Download Lecture Notes_Chapter 3

Chapter 3: Insurance, Collars,
and Other Strategies
FINA0301 Derivatives
Faculty of Business and Economics
University of Hong Kong
Dr. Huiyan Qiu
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Chapter Outline
Basic insurance strategies:
• insuring a long position: floors;
• insuring a short position: caps;
• selling insurance
Synthetic forwards: put-call parity
Spreads and collars: bull and bear spreads;
box spreads; ratio spreads; collars
Speculating on volatility: straddles;
strangles; butterfly spreads; asymmetric
butterfly spreads
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Long / Short Call / Put Options
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Basic Insurance Strategies
Insurance strategies using options:
• Used to insure long positions
• Buying put options
• Used to insure short positions
• Buying call options
• Written against asset positions (selling insurance)
• Covered call
• Covered put
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Insuring a Long Position
A long position in the underlying asset combined
with a put option
Goal: to insure against a fall in the price of the
underlying asset
At time 0
• Buy one stock at cost S0 (long position in the asset)
• Buy a put on the stock with a premium p
An insured long position (buy an asset and a put)
looks like a call!
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Example: S&R index and a S&R put option
with a strike price of $1,000 together
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Combined Payoff / Profit
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Protective Puts
The portfolio consisting of a long asset position
and a long put position is often called
“Protective Put”.
Protective puts are the classic “insurance” use of
options.
The protective put in the portfolio ensures a floor
value (strike price of put) for the portfolio. That
is, the asset can be sold for at least the strike
price at expiration.
Varying the strike price varies the insurance
cost.
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Insuring a Short Position
A call option is combined with a short position in
the underlying asset
Goal: to insure against an increase in the price
of the underlying asset
At time 0
• Short one stock at price S0
• Buy a call on the stock with a premium c
An insured short position (short an asset and
buy a call) looks like a put
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Example: short-selling the S&R index and holding
a S&R call option with a strike price of $1,000
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Combined Payoff / Profit
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Selling Insurance
For every insurance buyer there must be an
insurance seller
Naked writing is writing an option when the
writer does not have a position in the asset
Covered writing is writing an option when
there is a corresponding position in the
underlying asset
• Write a call and long the asset
• Write a put and short the asset
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Covered Writing
Covered calls: write a call option and hold the
underlying asset. (The long asset position
“covers” the writer of the call if the option is
exercised.)
• A covered call looks like a short put
Covered puts: write a put option and short the
underlying asset
• A covered put looks like a short call
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Covered Writing: Covered Calls
Example: holding the S&R index and writing a S&R
call option with a strike price of $1,000
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Combined Payoff / Profit
Writing a covered call generates the same profit as selling a put!
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Covered Writing: Covered Puts
Example: shorting the S&R index and writing a S&R
put option with a strike price of $1,000
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Combined Payoff / Profit
Writing a covered put generates the same profit as writing a call!
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Insurance vs. Pure Option Position
Buying an asset and a put generates the same
profit as buying a call
Short-selling an asset and buying a call generates
the same profit as buying a put
Writing a covered call generates the same profit as
selling a put
Writing a covered put generates the same profit as
selling a call
How to make the positions equivalent?
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Insurance vs. Pure Option Position
To make positions equivalent, borrowing or lending
has to be involved. Following table summarizes the
equivalent positions.
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Options Combined
Underlying asset: S&R Index, spot price = $1,000
6-month Forward: forward price = $1,020
6-month 1,000-strike call: call premium = $93.81
6-month 1,000-strike put: put premium = $74.20
Effective interest rate over 6 month = 2%
Positions: long call + short put
• Time-0 cash flow: – 93.81 + 74.20 = – 19.61
• What happens 6 months later?
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Long Call + Short Put
Outcome at expiration: pay the strike price of
$1,000 and own the asset
ST >1000
ST < 1000
Pay 1000, get asset
(ST – 1000)
Nothing (0)
Short Put
Nothing (0)
Pay 1000, get asset
(ST – 1000)
Total
Pay 1000, get asset
(ST – 1000)
Pay 1000, get asset
(ST – 1000)
Long Call
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Synthetic Forwards
A synthetic long forward contract: buying a call and
selling a put on the same underlying asset, with each
option having the same strike price and time to
expiration
Example: buy the $1,000strike S&R call and sell
the $1,000-strike S&R
put, each with 6 months
to expiration
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Synthetic Forwards (cont’d)
Both synthetic long forward contract and actual
forward contract result in owning the asset at
the expiration.
Differences
• The forward contract has a zero premium, while
the synthetic forward requires that we pay the net
option premium
• With the forward contract, we pay the forward
price, while with the synthetic forward we pay the
strike price
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Put-Call Parity
The net cost of buying the index using options
(synthetic forward contract) must equal the net
cost of buying the index using a forward contract
Call (K, t) – Put (K, t) = PV (F0,t – K)
• Call (K, t) and Put (K, t) denote the premiums of
options with strike price K and time t until
expiration
• PV (F0,t ) is the present value of the forward price
This is one of the most important relations in
options!
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More Option Strategies
Combined option positions can be taken to
speculate on price direction or on volatility.
