CFA level 3 CFA level 3 CFA level 3 CFA level 3 CFA level 3 finquiz curriculum note, study session 15, reading 29

Risk Management Applications of Option Strategies

It is important to note that options do not always
automatically increase risk.
2.2

Risk Management Strategies with Options and
the Underlying

An investor can reduce exposure without selling the
underlying by:
1) Selling a call on the underlying i.e. covered call.
2) Buying a put i.e. protective put.
2.2.1) Covered Calls
Covered Call = Long stock position + Short call position
Covered Call is appropriate to use when an investor:
• Owns the stock and
• Expects that stock price will neither increase nor
decrease in near future.
Characteristics:
• It is an imperfect form of portfolio protection. It
provides only limited downside protection.
• It exchanges “upside” potential for current income
in the form of option premium.
• It generates cash up front in the form of option
premium but removes some of the upside
potential.
• It reduces both the overall risk and the expected
return compared with simply holding the
underlying. This loss in potential upside gains is
compensated by option premium received by

selling a call.
• Writers of covered call options make small
amounts of money, but make it often; because
expected profits come from rare but large payoffs.

• Selling a call option on a stock already owned by
an investor reduces the overall risk.
• Selling a call without owning the stock exposes
the investor to unlimited loss potential.
Thus, covered call should not be viewed as a
conservative strategy.
Relationship between exercise price and potential
upside gains for the short call:
• The higher the exercise price of the call option, the
lower the price of an option and thus short call
receives lower premium. However, in this case,
short call has a greater opportunity to gain from
the upside.
NOTE:
Current value of the asset should be viewed as an
opportunity cost of an investor.

Practice: Example 3,
Volume 5, Reading 29.

2.2.2) Protective Puts
Protective Put = Long stock position + Long Put position
This provides protection against a decline in value.
It is similar to “insurance" i.e. buying insurance in the form
of the put, paying a premium to the seller of the

insurance, the put writer.
Characteristics:

To summarize:
a) Value at expiration = Value of the underlying +
Value of the short call = VT = ST – max (0, ST – X)
b) Profit = Profit from buying the underlying + Profit
from selling the call = VT – S0 + c0
c) Maximum Profit = X – S0 + c0
d) Max loss would occur when ST = 0. Thus, Maximum
Loss = S0 – c0
e) Breakeven =ST* = S0 – c0

• It provides downside protection while retaining the
upside potential.
• It requires the payment of cash up front in the form
of option premium.
• The higher the exercise price of a put option, the
more expensive the put will be and consequently
the more expensive will be the downside
protection.
Protective put is appropriate to use when:
• An investor owns a stock and does not want to sell
it.
• An investor expects a decline in the value of the
stock in the near future but wants to preserve
upside potential.
To summarize:
a) Value at expiration: VT = ST + max (0, X - ST)

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Reading 29


Reading 29

Risk Management Applications of Option Strategies

b) Profit = VT – S0 - p0
c) Maximum Profit = ∞
d) The maximum loss would occur when underlying
asset is sold at exercise price. Thus,
Maximum Loss = S0 + p0 – X
e) In order to breakeven, the underlying must be at
least as high as the amount paid up front to
establish the position. Thus, Breakeven =ST* = S0 +
p0

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but a different time to expiration. Time spread can
benefit from high volatility.
Money spreads: When the options have different
exercise price, the spread is called a money spread e.g.
an investor buys an option with a given expiration and
exercise price and sells an option with the same
expiration but a different exercise price.

• Note that the options are on the same underlying
asset.

Example:
2.3.1) Bull Spreads

Strike price = X = $45
Option cost = p0 = $6.
• The maximum possible loss is $6
• The potential gain is unlimited

A. Bull Call Spread: This strategy involves a combination
of a long position in a call with a lower exercise price
and a short position in a call with a higher exercise
price i.e.
• Buy a call (X1) with option cost c1 and sell a call
(X2) with option cost c2, where X1< X2 and c1 > c2.
Note that the lower the exercise price of a call option,
the more expensive it is.
Rationale to use Bull Call Spread: Bull call spread is used
when investor expects that the stock price or underlying
asset price will increase in the near future.
Characteristics:

Put v/s Insurance:
The exercise price of the put is like the insurance
deductible because the magnitude of the exercise
price reflects the risk assumed by the party who owns
the underlying. A higher exercise price of the put option
is equivalent to a lower insurance deductible.

• The higher the exercise price, the higher the option
premium and the less risk assumed by the holder of
the underlying and the more risk assumed by the
put seller.
• In insurance, the higher the deductible, the more
risk assumed by the insured party and the less risk
assumed by the insurer.

Practice: Example 4,
Volume 5, Reading 29.

2.3

Money Spreads

A spread is a strategy that involves buying one option
and selling another identical option but either with
different exercise price or different time to expiration.
Time Spread: When the options have different time to
expiration, the spread is called a time spread. Time
spread strategies are used to exploit differences in
perceptions of volatility of the underlying e.g. an investor
buys an option with a given expiration and exercise
price and sells an option with the same exercise price

• This strategy gains when stock price rises/ market
goes up.
• Like covered call, it provides protection against
downside risk but provides limited gain i.e. upside
potential.

