Trading Strategies

Portfolios of Options

Introduction

• Put-call parity: replication of positions in the underlying.

• Allow for very flexible types of payoffs (portfolios of options).

• Risk management:– Hedging risks.– Portfolio insurance: Stock + Put.

Relationship with Underlying

• Remember the put-call parity:

P + S = C + PV(X)

• PV (X) is independent of underlying.

• Long Stock + Long Put ~ Call.

• Short Stock + Short Put ~ Short Call.

• Short Stock + Long Call ~ Long Put.

• Long Stock + Short Call ~ Short Put.

Example

S

Payoff

0

Long Stock+ Long Put ~ Long Call

Long Put

Long StockLong Call

Spreads

• They involve investing in options of the same type, but with different strike price.

• Bull spread (1):– Buy call with low X.– Write call with high X.

• Bull spread (2):– Buy put with low X.– Write put with high X.

Bull Spread (1)

S

Payoff

0

Long Call (X1) + Short Call (X2)

Long Call

X2

X1

Bull Spread (2)

S

Payoff

0

Long Put (X1) + Short Put (X2)

X2

X1

Bull Spread with Calls (Example)

• Prices of securities:– S = 47.

– C(X=45) = 3.

– C(X=50) = 0.54.

• Buy a call with strike price 45 and write a call with strike price 50. Cost: 3-0.54 = 2.46.

• When S is lower than 45, both calls are out of the money and we lose the investment in the portfolio.

• When S is above 45, but below 50, only the call we bought is in the money. Net payoff:

S - 45 - 2.46 = S - 47.46

Bull Spread (Example, cont.)

• When the stock price is above 50, both options are in the money. The net payoff is,

S – 45 – (S – 50) – 2.46 = 2.54•  In summary,

Net Payoff S < 45 -2.46

45 < S < 50 S - 47.46 50 < S 2.54

• And the break-even point is S = 47.46

Spreads (cont.)

• Bear spread (1):– Buy call with high X.– Write call with low X.

• Bear spread (2):– Buy put with high X.– Write put with low X.

Bear Spread (1)

S

Payoff

0

Short Call (X1) + Long Call (X2)

Long CallX2X1

Bear Spread (2)

S

Payoff

0

Short Put (X1) + Long Put (X2)

X2

X1

Butterfly Spreads

• Positions in options with three different strike prices.

• Butterfly spread with calls:– Buy one call with low X and one with high X.– Write two calls with a X halfway in between.

• Butterfly spread with puts:– Buy one put with low X and one with high X.– Write two puts with a X halfway in between.

Calendar Spreads

• Positions in options with same strike price and different maturities.

• Options with longer maturities are more expensive.

• Bullish or bearish, depending on the relationship of the strike price to the price of the stock today.

Butterfly Spread (1)

S

Payoff

0

Long Call (X1) + 2 Short Calls (X2) + Long Call (X3)

X2

X1 X3

Butterfly Spread (2)

Long Put (X1) + 2 Short Puts (X2) + Long Put (X3)

S

Payoff

0

X2

X1 X3

Calendar Spread (1)

S

Payoff

0

Short Call (T1) + Long Call (T2)

X

Calendar Spread (2)

S

Payoff

0

Short Put (T1) + Long Put (T2)

X

Combinations

• Position in both calls and puts.

• Straddle: buy a put and a call with same strike price.

• Strangle: buy a put and a call with different strike prices.

• Strip: buy one call and two puts with same strike price.

• Strap: buy one put and two calls with same strike price.

Straddle

S

Payoff

0

Long Put (X) + Long Call (X)

X

Strangle

S

Payoff

0

Long Put (X1) + Long Call (X2)

X1 X2

Strip

S

Payoff

0

2 Long Puts (X) + Long Call (X)

X

Strap

S

Payoff

0

Long Put (X) + 2 Long Calls (X)

X