option Strategies
In interest rate options the 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. Similarly a put option will make money if the option expires with the underlying interest
rate below the exercise rate.
The payoff of an interest rate call option is:
(𝑁𝑜𝑡𝑖𝑜𝑛𝑎𝑙 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙)𝑚𝑎𝑥(0, 𝑈𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔 𝑟𝑎𝑡𝑒 𝑎𝑡 𝑒𝑥𝑝𝑖𝑟𝑎𝑡𝑖𝑜𝑛
𝐷𝑎𝑦𝑠 𝑖𝑛 𝑢𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔 𝑟 �𝑡𝑡
− 𝐸𝑥𝑒𝑟𝑐𝑖𝑠𝑒 𝑟𝑎𝑡𝑒) (
)
360
Similarly the payoff of an interest rate put option is:
(𝑁𝑜𝑡𝑖𝑜𝑛𝑎𝑙 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙)𝑚𝑎𝑥(0, 𝐸𝑥𝑒𝑟𝑐𝑖𝑠𝑒 𝑟𝑎𝑡𝑒
− 𝑈𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔 𝑟𝑎𝑡𝑒 𝑎𝑡 𝑒𝑥𝑝𝑖𝑟𝑎𝑡𝑖𝑜𝑛) (

𝐷𝑎𝑦𝑠 𝑖𝑛 𝑢𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔 𝑟𝑎𝑡𝑒
)
360

LO.c: Calculate the effective annual rate for a given interest rate outcome when a borrower (lender)
manages the risk of an anticipated loan using an interest rate call (put) option
This LO is covered in sections 3.1 and 3.2

3.1. Using Interest Rate Calls with Borrowing
If an entity wants to take out a loan in the future and is concerned that interest rates would have gone
up by the time it takes out the loan, then it can use an interest rate call to establish a maximum interest
rate for the loan. If interest rates rise above the exercise price, then the call payoff will compensate for
the higher interest rate that the entity has to pay on the loan.
Exhibit 13 demonstrates this scenario.
Exhibit 13. Outcomes for an Anticipated Loan Protected with an Interest Rate Call
Scenario (14 April)
Global Computer Technology (GCT) is a US corporation that occasionally undertakes short-term
borrowings in US dollars with the rate tied to Libor. To facilitate its cash flow planning, it buys an
interest rate call to put a ceiling on the rate it pays while enabling it to benefit if rates fall. A call gives
GCT the right to receive the difference between Libor on the expiration date and the exercise rate it
chooses when it purchases the option. The payoff of the call is determined on the expiration date, but
the payment is not received until a certain number of days later, corresponding to the maturity of the
underlying Libor. This feature matches the timing of the interest payment on the loan.
Action

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GCT determines that it will borrow $40 million at Libor plus 200 basis points on 20 August. The loan will
be repaid with a single payment of principal and interest 180 days later on 16 February.

To protect against increases in Libor between 14 April and 20 August, GCT buys a call option on Libor
with an exercise rate of 5 percent to expire on 20 August with the underlying being 180-day Libor. The
call premium is $100,000. We summarize the information as follows:
Loan amount

$40,000,000

Underlying

180-day Libor

Spread

200 basis points over Libor

Current Libor

5.5 percent

Expiration

20 August (128 days later)

Exercise rate

5 percent

Call premium

$100,000

Scenario (20 August)
Libor on 20 August is 8 percent.
Outcome and Analysis
For any Libor, the call payoff at expiration is given below and will be received 180 days later:
180
$40,000,000𝑚𝑎𝑥(0, 𝐿𝑖𝑏𝑜𝑟 − 0.05) (
)
360
For Libor of 8 percent, the payoff is
180
$40,000,000𝑚𝑎𝑥(0,0.08 − 0.05) (
) = $600,000
360
The premium compounded from 14 April to 20 August at the original Libor of 5.5 percent plus 200 basis
points is:
128
$100,000 [1 + (0.055 + 0.02) (
)] = $102,667
360
So the call costs $100,000 on 14 April, which is equivalent to $102,667 on 20 August. The effective loan
proceeds are $40,000,000 – $102,667 = $39,897,333. The loan interest is:
$40,000,000(Liboron20August+200Basispoints)(180/360)
For Libor of 8 percent, the loan interest is:
$40,000,000(0.08+0.02)(180/360)=$2,000,000
The call payoff was given above. The loan interest minus the call payoff is the effective interest. The
effective rate on the loan is:

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Risk Management Applications of Option Strategies

$40,000,000 𝑝𝑙𝑢𝑠 𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
(
)
$39,897,333
=(

IFT Notes

365/180

−1

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

365/180

− 1 = 0.0779

or 7.79 percent.
The results are shown below for a range of Libors on 20 August.
Libor on 20

August

Loan
Rate

Loan Interest Paid on 16
February

Call
Payoff

Effective
Interest

Effective Loan
Rate

0.010

0.030

$600,000

$0

$600,000

0.0360

0.015

0.035

700,000

0

700,000

0.0412

0.020

0.040

800,000

0

800,000

0.0464

0.025

0.045

900,000

0

900,000

0.0516

0.030

0.050

1,000,000

0

1,000,000

0.0568

0.035

0.055

1,100,000

0

1,100,000

0.0621

0.040

0.060

1,200,000

0

1,200,000

0.0673

0.045

0.065

1,300,000

0

1,300,000

0.0726

0.050

0.070

1,400,000

0

1,400,000

0.0779

0.055

0.075

1,500,000

100,000

1,400,000

0.0779

0.060

0.080

1,600,000

200,000

1,400,000

0.0779

0.065

0.085

1,700,000

300,000

1,400,000

0.0779

0.070

0.090

1,800,000

400,000

1,400,000

0.0779

0.075

0.095

1,900,000

500,000

1,400,000

0.0779

0.080

0.100

2,000,000

600,000

1,400,000

0.0779

0.085

0.105

2,100,000

700,000

1,400,000

0.0779

0.090

0.110

2,200,000

800,000

1,400,000

0.0779

Exhibit 14 shows the effective rate on an anticipated future loan protected using an interest rate call
option.

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Refer to Example 11 from the curriculum.

3.2. Using Interest Rate Puts with Lending
If an entity wants to lend money in the future and is concerned that interest rates will go down, it can
buy an interest rate put to establish a minimum interest rate for the loan. If interest rates fall below the

exercise price, then the payoff from the put will compensate the entity for the lower interest rate on the
loan.
Exhibit 15 demonstrates this scenario.
Exhibit 15. Outcomes for an Anticipated Loan Protected with an Interest Rate Put
Scenario (15 March)
Arbitrage Bank Inc. (ABInc) is a US bank that makes loan commitments to corporations. When ABInc
makes these commitments, it recognizes the risk that Libor will fall by the date the loan is taken out.
ABInc protects itself against interest rate decreases by purchasing interest rate puts, which give it the
right to receive the difference between the exercise rate it chooses and Libor at expiration. Libor is
currently 7.25 percent.
Action
ABInc commits to lending $50 million to a company at 90-day Libor plus 250 basis points. The loan will
be a single-payment loan, meaning that it will be made on 1 May and the principal and interest will be
repaid 90 days later on 30 July.
To protect against decreases in Libor between 15 March and 1 May, ABInc buys a put option with an
exercise rate of 7 percent to expire on 1 May with the underlying being 90-day Libor. The put premium

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is $62,500. We summarize the information as follows:
Loan amount

$50,000,000

Underlying

90-day Libor

Spread

250 basis points over Libor

Current Libor

7.25 percent

Expiration

1 May

Exercise rate

7 percent

Put premium

$62,500

Scenario (1 May)
Libor is now 6 percent.
Outcome and Analysis

For any Libor, the payoff at expiration is given below and will be received 90 days later:
$50,000,000max (0, 0.07−Libor) (90/360)
For Libor of 6 percent, the payoff is:
$50,000,000max (0, 0.07−0.060) (90/360) =$125,000
The premium compounded from 15 March to 1 May at current Libor plus 250 basis points is:
$62,500[1+ (0.0725+0.025) (47/360)] =$63,296
So the 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. For any Libor, the loan interest is:
$50,000,000[Libor on 1May plus 250 Basis points (90/360)]
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:
𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 𝑝𝑙𝑢𝑠 𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 365/90
(
)
−1
$50,063,296
365/90
$50,000,000 + $1,062,500 + $125,000
=(
− 1 = 0.0942
)
$50,063,296
or 9.42 percent. The results that follow are for a range of Libors on 1 May.