• It is similar to Covered call strategy i.e.
o In covered call, short position in call is covered
by long position in underlying.
o In bull call spread, the short position in the call
with a higher exercise price is covered by long
position in the call with a lower exercise price.
To summarize:
a) The initial value of the Bull call spread = V0 = c1 –
c2
b) Value at expiration: VT = value of long call – Value
of short call = max (0, ST – X1) - max (0, ST – X2)
c) Profit = Profit from long call + profit from short call.
Thus,
Profit = VT – c1 + c2
d) Maximum Profit = X2 – X1 – c1 + c2
e) Maximum Loss = c1 – c2
f) Breakeven =ST* = X1 + c1 – c2


Reading 29

Risk Management Applications of Option Strategies

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Maximum Profit = X2 – X1 – p2 + p1
e) Maximum Loss occurs when both puts expire outof-the-money and investor loses net premium i.e.
when ST> X2. Thus,
Maximum Loss = p2 – p1
f) Breakeven =ST* = X2 – p2 + p1

B. Bull Put spread: In bull put spread, investor buys a put
with a lower exercise price and sells an otherwise
identical put with a higher strike price.
• Buy a put (X1) and sell a put (X2), with X1< X2.
• Since put with a higher exercise price (X2) is
expensive than a put with a lower exercise price
(X1), bull put spread generates cash inflow at
initiation of the position.
• Profit occurs when both put options expire out-ofthe-money i.e. investor will earn net premium.
• It is identical to the sale of Bear put spread.
• Bull put spread pay-off diagram is the mirror-image
of the pay-off diagram of bear put spread.

Practice: Example 5,
Volume 5, Reading 29.

2.3.2) Bear Spreads
A. Bear Put Spread: This strategy involves a combination
of a long position in a put with a higher exercise price
and a short position in a put with a lower exercise
price i.e.
• Buy a put (X2) with option cost p2 and sell a put (X1)
with option cost p1, where X1< X2 and p1 < p2.
Note that the higher the exercise price of a put option,
the more expensive it is.
Rationale to use Bear Put Spread: Bear Put spread is used
when investor expects that the stock price or underlying
asset price will decrease in the future.
To summarize:

a) The initial value of the bear put spread = V0 = p2 –
p1
b) Value at expiration: VT = value of long put – Value
of short put = max (0, X2 - ST) - max (0, X1 - ST)
c) Profit = Profit from long put + profit from short put.
Thus,
Profit = VT – p2 + p1
d) Maximum Profit occurs when both puts expire inthe-money i.e. when underlying price ≤ short put
exercise price (ST ≤ X1),
• Short put is exercised and investor will buy an
asset at X1 and
• This asset is sold at X2 when long put is exercised.
Thus,

B. Bear Call Spread: In bear call spread, investor sells a
call with a lower exercise price and buys an otherwise
identical call with a higher strike price.
• Sell a call (X1) and buy a call (X2), with X1< X2.
• Since call with a lower exercise price (X1) is
expensive than a call with a higher exercise price
(X2), bear call spread generates cash inflow at
initiation of the position.
• Profit occurs when both call options expire out-ofthe-money i.e. investor will earn net premium.
• It is identical to the sale of a bull call spread i.e. it is
used when investor expects a decline in stock
price.
• Bear call spread pay-off diagram is the mirrorimage of the pay-off diagram of bull call spread.

Practice: Example 6,
Volume 5, Reading 29.

2.3.3) Butterfly Spreads
Butterfly spread strategy is a combination of a bull and
bear spread. Butterfly spreads perform based on the
volatility of the underlying.
A. Long Butterfly Spread (Using Call):
Long Butterfly Spread = Long Bull call spread + Short Bull
call spread (or Long Bear call spread)
Long Butterfly Spread = (Buy the call with exercise price
of X1 and sell the call with exercise price of X2) + (Buy the
call with exercise price of X3 and sell the call with
exercise price of X2)
where,
X1< X2 < X3
Cost of X1 (c1) > Cost of X2 (c2) > Cost of X3 (c3)
NOTE:
Long Butterfly spread requires cash outlay at initiation
because bull spread purchased by an investor is
expensive than a bull spread that is sold.


Reading 29

Risk Management Applications of Option Strategies

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Rationale to use Long Butterfly Spread: It is used when
investor expects that the volatility of the underlying will
be relatively low compared to what market expects i.e.

the underlying asset will trade near the middle exercise
price.
• When market is highly volatile, butterfly spread
strategy is not profitable and generates losses.
To summarize:
a) Value at expiration: VT = max (0, ST – X1) – 2 max
(0, ST – X2) + max (0, ST – X3)
b) Profit = VT – c1 + 2c2 - c3
c) Maximum Profit occurs when price of underlying is
close to the middle exercise price i.e. when ST =
X2. Thus,
Maximum Profit = X2 – X1 – c1 + 2c2 – c3
d) Maximum Loss occurs when price of underlying <
lower strike price or > upper strike price and
investor loses net premium. Thus, Maximum Loss =
c1 – 2c2 + c3
e) There are two breakeven points i.e.
i. Breakeven =ST* = X1 + net premium = X1 + c1 –
2c2 + c3
ii. Breakeven = ST* = 2X2 – X1 – Net premium = 2X2
– X1 – (c1 – 2c2 + c3 ) = 2X2 – X1 – c1 + 2c2 - c3

C. Long Butterfly Spread (Using Puts):
Butterfly Spread = Long Bear put spread + Short bear put
spread (or Long Bull put spread)
Long Butterfly Spread = (Buy the put with exercise price
of X3 and sell the put with exercise price of X2) + (Buy the
put with exercise price of X1 and sell the put with
exercise price of X2)
where,

X1< X2 < X3
Cost of X1 (p1) < Cost of X2 (p2) D. Short Butterfly Spread (Using Puts): It refers to selling
the butterfly spread i.e.
Short butterfly spread = Selling the puts with exercise
prices of X1 and X3 and buying two puts with exercise
prices of X2.
• The maximum profit occurs when either all four of
the options are out-of-the money or all four are inthe-money i.e. investor earns net premium.
Maximum Profit = p3 + p1 – 2p2
NOTE:

NOTE:
• Purple line represents after 1 month.
• Light Green line represents after 3 months.
• Dark Green line represents at expiry.
B. Short Butterfly Spread(Using Call): It refers to selling the
butterfly spread i.e.
Short butterfly spread = Selling the calls with exercise
prices of X1 and X3 and buying two calls with exercise
prices of X2.
Rationale to use Short butterfly Spread: This strategy is
preferably used when investor expects that the volatility
of the underlying will be relatively high compared to
what market expects.
• The maximum profit occurs when either all four of
the options are out-of-the money or all four are inthe-money i.e. investor earns net premium.
Maximum Profit = c1 + c3 – 2c2

When options are priced correctly, butterfly spread using

calls will provide the same result as butterfly using puts.

