4_Trading Strategies; Slope and Convexity Restrictions

Lecture 4:

Trading Strategies and Slope/ConvexityRestrictions

This lecture studies elementary options tradingstrategies. In the process, we derive no-arbitragerestrictions for options that are identical except for theirstrike price. We restrict how quickly the option pricecan change with the strike price (slope restrictions) andhow quickly this slope can change with the strike price(convexity restrictions).

I. Motivation

II. Definitions and Notation

III. Trading Strategies

A. HedgesB. European SpreadsC. American SpreadsD. European ButterflysE. American ButterflysF. Other Combinations

Trading Strategies and Slope/Convexity Restrictions

I. Motivation

Suppose a stock is trading at $100.

� You see four month European calls priced

– at $8 for K D $100, and

– at $19 for K D $90.

� The simple 4 month risk-free is 5.26%.

– I.e., B(t, t C 1=3) D $95, and

Questions:

� Does the $8 price of the K D $100 call satisfy theno-arbitrage bounds from Lecture 3?

� Does the $19 price of the K D $90 call satisfy theno-arbitrage bounds from Lecture 3?

� Does this mean that we cannot have arbitrage?

– How do recognize that there is an arbitrageopportunity?

� Pricing “restrictions”

– How can we exploit it?

� I.e., with what trading strategy?

Bus 35100 Page 2 Robert Novy-Marx


II. Definitions and Notation

Traders have names for common options portfolios.

� These portfolios are typically specific “bets” onwhat will happen to the prices and/or volatilities ofthe underlying securities.

� We use portfolios of options to illustrate further no-arbitrage restrictions and to generate profits whenthese restrictions are violated.

We describe a portfolio of options by the equation forthe current price of the portfolio. However, we dropsubscripts which are common to all securities.

Examples:

� c(K1) � c(K2) is a portfolio of a bought call withexercise price K1 and a written call with exerciseprice K2 and with the same maturity date.

� p(T1) � p(T2) is a portfolio of a bought put withmaturity T1 and a written put with maturity T2

and with the same exercise price.



III. Trading Strategies

A. Hedges: Combine options + underlying; protectsthe underlying against a loss, or vice-versa.

Example: A covered call– long stock / short a call on the stock

S(T)

Profit

K

Covered Call

Payoff is always positive, so no-arbitrage H)

price of the covered call is positive, i.e.,

S(t) � c(K) � 0 or c(K) � S(t)



Another Example: A protective put

– long stock / long a put on the stock

* the put provides insurance

* so you must pay an “insurance premium”

S(T)

Profit

Protected Put

K



B. Spreads

Combine options of the same type, with

– different strikes (a “vertical spread”), or

– different maturities (a “horizontal spread”)

1. Example: A bear spread

– Buy an ITM put (strike K2)

– Sell an OTM put (strike K1 < K2)

S(T)

Profit

Bear Spread

K2K

1

Payoff always positive ) p(K2) � p(K1) � 0



2. Another Example: A “bull spread” made with puts

– Buy an OTM put (strike K1)

– Sell an ITM put (strike K2 > K1)

* Technically a “short bear spread”

* We’ll do a real bull spread in a second

S(T)

Profit

Short Bear Spread

K2K1



3. Another Example: Bullish spread with puts, plusbonds

– Buy an OTM put (strike K1)

– Sell an ITM put (strike K2 > K1)

– Buy K2 � K1 face-value bonds

S(T)

Payoff

K2

K1

pK1(T) – pK2

(T) + (K2 – K1)

Portfolio value at time t � T

pK1(t, T ) � pK2(t, T ) C (K2 � K1) B(t, T )



Slope Restriction for Puts:

Can we say anything about the sensitivity of putsto the strike?

Payoff to this last spread is always positive, so

pK1(t, T ) � pK2

(t, T ) C (K2 � K1)B(t, T ) � 0.

So

0 �pK2

(t, T ) � pK1(t, T )

K2 � K1

� B(t, T ).

(LHS follows from pK1(t, T ) � pK2

(t, T ) � 0)

True for arbitrary K1 and K2 > K1, so

0 �@p

@K� B(t, T )

� A $1 dollar increase in the strike causes the putprice to increase by no more than the PV of $1

– Simple intuition?



4. Another Example: A true bull spread

– Buy an ITM call (strike K1)

– Sell an OTM call (strike K2 > K1)

S(T)

Profit

Bull Spread

K2K

1

What’s the difference between:

i. a bull spread, and

ii. a bullish spread made with puts?