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Libor on 1
May

Loan
Rate

Loan Interest Paid on 30
July

Put
Payoff

Effective
Interest

Effective Loan
Rate

0.030

0.055

$687,500

$500,000

$1,187,500

0.0942

0.035

0.060

750,000

437,500

1,187,500

0.0942

0.040

0.065

812,500

375,000

1,187,500

0.0942

0.045

0.070

875,000

312,500

1,187,500

0.0942

0.050

0.075

937,500

250,000

1,187,500

0.0942

0.055

0.080

1,000,000

187,500

1,187,500

0.0942

0.060

0.085

1,062,500

125,000

1,187,500

0.0942

0.065

0.090

1,125,000

62,500

1,187,500

0.0942

0.070

0.095

1,187,500

0

1,187,500

0.0942

0.075

0.100

1,250,000

0

1,250,000

0.0997

0.080

0.105

1,312,500

0

1,312,500

0.1051

0.085

0.110

1,375,000

0

1,375,000

0.1106

0.090

0.115

1,437,500

0

1,437,500

0.1161

0.095

0.120

1,500,000

0

1,500,000

0.1216

0.100

0.125

1,562,500

0

1,562,500

0.1271

0.105

0.130

1,625,000

0

1,625,000

0.1327

0.110

0.135

1,687,500

0

1,687,500

0.1382

Exhibit 16 shows the effective rate on an anticipated loan with an interest rate put option.

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Refer to Example 12 from the curriculum.
LO.d: Calculate the payoffs for a series of interest rate outcomes when a floating rate loan is
combined with 1) an interest rate cap, 2) an interest rate floor, or 3) an interest rate collar
This LO is covered in sections 3.3, 3.4 and 3.5.

3.3. Using an Interest Rate Cap with a Floating-Rate Loan
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.
Exhibit 17 demonstrates this scenario.
Exhibit 17. Interest Rate Cap
Scenario (15 April)
Measure Technology (MesTech) is a corporation that borrows in the floating-rate instrument market. It
typically takes out a loan for several years at a spread over Libor. MesTech pays the interest
semiannually and the full principal at the end.
To protect against rising interest rates over the life of the loan, MesTech usually buys an interest rate
cap in which the component caplets expire on the dates on which the loan rate is reset. The cap seller is
a derivatives dealer.
Action
MesTech takes out a $10 million three-year loan at 100 basis points over Libor. The payments will be
made semiannually. The lender is SenBank. Current Libor is 9 percent, which means that the first rate
will be at 10 percent. Interest will be based on 1/360 of the exact number of days in the six-month
period. MesTech selects an exercise rate of 8 percent. The caplets will expire on 15 October, 15 April of
the following year, and so on for three years, but the caplet payoffs will occur on the next payment date
to correspond with the interest payment based on Libor that determines the cap payoff. The cap
premium is $75,000. We thus have the following information:
Loan amount

$10,000,000

Underlying

180-day Libor

Spread

100 basis points over Libor

Current Libor

9 percent

Interest based on

actual days/360

Component caplets

five caplets expiring 15 October, 15 April, etc.

Exercise rate

8 percent

Cap premium

$75,000

Scenario (Various Dates throughout the Loan)

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Shown below is one particular set of outcomes for Libor:
8.50 percent on 15 October
7.25 percent on 15 April the following year
7.00 percent on the following 15 October
6.90 percent on the following 15 April
8.75 percent on the following 15 October
Outcome and Analysis
The loan interest due is computed as
𝐷𝑎𝑦𝑠 𝑖𝑛 𝑠𝑒𝑡𝑡𝑙𝑒𝑚𝑒𝑛𝑡 𝑝𝑒𝑟𝑖𝑜𝑑
$10,000,000(𝐿𝑖𝑏𝑜𝑟 𝑜𝑛 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑟𝑒𝑠𝑒𝑡 𝑑𝑎𝑡𝑒 + 100 𝐵𝑎𝑠𝑖𝑠 𝑝𝑜𝑖𝑛𝑡𝑠) × (
)
360
The caplet payoff is
𝐷𝑎𝑦𝑠 𝑖𝑛 𝑠𝑒𝑡𝑡𝑙𝑒𝑚𝑒𝑛𝑡 𝑝𝑒𝑟𝑖𝑜𝑑
$10,000,000𝑚𝑎𝑥(0, 𝐿𝑖𝑏𝑜𝑟 𝑜𝑛 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑟𝑒𝑠𝑒𝑡 𝑑𝑎𝑡𝑒 − 0.08) × (
)
360