Practice: Example 7,
Volume 5, Reading 29.

2.4

Combinations of Calls and Puts
2.4.1) Collars

Collar refer to the strategy in which the cost of buying
put option can be reduced by selling a call option.
• When call option premium is equal to put option
premium, no net premium is required up front. This
strategy is known as a Zero-Cost Collar *.
• This strategy provides downside protection at the
expense of giving up upside potential. Therefore,
zero-cost only refers to the fact the no cash is
required to be paid up front.


Reading 29

Risk Management Applications of Option Strategies

• In Zero-cost collar, first of all investor selects
exercise price of the put option. Then, the call
exercise price is set such that the call premium
offsets the put premium so that there is no initial
outlay for the options.

• Typically,
o Put exercise price (e.g. X1) < current value of the
underlying.
o Call exercise price (e.g. X2) must be > current
value of the underlying.
• When price < X1, put provides protection against
loss.
• When price > X2, short call reduces gains.
• When price lies between X1 and X2, both put and
call are out-of-the-money.

• Like forwards, collar requires no initial outlay
except the underlying price.
• Unlike forwards, collar payoff represents a range as
it is shown in the figure above that it breaks at the
two exercise prices.
• Collars represent directional strategies i.e. their
performance is based on the direction of the
movement in the underlying.

Practice: Example 8,
Volume 5, Reading 29.

*NOTE:
Typically, in a collar, the call and put premiums offset
each other. However, it is not necessarily always the
case i.e. call premium can be > put premium.
Important to note:
• Put premium decreases when put exercise price is
lowered.

• To offset this lower put premium, investor can sell
call option with a higher exercise price.
• Decreasing put exercise price and increasing call
exercise price results in increase in both the upside
potential and downside risk.
Collar v/s Bull Spread: The collar is quite similar to a bull
spread i.e. both have a cap on the gain and a floor on
the loss. However, bull spread does not involve actually
holding the underlying.

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2.4.2) Straddle
A. Long straddle: It involves buying a put and a call with
same strike price on the same underlying with the
same expiration; both options are at-the-money.
• In this strategy, an investor can make profit from
upside or downside movement of the underlying
price.
• Due to call option, the gain on upside is unlimited
and due to put option, downside gain is quite
large but limited.
• Straddle is a strategy that is based on the volatility
of the underlying. It benefits from high volatility.
• Straddle is a costly strategy.
Rationale to use Straddle: Straddle is to be used only
when the investor expects that volatility of the underlying
will be relatively higher than what market expects but is
not certain regarding the direction of the movement of
the underlying price.

To summarize:
To summarize:
(For zero-cost collar)
a) Initial value of the position = value of the
underlying asset = V0 = S0
b) Value at expiration: VT = Value of underlying ST +
Value of the put option + Value of the short call
option = ST + max (0, X1 - ST) – max (0, ST – X2)
c) Profit = VT – V0 = VT –S0
d) Maximum Profit = X2 – S0
e) Maximum Loss = S0 – X1
f) Breakeven =ST* = S0

Range forwards and risk reversals: Collars are also known
as range forwards and risk reversals.

a) Value at expiration: VT = max (0, ST -X) + max (0, X–
ST)
b) Profit = VT –p0 - c0
c) Maximum Profit = ∞
d) Maximum Loss occurs when both call and put
options expire at-the money and investor loses
premiums on both options i.e.
Maximum Loss = p0 + c0
e) Breakeven = ST* = X ± (p0 + c0)

B. Short Straddle: It involves selling a put and a call with
same strike price on the same underlying with the
same expiration; both options are at-the-money.

Reading 29

Risk Management Applications of Option Strategies

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• This strategy is preferably used when investor has
neutral view of the volatility or when investor
expects a decrease in volatility.
• This strategy gains when both the options expire
at-the money i.e. investor earns call and put
premium.
• This strategy has unlimited loss potential.

Practice: Example 9,
Volume 5, Reading 29.

2.4.3) Box Spreads
A box spread is a combination of a bull spread and a
bear spread i.e.
Box-spread = Bull spread + Bear spread
Variations of Straddle: When investor has any specific
outlook regarding direction of price movement, then
either a call or a put can be added to the straddle.
• Adding call option to a straddle is known as
“Strap”.
• Adding put option to a straddle is known as “Strip”.
• These strategies generate greater gains when

price movement occurs in the expected direction;
however, these strategies are more complex than
a straddle.
a) Long Strangle: It is a variation of the straddle. This
strategy involves buying the put and call on the same
underlying with the same expiration but with different
exercise prices. This strategy is used if investor view is
that volatility will increase.

A. Long Box-spread= (buy the call with exercise price X1
and sell the call with exercise price X2) + (buy the put
with exercise X2 and sell the put with exercise X1)
• Box spread pay-off (i.e. profit) is always the same
i.e. it is must be risk-free when the options are
priced correctly. * In simple words, box-spread
always results in buying the underlying at X1 and
selling it at X2. Since this outcome is known to an
investor at the start, a box-spread can be viewed
as a riskless strategy.
• Since transaction is risk free, the PV of the pay-off,
discounted at risk-free rate should be equal to the
initial outlay (net premium) i.e. we should have:
(X2 – X1) / (1 + r) r = c1 – c2 + p2 – p1
o When PV of the pay-off > net premium, the box
spread is underpriced and it should be
purchased. Buying a box-spread is referred to as
long box-spread.
o When PV of the pay-off < net premium, the box
spread is overpriced and it should be sold. It is
referred to as short box-spread.

*Arbitrage opportunity is available when options are not
priced correctly.

b) Short Strangle: This strategy involves selling the put
and call on the same underlying with the same
expiration but with different exercise prices. This
strategy is used if investor has a neutral view about
volatility or he/she expects that volatility will decrease.

