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

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

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

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

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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

The important points are:

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

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:

The important points to note are:

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

The important points to note are:

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

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

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:



 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)

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

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)

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

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

The important points to note are:

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

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

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

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:

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

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:

$40,000,000𝑚𝑎𝑥(0, 𝐿𝑖𝑏𝑜𝑟 − 0.05) (180

360) For Libor of 8 percent, the payoff is

$40,000,000𝑚𝑎𝑥(0,0.08 − 0.05) (180

360) = $600,000 The premium compounded from 14 April to 20 August at the original Libor of 5.5 percent plus 200 basis

points is:

$100,000 [1 + (0.055 + 0.02) (128

360)] = $102,667

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:

Loan Interest Paid on 16 February

Call Payoff

Effective Interest

Effective Loan Rate

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

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:

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:

Libor on 1

May

Loan Rate

Loan Interest Paid on 30 July

Put Payoff

Effective Interest

Effective Loan Rate

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

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 due is computed as

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 Caplet Payoffs Effective Interest

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

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

six-month 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:

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

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:

Loan Rate

Days in Period

Interest Due

Floorlet Payoffs

Effective Interest

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

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

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:

$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

Caplet Payoffs

Floorlet Payoffs

Effective Interest

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