The previous reset date is the expiration date of the caplet. The effective interest is the interest due
minus the caplet payoff.
The first caplet expires on the first 15 October and pays off the following April, because Libor on 15
October was 8.5 percent. The payoff is computed as:
$10,000,000𝑚𝑎𝑥(0,0.085 − 0.08)(182/360) = $10,000,000(0.005)(182/360) = $25,278
which is based on 182 days between 15 October and 15 April. The following table shows the payments
on the loan and cap:
Date

Libor

Loan Rate

Days in Period

Interest Due

15 April

0.0900

0.1000

15 October

0.0850

15 April

0.0950

183

$508,333

0.0725

0.0825

182

480,278

$25,278

455,000

15 October

0.0700

0.0800

183

419,375

0

419,375

15 April

0.0690

0.0790

182

404,444

0

404,444

15 October

0.0875

0.0975

183

401,583

0

401,583

182

492,917

37,917

455,000

15 April

Caplet Payoffs

Effective Interest
$508,333

Refer to Example 13 from the curriculum.

3.4. Using an Interest Rate Floor with a Floating-Rate Loan
An interest rate floor is a series of interest rate put options that expire on various interest rate reset

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dates. Each component is called a floorlet. It provides protection to the lender against falling interest
rates.
Exhibit 18 demonstrates this scenario.
Exhibit 18. Interest Rate Floor
Scenario (15 April)
SenBank lends in the floating-rate instrument market. Often it uses floating-rate financing, thereby
protecting itself against decreases in the floating rates on its loans. Sometimes, however, it finds it can
get a better rate with fixed-rate financing, but it then leaves itself exposed to interest rate decreases on
its floating-rate loans. Its loans are typically for several years at a spread over Libor with interest paid
semiannually and the full principal paid at the end.
To protect against falling interest rates over the life of the loan, SenBank buys an interest rate floor in
which the component floorlets expire on the dates on which the loan rate is reset. The floor seller is a
derivatives dealer.
Action
SenBank makes a $10 million three-year loan at 100 basis points over Libor to MesTech (see cap
example). The payments will be made semiannually. Current Libor is 9 percent, which means that the
first interest payment will be at 10 percent. Interest will be based on the exact number of days in the sixmonth period divided by 360. SenBank selects an exercise rate of 8 percent. The floorlets will expire on
15 October, 15 April of the following year, and so on for three years, but the floorlet payoffs will occur
on the next payment date so as to correspond with the interest payment based on Libor that
determines the floorlet payoff. The floor premium is $72,500. We thus have the following information:
Loan amount

$10,000,000

Underlying

180-day Libor

Spread

100 basis points over Libor

Current Libor

9 percent

Interest based on

actual days/360

Component floorlets

five floorlets expiring 15 October, 15 April, etc.

Exercise rate

8 percent

Floor premium

$72,500

Outcomes (Various Dates throughout the Loan)
Shown below is one particular set of outcomes for Libor:
8.50 percent on 15 October
7.25 percent on 15 April the following year
7.00 percent on the following 15 October

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6.90 percent on the following 15 April
8.75 percent on the following 15 October
Outcome and Analysis
The loan interest is computed as
$10,000,000(Libor on previous reset date+100 Basis points) × (Days in settlement period/360)
The floorlet payoff is:
$10,000,000max (0, 0.08−Libor on previous reset date) × (Days in settlement period/360)
The effective interest is the interest due plus the floorlet payoff. The following table shows the
payments on the loan and floor:

Date

Libor

Loan
Rate

15 April

0.0900

0.1000

15 October

0.0850

0.0950

183

$508,333

15 April

0.0725

0.0825

182

480,278

$0

480,278

15 October

0.0700

0.0800

183

419,375

38,125

457,500

15 April

0.0690

0.0790

182

404,444

50,556

455,000

15 October

0.0875

0.0975

183

401,583

55,917

457,500

182

492,917

0

492,917

15 April

Days in
Period

Interest
Due

Floorlet
Payoffs

Effective
Interest
$508,333

Refer to Example 14 from the curriculum.

