MFIN6003 Derivative Securities
Lecture Note Three
Faculty of Business and Economics
University of Hong Kong
Dr. Huiyan Qiu
3-1
Outline
Options as basic insurance strategies
Options and views on direction and volatility
Spreads and collars: bull and bear spreads;
ratio spreads; collars; box spreads
Speculating on volatility: straddles;
strangles; butterfly spreads; asymmetric
butterfly spreads
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Basic Insurance Strategies
Insurance strategies using options:
• Insure a long asset position
• Buy put options
• Insure a short asset position
• Buy 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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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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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 and Directional Views
Call option and put option can also be used for
speculation.
The implied market view of option position on the
direction of price movement:
Bullish on Direction
Bearish on Direction
Long Call
Long Put
Short Put
Short Call
3-15
Options and Volatility
Volatility is a measure of uncertainty in price
movements; roughly, more volatility means that
larger price swings may occur.
Options react to volatility! Option values
depend on how much uncertainty one expects in
the price of the underlying over the life of the
option.
Both long call and long put benefit from
volatility.
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Options and Volatility (cont’d)
Example: consider a call option on a stock with
strike price K = 10.
• Case 1: ST is 11 or 9 with equal probability
• Case 2: ST is 12 or 8 with equal probability
The option payoffs are:
• Case 1: CT is 1 or 0 with equal probability
• Case 2: CT is 2 or 0 with equal probability
Stock price in case 2 has the same mean but
greater volatility. Option buyer would be willing
to pay more for the higher payoff.
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Pure Option Strategies
Each option position corresponds to a unique
combination of views on market direction and
market volatility  pure option strategy
Bullish on
Direction
Bearish on
Direction
Bullish on Volatility
Long Call
Long Put
Bearish on Volatility
Short Put
Short Call
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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
Synthetic forward; 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
35
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
3-23
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
• 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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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<ST3-32
≤K3
= –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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Synthetic Forwards
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
2-37
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
2-38
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
Synthetic Forward
Actual Forward
–C+P
0
–K
–F
 – C + P – PV(K) = – PV(F)
Call (K, T) – Put (K, T) = PV (F0,T – K)
One of the most important relations in options!
2-40
Off-Market Forward
Forward by definition has a zero premium.
A forward contract with a nonzero premium
must have a forward price which is “off the
market (forward) price”. Thus, it is sometimes
called an off-market forward.
Unless the strike price equals the forward price,
buying a call and selling a put creates an offmarket forward.
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Box Spreads
A box spread is accomplished by using options to
create a synthetic long forward at one price (long
call and short put with strike price K1) and a
synthetic short forward at a different price (short
call and long put with strike price K2 ≠ K1).
At time 0, cash flow:
• – C(K1, T) + P(K1, T) + C(K2, T) – P(K2, T)
At expiration,
• Synthetic long forward: pay K1 to buy asset
• Synthetic short forward: sell asset for K2
•  fixed cash flow K2 – K1
3-42
Box Spreads
A box spread is a means of borrowing or lending
money. It has no stock price risk!
Box spread can be a source of funds.
• However, it works usually for option marketmakers only because they have relatively low
transaction costs.
Before 1993, box spreads also provided a tax
benefit for some investors in the US stock market.
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End of the Notes!
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