� i.e., a short bear spread



5. Slope Restriction for Calls:

A bull spread with a short position in bonds

– Buy an ITM call (strike K1)

– Sell an OTM call (strike K2 > K1)

– Sell K2 � K1 face-value bonds

S(T)

Payoff

K2

K1

cK1(T) – cK2

(T) – (K2– K1 )

Portfolio value at time t � T is

cK1(t, T ) � cK2

(t, T ) � (K2 � K1) B(t, T )



Slope Restriction for Calls:

Can we say anything about the sensitivity of callsto the strike?

Payoff to the last spread is always negative, so

cK1(t, T ) � cK2

(t, T ) � (K2 � K1)B(t, T ) � 0.

So

�B(t, T ) �cK2

(t, T ) � cK1(t, T )

K2 � K1

� 0.

(LHS follows from cK1(t, T ) � cK2

(t, T ) � 0)

Again, true for arbitrary K1 and K2 > K1, so

�B(t, T ) �@c

@K� 0

� A $1 dollar increase in the strike causes the callprice to decrease by no more than the PV of $1

– Simple intuition?



If the restriction is violated ))) Arbitrage

Example: European calls with 4 months to maturity

� S(t) D $100.

� Calls priced

– at $8 for K D $100, and

– at $19 for K D $90.

� The simple 4 month risk-free is 5.26%.

– I.e., B(t, t C 1=3) D $95, and

Is there an arbitrage opportunity?

How can it be exploited?

Payoff Payoff at T

Transaction at t S(T )<90 90<S(T )<100 S(T )>100

What is the payoff diagram for this position?



Profit diagram for the arbitrage portfolio:

S(T)

Payoff

$90 $110

-$10

$10



� Suppose one year European puts are priced:

– at $212

for K D $100, and

– at $1214

for K D $110.

� The yield-to-maturity on a one-year risk-freezero coupon bond is 5%.

Can we exploit this with an arbitrage?

Payoff Payoff at T

Transaction at t S(T )<100 100<S(T )<110 S(T )>110



Payoff Diagram:

S(T)

Payoff

$90 $110

-$10

$10

Question: is it still an arbitrage if the prices werefor American puts?



C. Slope Restrictions on American Option

The argument is basically the same: constructspreads and see what NA implies for prices.

One big difference: written options can beexercised against you at any time. If this happens:

� you’ll have to close out the whole position, and

� you may not gain the interest on the exerciseprice.

So no-arbitrage implies that if the strike pricechanges from K1 to K2 the price of the optionchanges by no more than jK2 � K1j. (Why?)



The slope restriction for American calls is therefore

�(K2 � K1) � C(K2) � C(K1) � 0,

or

�1 �@C

@K� 0.

The slope restriction for American puts is:

0 � P(K2) � P(K1) � (K2 � K1),

or

0 �@P

@K� 1.



D. Convexity Restrictions

A butterfly spread combines three options of thesame type but with different strike prices.

1. An Example: BS w/ European calls

� Long an ITM call (strike K1)

� Long an OTM call (strike K3 > K1)

� Short two ATM calls (strikes K2 DK1CK3

2)

It’s a combination of bullish and bearish verticalcall spreads at different exercise prices:

[c(K3) � c(K2)] � [c(K2) � c(K1)]

� It’s neither bullish or bearish.

� What is it a bet on?

It’s easier to see in the payoff diagram.



Butterfly Spread Payoff Diagram

S(T)

Payoff

K3K1 K2

Butterfly

Spread

It’s a “bet” on volatility.

� Why?

– And in which direction?



Convexity Restriction for Calls:

The payoff to this butterfly spread is alwayspositive. NA ) its price is positive.

c(K1) � 2c(K2) C c(K3) D

[c(K3) � c(K2)] � [c(K2) � c(K1)] � 0.

Since K3 � K2 D K2 � K1 > 0 we have

c(K3) � c(K2)

K3 � K2

�c(K2) � c(K1)

K2 � K1

� 0.

This implies that the call option price c(S, K, t, T )

is a convex function of the strike price K:

@2c(S, K, t, T )

@K2D

@

@K

�

@c(S, K, t, T )

@K

�

� 0

The sensitivity of the call price to the strikedecreases with the strike. (Remember, @c

@K� 0!)



Graphically, the call price looks like this, as afunction of the strike

� Here St D 100, r D 0.05, � D 0.4, and T � t = 1 year.

60 80 120 140Strike HKL

10

20

30

40

50

Call Price

Question:

� Is there a more intuitive way to think about thisconvexity?



Let’s find another way to think about theconvexity restriction

Butterfly Spread Profit Diagram

S(T)

Profit

K3K1 K2

Butterfly

Spread

How do we know that we must loose money forbig moves in the underlying?



