2013, Study Session # 15, Reading # 37

“RISK MANAGEMENT APPLICATIONS OF OPTION STRATEGIES”
IR = Interest Rate
NP = Notional Principal

2.2 Risk Management Strategies with Options and the Underlying

An investor can exposure without selling the underlying
through:
Covered call.
Protective put.

2.2.1 Covered Calls

Long stock + short call.
Appropriate when stock price neither nor in near future.
Limited upside potential & downside protection.
Reduces both overall risk & the expected return.
்ܸ = ்ܵ − ‫ݔܽܯ‬ሺ0, ்ܵ − ܺሻ
ܲ‫ ்ܸ = ݐ݂݅݋ݎ‬− ܵ଴ + ‫ܥ‬଴
‫ܵ = ݏݏ݋݈ݔܽܯ‬଴ − ‫ܥ‬଴
‫ܵ = ݊݁ݒ݁݇ܽ݁ݎܤ‬଴ − ‫ܥ‬଴
the X, lower the option premium.

2.2.2 Protective Puts

Long stock + long put.
Provide protection against in value (similar to insurance).
Requires upfront option cost.
Appropriate when an investor expects a in value of the stock in

near future.
்ܸ = ்ܵ + ‫ݔܽܯ‬ሺ0, ܺ − ்ܵ ሻ
Profit =்ܸ − ܵ଴ − ܲ଴ &‫∞ = ݐ݂݅݋ݎ݌ݔܽܯ‬.
Max loss =ܵ଴ + ܲ଴ − ܺ.
Break even = ܵ଴ + ܲ଴

2.3 Money Spreads

Spread ⇒ strategy which involves buying one option & selling
another identical option but either with different X or different
time to expiration.
Time spread ⇒ different time to expiration.
Money spread ⇒ different exercise price.

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2013, Study Session # 15, Reading # 37

2.3.1 Bull Spreads

Buying a call with a lower X & selling another with X.
Rationale ⇒ when investor expects an in stock price in the near
future.
Similar to covered call it provides protection against downside risk
& limited upside potential.
ܸ଴ = ‫ܥ‬ଵ − ‫ܥ‬ଶ
Where ‫ܥ‬ଵ &‫ܥ‬ଶ are option premiums for the lower X & higher X

respectively.
்ܸ = Value of long call - value of short call.
Profit =்ܸ = ‫ܥ‬ଵ + ‫ܥ‬ଶ
Max profit = ܺଵ − ܺଶ − ‫ܥ‬ଵ + ‫ܥ‬ଶ
Max loss = ‫ܥ‬ଵ −‫ܥ‬ଶ
Breakeven = ܺଵ + ‫ܥ‬ଵ − ‫ܥ‬ଶ

Bull Put Spread

Buys a put with a lower X & sells an identical put with a higher X.
Cash inflows at initiation of the position.
Identical to the sale of bear put spread.
Profit occurs when both put options expires out-of-the money.

2.3.2 Bear Spreads

Bear Put Spread

Long position in a put with X & short position in a put with a X.
Rationale ⇒ investor expects that stock price will in the future.
ܸ଴ = ܲଶ − ܲଵ
Where
P2 = put premium on higher X.
்ܸ = value of long put - value of short put.
Profit = ்ܸ − ܲଶ + ܲଵ
Max profit = ܺଶ − ܺଵ − ܲଶ + ܲଵ
Max loss = ܲଶ − ܲଵ
Breakeven = ܺଶ − ܲଶ + ܲଵ

Bear Call Spread


Investor sells a call with a lower X & buys an otherwise identical call
with a higher X.
Investor will earn net premium when both call options expire outof-the money.
Identical to the sale of a bull call spread.

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2013, Study Session # 15, Reading # 37

2.3.3 Butterfly Spreads

Long Butterfly Spread

Long bull call spread + short bull call spread.
Require cash outlay at initiation because bull spread purchased by an investor is expensive
than a bull spread sold.
Rationale ⇒ useful when investor expects that the volatility of underlying will be low
relative to market expectations.
்ܸ = ‫ݔܽܯ‬ሺ0, ்ܵ − ܺଵ ሻ − 2‫ݔܽܯ‬ሺ0, ்ܵ − ܺଶ ሻ + ‫ݔܽܯ‬ሺ0, ்ܵ − ܺଷ ሻ
ܲ‫ ்ܸ = ݐ݂݅݋ݎ‬− ‫ܥ‬ଵ + 2‫ܥ‬ଶ − ‫ܥ‬ଷ
‫ܺ = ݐ݂݅݋ݎ݌ݔܽܯ‬ଶ − ܺଵ − ‫ܥ‬ଵ + 2‫ܥ‬ଶ − ‫ܥ‬ଷ
‫ܥ = ݏݏ݋݈ݔܽܯ‬ଵ − 2‫ܥ‬ଶ + ‫ܥ‬ଷ
Two breakeven points.
ܺଵ + ‫ܥ‬ଵ − 2‫ܥ‬ଶ + ‫ܥ‬ଷ
2ܺଶ + ܺଵ − ‫ܥ‬ଵ + 2‫ܥ‬ଶ − ‫ܥ‬ଷ


Short Butterfly Spread

Selling the calls with ܺଵ &ܺଷ & buying two calls with ܺଶ
Rationale ⇒ preferable when investor expects that the volatility of the underlying will be
relatively high compared to market expectations.

