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Insurance, Collars, and Other Strategies

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Chapter OutlineBasic 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

We assumed an index price of $1000, a 2% effective 6-month interest rate, and premiums of $93.809 for the 1000-strike 6-month call and $74.201 for the 1000-strike 6-month put.

Payoff and profit at expiration from purchasing the S&R index and a 1000-strike put option. Payoff is the sum of the first two columns. Cost plus interest for the position is ($1000 + $74.201) × 1.02 = $1095.68. Profit is pay off less $1095.68.

For example, if the index is $900 at expiration, we have

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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 PositionA 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

The level of the floor is −$95.68, which is the lowest possible profit.

Panel (a) shows the payoff diagram for a long position in the index (column 1 in Table ).

Panel (b) shows the payoff diagram for a purchased index put with a strike price of $1000 (column 2 in Table).

Panel (c) shows the combined payoff diagram for the index and put (column 3 in Table).

Panel (d) shows the combined profit diagram for the index and put, obtained by subtracting $1095.68 from the payoff diagram in panel (c) (column 5 in Table).

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. 10

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Insuring a Short PositionA 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 InsuranceFor 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

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

Long CallPay 1000, get asset

(ST – 1000) Nothing (0)

Short Put Nothing (0)Pay 1000, get asset

(ST – 1000)

TotalPay 1000, get asset

(ST – 1000)Pay 1000, get asset

(ST – 1000)

Synthetic Forwards

A synthetic long call is created when long stock position is combined with a long put of the same series. It is so named because the established position has the same profit potential as a long call.

Married put and protective put strategies are examples of synthetic long calls.

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Synthetic ForwardsA 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 $1,000-strike 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 ParityThe 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!

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:

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Strike Call Put

35 6.13 0.44

40 2.78 1.99

45 0.97 5.08

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Bull SpreadsA 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

PortfolioValue = 0 when ST ≤ K1

= –K1+ [1]ST when K1<ST ≤ K2= K2 – K1 when ST >K2

K2-K1

Long a call (strike price K1, premium c1)Value = 0 when ST ≤ K1

= – K1 + [1]ST when ST > K1

-K1

K1

Short a call (K2 > K1, c2 < c1)Value = 0 when ST ≤ K2

= K2 + [-1]ST when ST > K2

K2

K2

ST

Value

c1 > c2Initial cash flows = – c1 + c2 <0

FV(-c1+c2)

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Bear SpreadsA 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 SpreadsA 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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CollarsA 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 SpreadsA 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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StraddlesBuying a call and a put with the same strike price and time to expiration

A straddle is a bet that volatility will be high relative to the market’s assessment

Figure Combined profit diagram for a purchased 40-strike straddle.

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StranglesBuying an out-of-the-money call and put with the same time to expiration

A strangle can be used to reduce the high premium cost, associated with a straddle

Figure 40-strike straddle and strangle composed of 35-strike put and 45-strike call.

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Written StraddlesSelling a call and put with the same strike price and time to maturity

A written straddle is a bet that volatility will be low relative to the market’s assessment

Figure Profit at expiration from a written straddle: selling a 40-strike call and a 40-strike put.

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Butterfly SpreadsA butterfly spread is = write a straddle + add a strangle = insured written straddle

Figure Written 40-strike straddle, purchased 45-strike call, and purchased 35-strike put.

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Butterfly Spread

ST

Value

K2

Sell a call (Strike price K2, premium c2)

Sell a put (Strike price K2, premium p2)

Written straddleValue = –K2 + [1]ST when ST ≤ K2

= K2 + [-1]ST when ST >K2

Long a put (Strike price K1<K2,premium p1)

Long a call (Strike price K3>K2>k1, c3)

Long strangle Value = K1 + [-1]ST when ST ≤ K1

= 0 when K1<ST ≤ K3= –K3 + [1]ST 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= –K3 + K2 when ST >K3

0 K1 K3

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Butterfly SpreadValue

K1K2

K3

Initial option costs = c2 + p2 – p1 – c3= (c2 – c3) + (p2 – p1) > 0

0 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.