For big enough moves in the underlying, wemust loose money on the butterfly spread

� but the payoff at maturity is then zero )

c(K3) C c(K1) � 2c(K2),so

c(K2) �c(K3) C c(K1)

2.

60 80 120 140Strike HKL

10

20

30

40

50

Call Price

Otherwise, there’s an arbitrage. (Why?)



One more way to think about the restriction

� Remember: the butterfly is long one bullspread and short another

c(K1) � 2c(K2) C c(K3) D

[c(K1) � c(K2)] � [c(K2) � c(K3)]

� Each bull spread is like a “bond you only getsome of the time”

– Pays off when the underlying is high

� Above the strikes

� So the bull spread with low strikes is worthmore

– Pays off more of the time



2. Another Example: BS w/ European puts

� Long an OTM put (strike K1)

� Long an ITM put (strike K3 > K1)

� Short two ATM puts (strikes K2 DK1CK3

2)

It’s a combination of bullish and bearish verticalput spreads at different exercise prices:

p(K1) � 2p(K2) C p(K3)

D [p(K3) � p(K2)] � [p(K2) � p(K1)]



Butterfly Spread Payoff Diagram

S(T)

Payoff

K3K1

Butterfly

Spread

K2

The payoff is exactly the same as a BS madewith calls

� Why?

� Is there any difference?

– Think p-c parity.



Convexity Restriction for Puts:

Again, the payoff to the butterfly spread is alwayspositive. NA ) its price is positive.

p(K1) � 2p(K2) C p(K3) D

[p(K3) � p(K2)] � [p(K2) � p(K1)] � 0

Since K3 � K2 D K2 � K1 > 0 we have

p(K3) � p(K2)

K3 � K2

�p(K2) � p(K1)

K2 � K1

� 0

This implies the put price p(S, K, t, T ) is aconvex function of the strike price K:

@

@K

�

@p(S, K, t, T )

@K

�

D@2p(S, K, t, T )

@K2� 0

I.e., the sensitivity of the put price to the strikeincreases with the strike.



Graphically:

60 80 120 140Strike HKL

10

20

30

40

Put Price

Again, the convexity restriction just says that for bigenough moves in the underlying, we must loosemoney on the butterfly spread:

p(K3) C p(K1) � 2p(K2),

or

p(K2) �p(K3) C p(K1)

2.



Note:

No-arbitrage implies that European put and callbutterfly spreads have the exact same price.

� Why?

So

p(K1) � 2p(K2) C p(K3)

D c(K3) � 2c(K2) C c(K1),

which implies

@2p(S, K, t, T )

@K2D

@2c(S, K, t, T )

@K2.

Question: Do these convexity restrictions apply tooptions on dividend paying stocks?

� Absolutely!

– The relations were NA, based on payoffs atexpiration.



Arbitrage example:

Stock ABC is trading at $4818. 4-month European

put options are priced

– at $11 for K D 55,

– at $1534

for K D 60,

– and at $20 for K D 65.

The risk-free rate over the next 4 months is 3.25%.

� Is there an arbitrage opportunity?

– How can you tell?

– How can it be exploited?

Payoff Payoff at T

Transaction at t ST <55 55<ST <60 60<ST <65 ST >65



More general convexity restriction

� In all our examples K2 DK1CK3

2.

� This wasn’t necessary; it was just convienient.

More generally,

p(K2) �

�

K3�K2K3�K1

�

p(K1)C�

K2�K1K3�K1

�

p(K3).

If not, you get payed today to construct the portfolio

�

K3�K2

K3�K1

�

p(K1) C

�

K2�K1

K3�K1

�

p(K3) � p(K2).

Payoff Payoff at T

at t ST <K1 K1 <ST <K2 K2 <ST <K3

�K3�K2

K3�K1

p(K1)K3�K2

K3�K1

(K1 � ST ) 0 0

�K2�K1

K3�K1

p(K3)K2�K1

K3�K1

(K3 � ST ) K2�K1

K3�K1

(K3 � ST ) K2�K1

K3�K1

(K3 � ST )

p(K2) �(K2 � ST ) �(K2 � ST ) 0

� 0 0 � 0 � 0

If ST > K3, then all the options are OTM and thepayoff is zero.



D. Convexity Restrictions on American Options

� The convexity restrictions for American optionsare exactly the same as they are for Europeanoptions.

� The arbitrage arguments are also the same,except...

... a written American option may be exercisedagainst you.

– This will force you to liquidate the rest of theposition.

� Since the payoff at liquidation must always bepositive (why?), the spread must have a positivevalue.

In any event, we still have

@2P

@K2D

@2C

@K2� 0.