Long Butterfly Spread (using puts)

Long bear put spread + short bear put spread.
Cost of ܺଵ ሺܲଵ ሻ < ܿ‫݂ܺ݋ݐݏ݋‬ଶ ሺܲଶ ሻ < ܿ‫݂ܺ݋ݐݏ݋‬ଷ ሺܲଷ ሻ

Long Butterfly Spread (using puts)

Selling the puts with ܺଵ &ܺଷ & buying two puts with ܺଶ
Max profit = ܲଷ + ܲଵ − 2ܲଶ
If correctly priced, butterfly spread using calls will
provide the same result as butterfly using puts.

2.4 Combinations of calls and Puts

2.4.1 Collars

Strategy in which cost of buying put option can be reduced by selling a call
option.
Provide downside protection at the expense of giving up upside potential.
Zero cost collar ⇒ call option premium is equal to put option premium.
Put X & call X results is in both the upside & downside potential.
Quite similar to bull spread with respect to cap on gains & a floor on loss but
no underlying holdings in bull spread.
ܸ଴ = ܵ଴

்ܸ = ்ܵ + ‫ݔܽܯ‬ሺ0, ܺଵ − ்ܵ ሻ − ‫ݔܽܯ‬ሺ0, ்ܵ − ܺଶ ሻ
ܲ‫ ்ܸ = ݐ݂݅݋ݎ‬− ܸ଴
‫ܺ = ݐ݂݅݋ݎ݌ݔܽܯ‬ଶ − ܵ଴
‫ܵ = ݏݏ݋݈ݔܽܯ‬଴ − ܺଵ
Breakeven = ܵ଴
Collars are also known as range forwards & risk reversals.

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2013, Study Session # 15, Reading # 37

2.4.2 Straddle

Long Straddle

Buying at-the-money put & a call with same X on same underlying &
expiration.
Rationale ⇒ investor expects volatility than what market expects.
Costly strategy.
்ܸ = ‫ݔܽܯ‬ሺ0, ்ܵ − ܺሻ + ‫ݔܽܯ‬ሺ0, ܺ − ்ܵ ሻ
ܲ‫ ்ܸ = ݐ݂݅݋ݎ‬− ܲ଴ − ‫ܥ‬଴
‫ &∞ = ݐ݂݅݋ݎ݌ݔܽܯ‬max ݈‫ܲ = ݏݏ݋‬଴ +‫ܥ‬଴
Breakeven ܺ ± ሺܲ଴ +‫ܥ‬଴ ሻ

Short Straddle

Selling a put & a call with same X on the same underlying with the same

expiration.
Preferable when neutral view about volatility.
Unlimited loss potential.
This strategy gains when both the options expires out-of-the money.

Variation of Straddle

Adding cal (put) to a straddle is known as strap (strip).
Long strangle ⇒ variation of the straddle (buying put &
calls with different (X).
Short strangle ⇒ selling the put & call with different X.

2.4.3 Box Spread

Bull spread + bear spread

Long Box Spread

Long call with ܺଵ & short cal with ܺଶ + long put with ܺଶ & short put withܺଵ .
If options are correctly priced, the box spread payoff is always RF (riskless strategy).
PV of the payoff discounted at RF should be equal to initial outlay.
ܸ଴ = ‫ܥ‬ଵ − ‫ܥ‬ଶ + ܲଶ − ܲଵ
்ܸ = ܺଶ − ܺଵ
Profit & Max profit = ܺଶ − ܺଵ − ሺ‫ܥ‬ଵ − ‫ܥ‬ଶ + ܲଶ − ܲଵ ሻ
No breakeven & max loss.
Short box is also possible with opposite positions.
Benefits of box spread:
To exploit an arbitrage opportunity.
Does not require a volatility estimate.
Hold lower transaction costs.


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2013, Study Session # 15, Reading # 37

3. INTEREST RATE OPTION STRATEGIES

IR call & put options are used to protect against IR.
IR call option pay-off = N.P × Max (0, underlying rate at expiration –exercise rate)
× Days in underlying rate/360.
180 day LIBOR can be used as the underlying rate & underlying days could
be 180, 182 183 etc.
Rate is determined on the day when option expires & payment is made m
days later.
IR put option pay-off =NP × Max (0, X – underlying rate at expiration) × days in
underlying rate/360.