B. Short Box-spread = (Sell the call with exercise price X1
and buy the call with exercise price X2) + (Sell the put
with exercise X2 and buy the put with exercise X1)
Advantages:
• A box spread can be used to exploit an arbitrage
opportunity.
• A box spread does not require the binomial or
Black-Scholes-Merton model to hold.
• It does not require a volatility estimate and all the
transactions associated with box-spread strategy
can be executed within the options market.
• Box-spread is a simple strategy and has lower
transaction costs.


Reading 29

Risk Management Applications of Option Strategies

To summarize (for Long Box-spread):
a) Initial value of the box spread = Net premium = c1

– c2 + p2 – p1.
b) Value at expiration: VT = X2 –X1
c) Profit = X2 –X1 - (c1 – c2 + p2 – p1)
d) Maximum Profit = same as profit
e) Maximum Loss = no loss is possible given fair
option prices
f) Breakeven =ST* = no break-even; the transaction
always earns the risk-free rate, given fair option
prices.

3.

Practice: Example 10,
Volume 5, Reading 29.

Volatility
will
increase

Neutral view
on volatility

Volatility will
decrease

Price will
decrease

Buy Puts

Sell
Underlying

Sell Calls

Neutral
view on
price

Buy
Straddle

Do nothing

Sell Straddle

Price will
increase

Buy Calls

Buy
Underlying

Sell Puts

INTEREST RATE OPTION STRATEGIES

Interest rate call and put options are used to protect
against changes in interest rates.

• For dollar based interest rate options, generally,
the underlying rate is LIBOR.
• The underlying rate is always a specific rate i.e. the
rate on 90-day or 180-day underlying instrument.
• When the option is exercised, the pay-off is
determined using a specific notional principal.
• Traditionally, the pay-off on interest rate option
does not occur immediately upon exercise; rather,
it is paid on the date when payment on the
underlying instrument is due.
The pay-off of an interest rate Call Option= (Notional
principal) × max (0, Underlying rate at expiration –
ୈୟ୷ୱ୧୬୳୬ୢୣ୰୪୷୧୬୥୰ୟ୲ୣ
ቁ
Exercise rate) ì


ã 180-day LIBOR can be used as the underlying rate
and days in underlying could be 180 or perhaps
182, 183 etc.
• When an interest rate option is based on m-day
LIBOR, it is important to note that the rate is
determined on the day when the option expires
and payment is made m days later.
The pay-off of an interest rate Put Option= (Notional
principal) × max (0, Exercise rate - Underlying rate at
ୈୟ୷ୱ୧୬୳୬ୢୣ୰୪୷୧୬୥୰ୟ୲ୣ
ቁ
expiration) × ቀ
ଷ଺଴

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3.1

Using Interest Rate Calls with Borrowing

Interest rate call options are used by borrowers to
manage interest rate risk on floating-rate loans. In
interest rate call options, the following factors must be
considered.
1) Option expiration date is the same as when loan
starts.
2) Option pay-offs occur at the time when borrower
makes interest payments on loan.
3) Option premium is paid by the borrower today
(i.e. at time t0).
Example:
A company plans to borrow $40 million in 128 days at
180-day LIBOR plus 200 basis points. To manage the risk
associated with higher interest rate on a loan, it buys a
call option in which the underlying is the rate on 180-day
LIBOR.
•
•
•
•
•

The option expires in 128 days.

The exercise rate is 5%.
The notional principal is $40 million.
The company pays a premium of $100,000.
Current LIBOR = 5.5%


Reading 29

Risk Management Applications of Option Strategies

Solution:

loan, it buys a put option in which the underlying is the
rate on 90-day LIBOR.

a) The company will pay $100,000 up front in the
form of option premium.
• The rate the firm could earn if it invested the
$100,000 would be 5.5% (i.e. current LIBOR
given).
Thus, compounding premium at the
original/current LIBOR of 5.5% + 200 bps for 128
days =
$100,000[1 + (0.055+ 0.02) ì (128/360)] = $102,667
ã Thus, call premium of $100,000 is equivalent to
$102,667 at the time the loan is taken out.
• This increases the cost of the loan because by
paying this amount, the firm effectively receives
= $40 million - $102,667 = $39,897,333.
• Thus, effective loan proceeds = $39,897,333.

b) The option expires on the date when the loan is
taken out by the company and pays off =
ଵ଼଴
($40,000,000) × Max (0, LIBOR – 5%) × ቀ ቁ
ଷ଺଴

• Note that whenever LIBOR is below 5%, the
payoff is zero.
• Whenever LIBOR is > 5% e.g. when LIBOR is 8%,
the payoff is
ଵ଼଴
($40,000,000) × Max (0, 8% – 5%) × ቀ ቁ =
ଷ଺଴

$600,000
c) Loan interest = ($40,000,000) × (LIBOR on the date
ଵ଼଴
loan is taken out + 200 bps) × ቀ ቁ
ଷ଺଴

For LIBOR = 8%,
Loan interest = ($40,000,000) × (8% + 200 bps) ×
ଵ଼଴
ቀ ቁ = $2,000,000.

•
•
•
•
•

•

The option expires in 47days.
The exercise rate is 7%.
The notional principal is $50 million.
The company pays a premium of $62,500.
Current LIBOR = 7.25%
Assume LIBOR changes to 6%.

