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Trading Strategies Involving Options

──選擇權交易策略

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Hypothesis

The underlying asset is a stock Other underlying assets can apply to similar results

The options used in the strategies are European

American option may probably be exercised early that leading to different profit outcome

Ignore the time value of money To Simplify the exposition

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Hypothesis

Initial cost premium( 權利金 )-long side margin( 保證金 )-short side

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Types of Strategies

Hedge holding single option on a stock and the stock itself

Spread taking a position in two (or more) options of the same type.

Combination Taking a position in both calls and puts on the same stock

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Hedge strategy

Covered call( 備兌買權 )

Protective call( 保護性買權 )

Protective put ( 保護性賣權 )

Covered put ( 備兌賣權 )

holding single option on a stock and the stock itself

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Put – Call Parity

S + P = C + Ke-r*T + D

S : stock price ; P : the price of put ; C : the price of call ; K : strike price / exercise price ; r : risk-free interest rate ; T : the time to maturity ; D : the present value of the dividends

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Covered call long a stock & short a callS-C=Ke-rf*T+D–P

K

Profit

ST

+S

-C

C+C

-(K-C)

K-C

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Protective callshort a stock & long a call-S+C=-Ke-rf*T-D+P

K

Profit

ST

-S

+C

C

K-C

-C

K-C

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Protective putlong a stock & long a putS+P=Ke-rf*T+D+C

K

Profit

ST

+S

+P-P

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Covered putshort a stock & short a put-S-P=-Ke-rf*T-D-C

K

Profit

ST

-S

-P+P

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Spread trading strategy

Bull Spreads( 多頭價差 ) Bear Spreads( 空頭價差 ) Box Spreads( 箱形價差 ) Butterfly Spreads( 蝶形價差 ) Calendar Spreads( 時間價差 / 行事曆價差 /…) Diagonal Spreads( 對角價差 )

taking a position in two (or more) options of the same type (i.e., two calls or two puts)

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Definition

Type: two types—call & put Option series:1.options of the same type

2.different expiration dates, the same strike price;

different strike prices, the same expiration date.

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Bull spread

Both options have the same expiration date Hoping that the stock price will be up↑ Limit both the upside profit potential and the downsi

de risk i.e. limiting both sides Three types of bull spreads can be distinguished 1. both calls are initially out of the money 2. one call is initially in the money ; the other call is initially out of the money 3. both calls are initially in the money

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Bull Spread Using CallsBuy a call with a lower strike price and sell a call with a higher strike price

Profit

STK1

+C

K2-C

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Payoff from a bull spread created using callsStock price range

Payoff from long call option

Payoff from short call option

Total payoff

ST K≧ 2 ST - K1 -(ST – K2) K2 - K1

K1 < ST < K2 ST - K1 0 ST - K1

ST K≦ 1 0 0 0

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Example

An investor buys for $3 a call with a strike price of $30 and sells for $1 a call with a strike price of $35

The profit is therefore as follows:

Stock price range Profit

ST ≦ 30 -2

30 < ST < 35 ST – 32

ST ≧ 35 3

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Bull Spread Using PutsBuy a put with a lower strike price and sell a put with a higher strike price

Profit

STK1

+P

K2

-P

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Bear Spreads

Both options have the same expiration date Hoping that the stock price will decline Limit both the upside profit potential and the d

ownside risk

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Bear Spread Using PutsSell a put with a lower strike price and buy a put with a higher strike price

Profit

STK1

-P

K2

+P

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Payoff from a bear spread created using putsStock price range

Payoff from long put option

Payoff from short put option

Total payoff

ST K≧ 2 0 0 0

K1 < ST < K2 K2 - ST 0 K2 - ST

ST K≦ 1 K2 - ST -(K1 – ST) K2 - K1

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Example

An investor buys for $3 a put with a strike price of $35 and sells for $1 a put with a strike price of $30

The profit is therefore as follows:

Stock price range Profit

ST ≦ 30 3

30 < ST < 35 33 - ST

ST ≧ 35 -2

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Bear Spread Using CallsSell a call with a lower strike price and buy a call with a higher strike price

Profit

STK2

+c

K1

-c

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Box Spread

A combination of a bull call spread with strike prices k1 and k2 and a bear put spread with the same two strike prices

If all options are European a box spread is worth the present value of the difference between the strike prices( (K2 - K1)e-rT ) ; If they are American this is not necessarily so.