3.1 Using Interest Rate Calls with Borrowing

Used by borrowers to manage IR risk on floating rate loans.
Consider the following factors:
Option expiration date is the same as when loan starts.
Option pay-offs must occur at the time when borrower makes IR payments
on loan.

3.2 Using Interest Rate Puts with Lending


Used by lender to manage IR risk on
floating rate loans.

3.3 Using an Interest Rate Cap with a Floating-Rate Loan

Interest rate cap ⇒combination of IR call options.
Each option in a cap is called a caplet.
Each caplet has same X but its own expiration date.
Cap seller makes payments if IR < strike rate.
Payoff is determined on its expiration date but made on the next payment date.
Cap pay-off = NP × (0, LIBOR on previous reset date – X) X days in settlement
period / 360.
Effective interest = interest due on the loan – caplet pay-off.

3.4 Using an Interest Rate Floor with a Floating-Rate Loan

Interest rate floor ⇒ combination of IR put options.
Floorlet pay-off = NP X (0,X –LIBOR on previous reset date) × days in settlement
period/360
Effective interest = interest received on the loan + floorlet pay-off.

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2013, Study Session # 15, Reading # 37

3.5 Using an Interest Rate Collar with a Floating-Rate Loan


Combination of a long (short) position in a cap & a short (long) position in a floor.
The borrower (lender) can buy a cap (floor) to protect against rising (falling) IR &
sell the floor (cap) to finance the premium paid to buy a cap.
Initial cost of the hedge can be by call exercise rate & floor exercise rate.
Cost can also be by having NP for the cap & NP for the floor.
Borrower will benefit when IR & will be hurt when IR within the collar.

4. OPTION PORTFOLIO RISK MANAGEMENT STRATEGIES

Dealers provide liquidity to the market & take risk by trading in options.
Dealers use different hedging strategies to avoid risk.
If a dealer has sold a call, he can hedge his/her risk by:
Buying an identical call option.
Buying a put with same X & expiration, buying the asset & selling a bond
(static hedge).
Using delta hedging.
Size of the long position in underlying to offset the risk associated
with short position in option = -1/ delta
Three complicating issues in delta hedging:
Delta is an approximate for small changes only.
Delta changes with the change in the price of the underlying & or
time.
Small amount of imprecision due to rounding the no. of unit of
underlying.

4.1 Delta Hedging an Option over Time

Dynamic hedging ⇒ delta-hedged position needs to be rebalanced as underlying
priced ∆ or with the passage of time.
Delta of in-the-money (out-of-the money) call option will ( ) towards 1 (0)

near expiration.
Delta hedges are most difficult to maintain for-at-the-money option & /or near
expiration.

Hedging Using Non-Identical Option

ܸ = ܰଵ ‫ܥ‬ଵ + ܰଶ ‫ܥ‬ଶ
Where
ܰଵ &ܰଶ = quantity of each option that hedges the value of one of the options in
a portfolio.
‫ܥ‬ଵ &‫ܥ‬ଶ = price of option 1&2.
desired quantity of option 1 relative to option 2:
‫݊݋݅ݐ݌݋݂݋ܽݐ݈݁ܦ‬2
−∆‫ܥ‬ଶ
ܰ
ܴܽ‫ = ݀ܽ݁ݎ݌ݏ݋݅ݐ‬
= ଵൗܰ =
ଶ
∆‫ܥ‬ଵ
‫݊݋݅ݐ݌݋݂݋ܽݐ݈݁ܦ‬1

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2013, Study Session # 15, Reading # 37

4.2 Gamma and the Risk of Delta


‫ = ܽ݉݉ܽܩ‬

∆௜௡ௗ௘௟௧௔

∆௜௡௨௡ௗ௘௥௟௬௜௡௚௣௥௜௖௘

Larger the gamma greater will be the risk.
Gamma is largest for at-the-money options & /or near expiration.
Gamma hedge ⇒ position in underlying + position in two options.

4.3 Vega and Volatility Risk

ܸ݁݃ܽ =

∆௜௡௢௣௧௜௢௡௣௥௜௖௘

∆௜௡௩௢௟௔௧௜௟௜௧௬௢௙௧௛௘௨௡ௗ௘௥௟௬௜௡௚

At-the-money option has greater sensitivity to ∆ in volatility.
Volatility is unobservable, so it is difficult to estimate Vega.
Delta is required to manage Vega risk jointly with delta & gamma.

5. FINAL COMMENTS

Major difference b/w equity & bond option strategies are that
bond options must expire before the bond mature.
Bullish (bearish) investor buys puts (calls) on IR.
Bullish (bearish) equity or bond investors buy calls (puts).

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