Solution:
a) The bank will pay $62,500 up front in the form of
option premium.
• The rate the bank could earn on if it invested
the $62,500 would be 7.25% (i.e. current LIBOR
given).
Thus, compounding premium at the
original/current LIBOR of 7.25% + 250 bps for 47
days =$62,500 [1 + (0.0725+ 0.025) × (47 / 360)] =
$63,296
• Thus, put premium of $62,500 is equivalent to
$63,296 at the time the loan is made.
• Thus, by paying this amount, the bank loaned =
$50 million + $63,296 = $50,063,296.
• Thus, effective amount loaned = $50,063,296.
b) The option expires on the date when the loan is
made by the bank and pays off =
ଽ଴
($50,000,000) × Max (0, 0.07 – LIBOR) × ቀ ቁ
ଷ଺଴

• Note that whenever LIBOR is above 7%, the
payoff is zero.
• Whenever LIBOR is < 7% e.g. when LIBOR is 6%,
the payoff is
ଽ଴
($50,000,000) × Max (0, 0.07 – 0.06%) × ቀ ቁ =

ଷ଺଴

d) Effective Interest paid = $2,000,000 - $600,000 =
$1,400,000.
e) Effective rate on the loan = {(NP + Effective
interest) / effective loan proceeds} 365 / Days in
underlying rate – 1 = {($40m + $1,400,000) /
$39,897,333}365 / 180 – 1 = 0.0779 = 7.79%.

ଷ଺଴

$125,000.
c) Loan interest = ($50,000,000) × (LIBOR on the date
ଽ଴
loan is made + 250 bps) × ቀ ቁ
ଷ଺଴

For LIBOR = 6%,
Loan interest = ($50,000,000) × (6% + 250 bps) ×
ଽ଴
ቀ ቁ = $1,062,500.

Source: Curriculum, Reading 29, Exhibit 13.

Practice: Example 11,
Volume 5, Reading 29.

3.2

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ଷ଺଴

d) Effective Interest received = $1,062,500 + $125,000
= $1,187,500.
e) Effective rate on the loan = {(NP + Effective
interest) / effective amount loan loaned} 365 / Days
in underlying rate – 1 = {($50m + $1,187,500) /
$50,063,296}365 / 90 – 1 = 0.0942 = 9.42%.

Using Interest Rate Puts with Lending
Source: Reading 29, Exhibit 15.

Interest rate put options can be used by lenders to
manage interest rate risk on floating-rate loans i.e. when
interest rate falls below a specific level, interest rate put
option generates a pay-off for the lender and thus
compensates the lender (e.g. bank) for the lower
interest rate on the loan.
Example:
A Bank plans to lend $50 million in 47 days at 90-day
LIBOR plus 250 basis points. To manage the risk
associated with lower interest rate on a floating-rate

Practice: Example 12,
Volume 5, Reading 29.

3.3

Using an Interest Rate Cap with a Floating-Rate
Loan

Interest rate cap is a combination of interest rate call
options where each option’s pay-off occurs on the date


Reading 29

Risk Management Applications of Option Strategies

when the interest payments on a loan are due. Each
option in a cap is called a caplet.
• Each caplet has its own expiration date.
• Each caplet has the same Exercise rate.
• The cap seller makes payments to the borrower if
interest rates > strike rate during the term of the
cap.
• The pay-off of each caplet is determined on its
expiration date, but the caplet pay-off (if any) is
made on the next payment date i.e. the date on
which the loan interest is paid.
o This implies that if a loan has e.g. 6 interest
payments, the cap will contain only five caplets

because there will be only five risky payments as
the first rate on the loan is already set.
o A cap will contain six caplets only when the
borrower purchases the cap in advance of
taking out the loan i.e. the additional caplet can
be used to protect the 1st rate setting on a loan.
Effect of Notional principal amount and exercise
rate/strike rate on cost of cap:
• The cap can be used to protect the entire loan
amount or only a portion of the loan amount.
Reducing the dollar amount of the cap results in
reduction of the cost of the cap.
• Reducing the strike rate of a cap results in increase
in the cost of the cap.

3.4

FinQuiz.com

Using an Interest Rate Floor with a Floating-Rate
Loan

Interest rate floor is a combination of interest rate put
options where each option’s pay-off occurs on the date
when the interest payments on a loan are due to be
received. Each option in a floor is called a floorlet.
• Each floorlet has its own expiration date.
• Exercise rate on each floorlet is the same.

a) Loan Interest payment: It is computed as follows

Loan interest = Notional Principal × (LIBOR on previous
ୈୟ୷ୱ୧୬ୱୣ୲୲୪ୣ୫ୣ୬୲୮ୣ୰୧୭ୢ
reset date + 100 bps) × ቀ
ቁ
ଷ଺଴

b) Floorlet Pay-off: It is computed as follows
The floor pay-off = Notional Principal × (0, Exercise rate ୈୟ୷ୱ୧୬ୱୣ୲୲୪ୣ୫ୣ୬୲୮ୣ୰୧୭ୢ
LIBOR on previous reset date) × ቀ
ቁ
ଷ଺଴

c) Effective Interest = Interest received on the loan +
Floorlet pay-off
For detail calculations, refer to Exhibit 18, Reading 29.

Practice: Example 14,
Volume 5, Reading 29.

a) Loan Interest payment: It is computed as follows
Loan interest = Notional Principal × (LIBOR on previous
ୈୟ୷ୱ୧୬ୱୣ୲୲୪ୣ୫ୣ୬୲୮ୣ୰୧୭ୢ
reset date + 100 bps) × ቀ
ቁ
ଷ଺଴

b) Cap Pay-off: It is computed as follows
The cap pay-off = Notional Principal × (0, LIBOR on
previous reset date – Exercise rate) ×
ୈୟ୷ୱ୧୬ୱୣ୲୲୪ୣ୫ୣ୬୲୮ୣ୰୧୭ୢ

ቀ
ቁ

3.5

A collar is a combination of a long (short) position in a
cap and a short (long) position in a floor.
For borrower

For lender

The borrower can buy a
cap to protect against
rising interest rates and sell
the floor to finance the
premium paid to buy a
cap.

The lender can buy a floor
to protect against falling
interest rates and sell the
cap to finance the
premium paid to buy a
floor.