3.5. Using an Interest Rate Collar with a Floating-Rate Loan
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.
Exhibit 19 demonstrates this scenario.
Exhibit 19. Interest Rate Collar
Scenario (15 April)
Consider the Measure Technology (MesTech) scenario described in the cap and floor example in Exhibits
17 and 18. MesTech is a corporation that borrows in the floating-rate instrument market. It typically
takes out a loan for several years at a spread over Libor. MesTech pays the interest semiannually and
the full principal at the end.

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To protect against rising interest rates over the life of the loan, MesTech usually buys an interest rate
cap in which the component caplets expire on the dates on which the loan rate is reset. To pay for the

cost of the interest rate cap, MesTech can sell a floor at an exercise rate lower than the cap exercise
rate.
Action
Consider the $10 million three-year loan at 100 basis points over Libor. The payments are made
semiannually. Current Libor is 9 percent, which means that the first rate will be at 10 percent. Interest is
based on the exact number of days in the six-month period divided by 360. MesTech selects an exercise
rate of 8.625 percent for the cap. Generating a floor premium sufficient to offset the cap premium
requires a floor exercise rate of 7.5 percent. The caplets and floorlets will expire on 15 October, 15 April
of the following year, and so on for three years, but the payoffs will occur on the following payment
date to correspond with the interest payment based on Libor that determines the caplet and floorlet
payoffs. Thus, we have the following information:
Loan amount

$10,000,000

Underlying

180-day Libor

Spread

100 basis points over Libor

Current Libor

9 percent

Interest based on

actual days/360

Component options

five caplets and floorlets expiring 15 October, 15 April, etc.

Exercise rate

8.625 percent on cap, 7.5 percent on floor

Premium

no net premium

Scenario (Various Dates throughout the Loan)
Shown below is one particular set of outcomes for Libor:
8.50 percent on 15 October
7.25 percent on 15 April the following year
7.00 percent on the following 15 October
6.90 percent on the following 15 April
8.75 percent on the following 15 October
Outcome and Analysis
The loan interest is computed as:
$10,000,000(Libor on previous reset date+100Basispoints) × (Days in settlement period/360)
The caplet payoff is:

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$10,000,000max (0, Libor on previous reset date−0.08625) × (Days in settlement period/360)
The floorlet payoff is:
($10,000,000max (0, 0.075 – Libor on previous reset date) × (Days in settlement period/360)
The effective interest is the interest due minus the caplet payoff minus the floor-let payoff. Note that
because the floorlet was sold, the floorlet payoff is either negative (so we would subtract a negative
number, thereby adding an amount to obtain the total interest due) or zero.
The following table shows the payments on the loan and collar:

Date

Libor

Loan
Rate

Days in
Period

Interest
Due

15 April

0.0900

0.1000

15 October

0.0850

15 April

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

Refer to Example 15 from the curriculum.

4. Option Portfolio Risk Management Strategies
Many options are traded by dealers, who make the markets in these options. Dealers provide liquidity
by taking on risk then hedge their position in order to earn the bid-ask spread.
Static hedge: In this method there is no need to change position. If a dealer has sold a call, then he will
buy a similar call to offset his risk.
Another option available is to use the put-call parity relationship. The dealer could buy a synthetic call
by buying a put, buying the asset, and selling a bond.
LO.e: Explain why and how a dealer delta hedges an option position, why delta changes, and how
the dealer adjusts to maintain the delta hedge
Note: The remainder of this LO is covered in section 4.1.
Delta hedge: Often a static hedge is not a viable option because the necessary options may not be
available or may not be favourably priced.
Hence to offset the short call the dealer will go long in the underlying stock. He will hope to offset the
change in option price with the change in stock price. This method is called dynamic hedging because
the hedge needs to change with stock price and passage of time.

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