Speculating on direction: bull and bear
spreads; ratio spreads; collars
Speculating on volatility: straddles; strangles;
butterfly spreads; asymmetric butterfly spreads
Box spread
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Underlying Asset and Options
Underlying asset: XYZ stock with current stock
price of $40
8% continuous compounding annual interest rate
Prices of XYZ stock options with 91 days to
expiration:
Strike
Call
Put
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6.13
0.44
40
45
2.78
0.97
1.99
5.08
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Bull Spreads
A bull spread is a position with the following
profit shape.
It is a bet that the
price of the underlying
asset will increase,
but not too much
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Bull Spreads (cont’d)
A bull spread is to buy a call/put and sell an
otherwise identical call/put with a higher strike
price
Bull spread using call options:
• Long a call with no downside risk, and
• Short a call with higher strike price to eliminate
the upside potential
Bull spread using put options:
• Short a put to sacrifice upward potential, and
• Long a put with lower strike price to eliminate the
downside risk
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Bull Spread with Calls
Long a call (strike price K1, premium c1)
Value = 0
when ST ≤ K1
= – K1 + [1]ST when ST > K1
Value
K2
Short a call (K2 > K1, c2 < c1)
Value = 0
when ST ≤ K2
= K2 + [-1]ST when ST > K2
K2-K1
FV(-c1+c2)
-K1
K1
K2
ST
Portfolio
Value = 0
= –K1+ [1]ST
= K2 – K1
when ST ≤ K1
when K1<ST ≤ K2
when ST >K2
c1 > c2
Initial cash flows = – c1 + c2 <0
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Bear Spreads
A bear spread is a position in which one sells a
call (or a put) and buys an otherwise identical call
(or put) with a higher strike price. Opposite of a
bull spread.
• Example: short 40-strike call and long 45-strike put
It is a bet that the price of the underlying asset
will decrease, but not too much
• Option traders trading bear spreads are moderately
bearish on the underlying asset
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Ratio Spreads
A ratio spread is constructed by buying a
number of calls ( puts) and selling a different
number of calls (puts) with different strike price
Figure: profit diagram of a
ratio spread constructed by
buying a low-strike call and
selling two higher-strike calls.
Limited profit and unlimited
risk. To bet that the stock will
experience little volatility.
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Collars
A collar is a long put combined with a short call with
higher strike price
• It resembles a short forward with a flat middle
• To bet that the price of the underlying asset will
decrease significantly
A zero-cost collar
can be created when
the premiums of the
call and put exactly
offset one another
Long 40-strike put and
short 45-strike call
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Box Spreads
A box spread is accomplished by using options to
create a synthetic long forward at one price and a
synthetic short forward at a different price
Synthetic long forward: long a call and short a
put with the same strike price
The combination of payoff diagrams of a synthetic
long forward and a synthetic short forward is a
horizontal line.
A box spread is a means of borrowing or lending
money. It has no stock price risk!
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Speculating on Volatility
Non-directional speculations:
• Straddles
• Strangles
• Butterfly spreads
• Asymmetric butterfly spreads
Who would use non-directional positions?
• Investors who have a view on volatility but are
neutral on price direction
• Speculating on volatility
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Straddles
Buying a call and a put with the same strike
price and time to expiration
Figure
Combined profit
diagram for a
purchased 40strike straddle.
A straddle is a bet that volatility will be high
relative to the market’s assessment
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Strangles
Buying an out-of-the-money call and put with the same
time to expiration
Figure 40-strike
straddle and
strangle composed
of 35-strike put and
45-strike call.
A strangle can be used to reduce the high premium
cost, associated with a straddle
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Written Straddles
Selling a call and put with the same strike price
and time to maturity
Figure Profit at
expiration from a
written straddle:
selling a 40-strike
call and a 40-strike
put.
A written straddle is a bet that volatility will be
low relative to the market’s assessment
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Butterfly Spreads
A butterfly spread is = write a straddle + add a
strangle = insured written straddle
Figure Written
40-strike straddle,
purchased 45strike call, and
purchased 35strike put.
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Butterfly Spread
Value
Sell a call (Strike price K2, premium c2)
Sell a put (Strike price K2, premium p2)
Written straddle
Value = –K2 + [1]ST
= K2 + [-1]ST
when ST ≤ K2
when ST >K2
Long a put (Strike price K1<K2,premium p1)
0
K3
K1
K2
Long a call (Strike price K3>K2>k1, c3)
ST
Long strangle
Value = K1 + [-1]ST
=0
= –K3 + [1]ST
when ST ≤ K1
when K1<ST ≤ K3
when ST > K3
Butterfly spread (K3 – K2 = K2 – K1)
Value = –K2 + K1
when ST ≤ K1
= –K2 + [1]ST when K1<ST ≤ K2
= K2 + [-1]ST when K2<ST ≤K3
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= –K3 + K2
when ST >K3
Butterfly Spread
Value
Initial option costs = c2 + p2 – p1 – c3
= (c2 – c3) + (p2 – p1)
>0
0
K1
K3
K2
A butterfly spread is an
insured written straddle.
Can be used to bet for
low volatility.
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Asymmetric Butterfly Spreads
By trading unequal units of options
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Summary of Various Strategies
Option strategy positions driven by the view on
the stock price and volatility directions.
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End of the Notes!
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