NOTE:

NOTE:

In this case, effective

interest paid in each
period will be:

In this case, effective
interest earned in each
period will be:

Effective interest paid =
actual interest paid – cap
pay-off + floor pay-off
• The premium received
by selling the floor can
be used to offset the
premium paid to buy a
cap.
• Buying a cap provides

Effective interest earned =
actual interest earned +
floor pay-off – cap pay-off
• The premium received
by selling the cap can
be used to offset the
premium paid to buy a
floor.
• Buying a floor provides

ଷ଺଴

c) Effective Interest = Interest due on the loan – Caplet

pay-off
NOTE:
Since loan has multiple payments, the effective rate on
a loan is similar to IRR on capital investment project or
YTM on a bond.
For detail calculations, refer to Exhibit 17, Reading 29.

Practice: Example 13,
Volume 5, Reading 29.

Using an Interest Rate Collar with a Floating-Rate
Loan


Reading 29

Risk Management Applications of Option Strategies

For borrower

For lender

protection against rising
interest rates but sale of
the floor results in the
borrower giving up any
gains from interest rates
falling below the
exercise rate on the
floor.

• Like Zero-cost collar in
equity options, in Zerocost interest rate collar,
first of all borrower
selects exercise rate of
the cap. Then, the floor
exercise rate is set such
that the floor premium
offsets the cap premium
so that there is no initial
outlay.

protection against
falling interest rates but
sale of the cap results in
the lender giving up any
gains from interest rates
rising above the
exercise rate on the
cap.
• Like Zero-cost collar in
equity options, in Zerocost interest rate collar,
first of all lender selects
exercise rate of the
floor. Then, the cap
exercise rate is set such
that the cap premium
offsets the floor premium
so that there is no initial
outlay.

• Typically, in a collar, the call and put premiums
offset each other. However, it is not necessarily
always the case.
• Zero-cost only means that there is no upfront cash
outlay.

FinQuiz.com

• Initial cost of the hedge can be reduced by
increasing the cap exercise rate and decreasing
the floor exercise rate; this will result in a decrease
in cost of the cap and generate income from
selling the floor. However, it will expose the buyer
of collar to more interest rate risk.
• Initial cost of the hedge can be reduced by
having lower notional principal for the cap and
higher notional principal for the floor.
A collar creates a band within which the buyer’s
effective interest rate fluctuates i.e.
• The borrower will benefit when interest rate falls
and will be hurt when interest rate increases within
that range/band. This implies that the borrower will
face risk within that range.
• Change in interest rate will have no net effect
when:
a) Interest rate > cap exercise rate
b) Interest rate < floor exercise rate.

Practice: Example 15,
Volume 5, Reading 29.

Effect of exercise rate and size of notional principal on
cost of hedge: For example in case of long cap & short
floor,
4.

OPTION PORTFOLIO RISK MANAGEMENT STRATEGIES

By trading in options, dealers provide liquidity to the
market and take risk. To earn the bid-ask spread without
taking risk, dealers can hedge their positions by using
hedging strategies. For example if a dealer has sold a
call, he can hedge his/her risk either by:
i. Buying an identical call option or
ii. Buying a put with the same exercise price and
expiration, buying the asset, and selling a bond or
taking out a loan with face value equal to the
exercise price and maturity equal to that of the
option’s expiration (it refers to put-call parity). This
hedge is static in nature i.e.no change in the position
is required as time passes.
iii. Using Delta Hedging: When necessary options are not
available or are not favorably priced, then the dealer
can hedge risk by taking a long position in a certain
number of units of the underlying asset. The size of
that long position is determined using option’s delta
i.e.
Delta =

۱‫܍܋ܑܚ۾ܖܗܑܜܘ۽ܖܑ܍܏ܖ܉ܐ‬

۱‫܍܋ܑܚ۾܏ܖܑܡܔܚ܍܌ܖ܃ܖܑ܍܏ܖ܉ܐ‬

=

∆۱
∆‫܁‬

• Delta is used to measure the sensitivity of the price
of an option to changes in the price of the
underlying asset.

• The delta usually lies between 0 and 1.
o Delta will be 1 only at expiration and only if the
option expires in-the-money.
o During the option’s life, if the option is in-themoney, delta will tend to be above 0.5.
o As expiration approaches, the deltas of in-themoney options will move slowly towards 1.0.
o Delta will be 0 only at expiration and only if the
option expires out-of-the money.
o During the option’s life, if the option is out-of-themoney, delta will tend to be below 0.5.
o As expiration approaches, the deltas of out-ofthe-money options will move slowly toward 0.
o Delta moves quickly towards 1 or 0 when delta is
at-the-money and/or near expiration.
o 0.5 is often viewed as an “average” delta.
o For calls: delta lies between 0 and 1.
o For puts: delta lies between -1 and 0.
NOTE:
The deltas of options that are very slightly in-the-money
will temporarily move down as expiration approaches.
But eventually they will move up towards 1.0.
How to determine size of the Long position: Delta can be

used to determine how many units of the underlying are


Reading 29

Risk Management Applications of Option Strategies

needed to offset the risk associated with short position in
option.
Nc / Ns = - 1 / (∆C / ∆S) = -1 / Delta
where,
Nc = Number of call options
Ns = Number of units of the underlying e.g. stocks
For example, if the dealer sells 100 calls, it will need to
own number of shares = 100 × (Delta).
• The loss on the underlying may be offset by the
gain on the options.
Rule: Buy (sell) delta shares for each option short (long).
Three complicating issues in delta hedging:
1) Delta represents only an approximate change in
the call price for a (small) change in the
underlying. Delta is not a perfect hedge because
(particularly for calls), the delta underestimates
the effects of increases in the underlying and
overestimates the effects of decreases in the
underlying.
2) Delta changes with the change in the price of the
underlying and/or time. The greater the change
in the price of the underlying, the worse the deltabased approximation.
• Increase in the underlying price leads to an

increase in delta.
• Delta decreases as time passes.
• Usually, effect of changes in underlying price
dominates the effect of time.
3) Rounding off the number of units of the
underlying per option results in a small amount of
imprecision in the balancing of the two offsetting
positions.
• When we round up, we have more units of the
underlying than needed. It negatively affects
the hedged position when the price of
underlying decreases.
• When we round down, we have fewer units of
the underlying than needed. It negatively
affects the hedged position when the price of
underlying increases.
4.1