E. Other Combinations

1. Straddles

� Buy a put and a call with the same strike

py g ( ) y

C=f(S,t)

S = 100

K = 100

t = 1

r = 1.15

d = 1.00

= . 3

Future Asset Price

Straddle Value = $18.84 + $5.80 = $24.64

50 75

25

125 150

-25 Strategy: Believe volatility of asset price will be high, but haveno clue about direction.

Strategy: Believe volatility of asset price will be high, but haveno clue about direction.

(BUY ATM CALL @ $18.84)(BUY ATM PUT @ $5.80)

Profit

Loss

Straddle:ATM CALL + ATM PUT

� Another bet on volatility

– a (long) straddle is a bet on high volatility

� if you think volatility is higher than the marketconsensus, buy a straddle - they’re cheap



2. Strangles

� A close cousin to the straddle.

� Buy a put stuck at K1 and a call at K2 > K1.

– E.g., buy both options OTM

Derivatives: A PowerPlus Picture Book I

Copyright©1999(Dec) by Mark Rubinstein

162

C=f(S,t)

S = 100

K1 = 90

K2 = 110

t = 1

r = 1.15

d = 1.00

= . 3

Future Asset Price

Strangle Value = $13.97 + $3.07 = $17.04

50 75

25

125 150

-25

Strategy: Similar to a straddle.Strategy: Similar to a straddle.

(BUY OTM PUT @ $3.07)

(BUY OTM CALL @ $13.97)

Profit

Loss

Strangle:OTM CALL + OTM PUT

� A more extreme bet than a straddle

– Requires bigger moves (i.e., higher volatility)to produce gains,

– but you don’t have to risk as much capital



3. Condor

� A close cousin to the Butterfly Spread.

� Buy deep in and out of the money calls (orputs)

� Sell near in and out of the money calls (or puts)



168

C=f(S,t)

S = 100

K1 = 90

K2 = 95

K3 = 105

K4 = 110

t = 1

r = 1.15

d = 1.00

= . 3

Future Asset PriceCondor Value = $0.82 =$24.81 - $21.69 - $16.27 + $13.97

50 75

25

125 150

-25 Strategy: Similar to a butterfly spread.

Strategy: Similar to a butterfly spread.

(BUY DITM CALL @ $24.81)

(SELL ITM CALL @ $21.69)

(SELL OTM CALL @ $16.27)

(BUY DOTM CALL @ $13.97)

Profit

Loss

Condor:DITM CALL - ITM CALL - OTM CALL + DOTM CALL

� A less “confident” bet than a butterfly

– You gain over a bigger price range,

– but if you gain, you gain less.



4. Bull Cylinder

� Like a “synthetic future”, only less optimistic

� Buy an out of-the-money call

� Sell an out of-the-money put

– Instead of the in-the-money in a future



160

C=f(S,t)

S = 100

K1 = 90

K2 = 110

t = 1

r = 1.15

d = 1.00

= . 3

Future Asset Price

Bull Cylinder Value = $13.97 - $3.07 = $10.90

50 75

25

125 150

-25

(SELL OTM PUT @ $3.07)

(BUY OTM CALL @ $13.97)

Profit

Loss

Bull Cylinder:OTM CALL - OTM PUT

� Takes on less downside risk than a future

– You’re less likely to pay on the put

– So you have to pay something today



Possible strategies are more or less endless.Some other common strategies go by names like

� Straps

– 2 ATM calls + ATM put

– A bet on high vol., with a bullish tilt

� Collars

– Underlying + OTM put - OTM call

– Caps the underlying’s up- and down-sides

� Range Forwards (or “Fences”)

– Forward + ITM put - OTM call

– Essentially a forward bull spread

� Back Spreads

– 2 OTM calls - ATM call

– A bear spread, plus a bet on upside

� Seagulls

– ATM call - OTM calls - OTM Put

– A bull spread with a dropping tail

Try drawing the profit diagrams for these.



5. European box spread: borrowing or lending inthe options market

� An Example: lending

– Buy ITM call and put (strikesK1 andK2>K1)

– Sell OTM call and put (strikes K2 and K1)

c(K1) � c(K2) C p(K2) � p(K1)

� It’s a portfolio that’s

– long a synthetic future with delivery K1, and

– short a synthetic future with delivery K2.

– At delivery you’ll buy at K1 and sell at K2,

So it must be priced at (K2 � K1) � B today.

� It’s also a portfolio that’s

– long a bull spread, and

– long a bear spread with the same strikes

� i.e. short a bullish put spread.

– The difference: CF timing.

� The positions transfer cash through time.


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4_Trading Strategies; Slope and Convexity Restrictions