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Box Spread

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Box Spread

If the market price of the box spread is too low(high),it is profitable to buy(sell) the box,called

“long box” or “short box” Commissions are important to be considered

when implementing this strategy coz the small profit may be easily offset by commissions

Alligator spread

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Butterfly Spread

Involves positions in options with three different strike prices

K1 : a relatively low strike price

K3 : a relatively high strike price

K2 : halfway between K1 and K3 , close to the current stock price

Large stock price moves are unlikely

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Butterfly Spread Using CallsBuy a call option with a relatively low K1 , buy a call option with a relatively high K3 , and sell two call options with K2

Profit

STK1

+C

K3

+C

K2

-C

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Payoff from a butterfly spread

Stock price range

Payoff from first long call

Payoff from second long call

Payoff from short calls

Total payoff

ST < K1 0 0 0 0

K1 < ST < K2 ST - K1 0 0 ST - K1

K2 < ST < K3 ST - K1 0 -2(ST – K2) K3 - ST

ST > K3 ST - K1 ST - K3-2(ST – K2) 0

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Example

The stock price is $61.An investor buys for $10 a call with a strike price of $55,$5 a call with a strike price of $65 and sells for $7 two puts with a strike price of $60

The profit is therefore as follows:

Stock price range Profit

ST < 55 -1

55 < ST < 60 ST - K1 -1

60 < ST < 65 K3 - ST -1

ST > 65 -1

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Butterfly Spread Using PutsBuy a put option with a relatively low K1 , buy a put option with a relatively high K3 , and sell two put options with K2

Profit

STK1

+P

K3

+P

K2

-P

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Calendar Spread

The options have the same strike price and different expiration dates

Neutral calendar spread : A strike price close to the current stock price is chosen

Bullish calendar spread : Involves a higher strike price

Bearish calendar spread : Involves a lower strike price

Reverse calendar spread : Buys a short-maturity option and sells a long-maturity option

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Calendar Spread Using CallsBuy a longer-maturity call option and Sell a call option with the same strike price

Profit

STK

+C

-C

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Calendar Spread Using PutsBuy a longer-maturity put option and Sell a put option with the same strike price

Profit

STK

+P

-P

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Brief Summary

Bull and Bear spreads: different strike prices and the same expiration date Calendar spreads: the same strike price and different expiration date

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Diagonal Spread

Both the expiration date and the strike price of the calls are different

Increases the range of profit patterns that are possible

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Diagonal Bull Spread

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Diagonal Bear Spread

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Combinations

Taking a position in both calls and puts on the same stock

Straddle( 跨式價差 ) Strips( 紙帶價差 ) Straps( 皮帶價差 ) Strangles( 勒式價差 )

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Straddle

Bottom straddle (straddle purchase) : Buying a call and put with the same strike price and expiration date ; expecting a large move in a stock price but does not know in which direction the move will be

Top straddle (straddle write) : Selling a call and put with the same strike price and expiration date ; large stock price moves are unlikely

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Bottom straddle

Profit

STK

+C

+P

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Payoff from a bottom straddle

Stock price range

Payoff from long call

Payoff from long put

Total payoff

ST ≤ K 0 K - ST K - ST

ST > K ST - K 0 ST - K

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Strip & Strap

Strip : Buy one call and two puts with the same strike price and expiration date ; Considers a decrease in the stock price to be a more likely than an increase

Strap : Buy two calls and one put with the same strike price and expiration date ; Considers a increase in the stock price to be a more likely than an decrease

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Strip & Strap

call put

Strip 1 2

Strap 2 1

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Strip & Strap

Profit

K ST

Profit

K ST

Strip Strap

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Strangle

Bottom vertical combination : Buy a put and a call with the same expiration date and different strike prices

Top vertical combination : Sell a put and a call with the same expiration date and different strike prices

The stock price has to move farther in a strangle than in a straddle for the investor to make a profit ; The downside risk is less than a straddle

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Bottom vertical combination

K1 K2

Profit

ST

+C

+P

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K1 K2

Profit

ST

+C

+P

Profit

STK

+C

+P

The stock price has to move farther in a strangle than in a straddle for the investor to make a profit ;

The downside risk is less than a straddle

Strangle

Straddle

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Payoff from a bottom vertical combination

Stock price range

Payoff from long call

Payoff from long put

Total payoff

ST K≦ 1 0 K1 - ST K1 - ST

K1 < ST < K2 0 0 0

ST K≧ 2 ST - K2 0 ST - K2

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Other Payoffs

All payoff functions (at time T) can be found:if Euro options can expire (at time T) with every single

possible strike price

Profit

STK1 K2 K3