Delta Hedging an Option over Time

A delta-hedged position needs to be rebalanced
whenever the underlying price changes and/or with the
passage of time. It is referred to as Dynamic hedging.
Guidelines for Perfectly hedged Short-call and long
underlying Portfolio:
• With the passage of time, both the delta & value
of short-call position decrease. In this situation (all
else equal), the dealer should sell shares of stock
and invest the proceeds at risk-free rate. Thus,

FinQuiz.com

value of portfolio will grow at risk-free rate over
time.
• A small decrease in the underlying price results in
decrease in delta; however, there would be no
change in the value of the portfolio. In this case,
dealer should sell some shares of stock and invest
the proceeds at risk-free rate.
• A small increase in the underlying price results in
increase in delta; however, there would be no
change in the value of the portfolio. In this case,
dealer should buy some shares of stock by
borrowing the required amount at risk-free rate.
• Delta of in-the-money call option will increase
towards 1 near expiration; whereas, delta of outof-the-money call option will decrease towards 0
near expiration.
Example:
Suppose a call option in which:
•
•
•
•
•
•
•

Underlying price = $1210.
Exercise price = $1200.
Continuously compounded risk-free rate = 2.75%.

Expiration time = 120 days.
Option price = $65.88
Delta = 0.5826
Number of call options sold = 1000
a) If underlying price changes to $1200, new
option price is computed as follows:
Option price + (∆S × delta) = 65.88 + (1200 –
1210) (0.5826) = 60.05
b) Number of shares of stock needed to delta
hedge = delta × number of options sold = 0.5826
× 1000 = 583
c) Value of portfolio = 583 ($1210) – 1,000 ($65.88)
= $639,550
• Thus, initially, we need to invest $639,550 to
delta hedge.
d) One day later, this amount should grow at a riskfree rate i.e. it should be = $639,550 exp (0.0275
/ 365) = $639,598. Thus, our benchmark value =
$639,598.

Suppose after 1 day,
•
•
•
•
•
•

Underlying price = $1215.
Exercise price = $1200.
Continuously compounded risk-free rate = 2.75%.

Expiration time = 119 days.
New option price = $68.55
New Delta = 0.5966
a) Now value of portfolio = 583 ($1215) – 1,000
($68.55) = $639,795.
b) Compare this amount with the benchmark
value: The amount we have in excess of
benchmark value = $639,795 - $639,598 = $197.
c) Number of shares of stock needed to delta
hedge = delta × number of options sold = 0.5966
ì 1000 = 597.
ã We have 583 shares, so now we need to buy
14 more shares (i.e. 597 – 583).
• To buy additional 14 shares, we need = 14 ×


Reading 29

Risk Management Applications of Option Strategies

$1215 = $17,010.
d) We will borrow $17,010 at risk-free rate.
e) Now value of portfolio is still = 597 ($1215) – 1,000
($68.55) - $17,010 = $639,795.
• This implies that, we cannot create or destroy
any wealth by just rearranging the position.
e) One day later, this amount should grow at a riskfree rate i.e. it should be = $639,795 exp (0.0275
/ 365) = $639,843. Thus, our new benchmark
value = $639,843.
f) The loan will grow at a risk-free rate to = $17,010

exp (0.0275 / 365) = $17,011.
Now suppose next day,
•
•
•
•
•

Underlying price = $1198.
Exercise price = $1200.
Continuously compounded risk-free rate = 2.75%.
New option price = $58.54
New Delta = 0.5479
g) New portfolio value = 597 ($1198) – 1,000
($58.54) - $17,011* = $639,655.
•
*The loan will grow to $17,010
exp(0.0275/365) = $17,011
h) Compare this amount with the benchmark
value i.e. = $639,655 - $639,843 = -$188.
f) Number of shares of stock needed to delta
hedge = delta × number of options sold = 0.5479
× 1000 = 548.
• We have 597 shares, so now we need to sell
49 shares (i.e. 597 – 548).
• By selling 49 shares, we will generate cash
inflow = 49 × $1198 = $58,702.
• Now we can pay back our debt i.e. $58,702 $17,011 = $41,691.
• This amount can be invested at risk-free rate.
i) New portfolio value is still = 548 ($1198) – 1,000

($58.54) + $41,691 = $639,655.
j) After one day, value of portfolio should grow at
a risk-free rate to $639,655 exp (0.0275 / 365) =
$693,703. Thus, our new benchmark value =
$693,703.
Source: Curriculum, Reading 29, Section 4.1.

Example:
Dealer has sold 500 call options on a stock currently
priced at $125.75, 60 days until expiration, a price of
$10.89 and a delta of 0.5649. Note that risk-free rate is
4%.
• Number of shares of stock needed to delta hedge
500 call options = 0.5649 × 500 = 282.45.
Suppose after 1 day, delta changes to 0.6564.
• Number of shares of stock needed to delta hedge
500 call options = 0.6564 × 500 = 328.20 ≈ 328.

FinQuiz.com

Suppose after 2 days, the following information applies:
•
•
•
•
•
•

Stock price = $122.75
Option price = $9.09

Delta = 0.5176
Number of options sold = 500
Number of shares purchased = 328
Bond Balance = -$6,072

When stock price = $122.75, stocks worth = $122.75 × 328
= $40,262.
When option price = $9.09, options worth = $9.09 ì (-500)
= -$4,545.
Bond balance (given) = -$6,072
ã Total market value of dealer’s total position =
$40,262 + (-$4,545) + (-$6,072) = $29,645.
• When delta = 0.5176, number of shares of stock
needed to delta hedge 500 call options = 0.5176 ×
500 = 258.80 ≈ 259.
• Dealer now needs 259 shares instead of 328; thus,
he must sell = 328 – 259 = 69 shares.
• Selling those 69 shares will generate = 69 ($122.75)
= $8,470.
• Dealer invests $8,470 in risk-free bonds.
• Since the bond balance was -$6,072, this amount
can be used to pay-off this debt i.e. $8,470 - $6,072
= $2,398 will be excess amount that can be
invested in risk-free bonds.
Now the total market value of dealer’s total position is
calculated as follows:
• When stock price = $122.75, stocks worth = $122.75
× 259 = $31,792.
• When option price = $9.09, options worth = $9.09 ×
(-500) = -$4,545.

• Bond balance (given) = $2,398
• Total market value = $31,792 + $4,545 + $2,398 =
$29,645.
Benchmark value = $29,645 exp (0.04 / 365) = $29,648
Value of bond after 1 day = $2,398 exp (0.04 / 365)
= $2,398
Now suppose after 3 days, the following information
applies:
Stock price = $120.50
Option price = $7.88
• When stock price = $120.50, stocks worth = $120.50
ì 259 = $31,210.
ã When option price = $7.88, options worth = $7.88 ì
(-500) = -$3,940.
ã Bond balance (given) = $2,398
• Total market value = $31,210+ (-$3,940) + $2,398 =
$29,668.
• Market value ($29,668) > benchmark value


Reading 29

Risk Management Applications of Option Strategies

($29,648) i.e. market value is $20 more than the
benchmark.
Source: Curriculum, Reading 29, example 16.

NOTE:
Delta hedges are most difficult to maintain for at-themoney options and/or near expiration.

Hedging using non-identical option:
Suppose two options are available on the same
underlying but are not identical i.e. they differ by
exercise price, expiration, or both.
• One option has a delta of ∆1.
• Other option has a delta of ∆2.
The value of the position is:
V = N1 c1 + N2c2

FinQuiz.com

• For a perfect delta hedge, the return to the delta
hedge would be the risk-free rate and gamma = 0.
Note that gamma would be zero when delta
would represent a perfectly linear slope i.e. ∆c =
∆S.
• When gamma = 0, dealer does not need to adjust
the delta hedge.
• When delta does not represent a perfectly linear
slope, gamma ≠ 0.
• The larger the gamma, the more the deltahedged position deviates from being risk free and
greater will be the risk. In this case, dealer should
gamma hedge his/her position.
• Gamma is largest for at-the-money options and/or
near expiration.
Gamma hedge involves adding a position in another
option so that both delta and gamma are zero i.e.
Gamma hedge = Position in underlying + Positions in two
options

where,
N1 and N2 represent the quantity of each option in a
portfolio that hedges the value of one of the options in a
portfolio.
c1 = price of option 1
c2 = price of option 2
To delta hedge, we get:

4.3

Vega and Volatility Risk

An option price is very sensitive to the changes in
volatility of the underlying. Vega is used to measure the
sensitivity of the option price to the volatility.
Vega =

େ୦ୟ୬୥ୣ୧୬୓୮୲୧୭୬୮୰୧ୡୣ
େ୦ୟ୬୥ୣ୧୬୚୭୪ୟ୲୧୪୧୲୷୭୤୲୦ୣ୳୬ୢୣ୰୪୷୧୬୥

Desired Quantity of option 1 relative to option 2 =
ୈୣ୪୲ୟ୭୤୭୮୲୧୭୬ଶ
ୈୣ୪୲ୟ୭୤୭୮୲୧୭୬ଵ

N1 / N2 = - ∆c2 / ∆c1
• It is known as Ratio Spread.
• The negative sign indicates that a long position in
one option will require a short position in the other.
• These deltas will change and will require
monitoring and modification of the position over

time.
4.2

Gamma and the Risk of Delta

Gamma measures the sensitivity of the delta to a
change in the underlying. In effect, it is the delta of the
delta.
େ୦ୟ୬୥ୣ୧୬ୢୣ୪୲ୟ
Gamma =
େ୦ୟ୬୥ୣ୧୬୳୬ୢୣ୰୪୷୧୬୥୮୰୧ୡୣ

• Like delta, gamma represents only an
approximate change in the delta for a (small)
change in the underlying.

• Like delta and gamma, Vega represents only an
approximate change in the option price for a
(small) change in the volatility.
• Price of at-the-money option has greater sensitivity
to changes in volatility.
• Volatility is the most critical variable because it is
the only unobservable variable. Thus, it is difficult to
estimate Vega.
• Value of delta-hedged position with a zero or
insignificant gamma can change by a large
amount when the volatility changes e.g. if dealer
has sold options to delta hedge a long position in
the underlying, then an increase in volatility leads
to increase in value of options and results in large

loss for the dealer.
How to manage Vega Risk: Vega risk cannot be
managed independently. The dealer is required to jointly
monitor and manage the risk associated with the delta,
gamma and Vega.


Reading 29

Risk Management Applications of Option Strategies

5.

FinQuiz.com

FINAL COMMENTS

Equity options v/s Bond options: Equity option strategies
apply similarly to bond option strategies. However, the
major difference is that the bond options must expire
before the bonds mature.
Interest rate options

Equity or Bond
Options

1) Bullish (bearish) investors buy
puts (calls) on interest rates.
Because being bullish
(bearish) on interest

rates means that
investor expects
decrease (increase)
in interest rates.
2) Interest rate options pay-offs
represent interest payments.
3) Interest rate options can be
used for hedging purposes
e.g. caps and floors can be
used for hedging floating-rate
loans.

1) Bullish (bearish)
equity or bond
investors buy calls
(puts).
2) Equity or bond
options pay-offs
represent sale or
purchase of
stocks or bonds
by the option
buyer.
3) Standard option
strategies i.e.
straddles and
spreads can be
used by investors
for the purpose of
hedging.

NOTE:
Interest rate swaps represent the most widely used
financial derivative by investors. However, they are less
widely used with currencies and equities when
compared to forwards, futures and options.

Practice: End of Chapter Practice
Problems for Reading 29 & FinQuiz
Item-set ID# 9280.