Trading Strategy Options

MGTC71 1J. Wei, Department of Management, UTSC

III. Option Trading StrategiesAnd

Properties of Stock Options

- Basics

- Trading Strategies single option holdings

straddle

strip

strap

strangle

bull spread

bear spread

butterfly spread

- Factors Affecting Option Prices

- Bounds for Option Prices

- Early Exercise of American Options

without dividends

with dividends


Basics

- Options: right but not obligation to buy or sell an asset at a fixed price.

- Time to Maturity: Time period within which the right is in effect.

- Exercise price / Strike price: the fixed price at which the asset can be sold or bought.

- European options: the right can be exercised only at maturity.

- American options: the right can be exercised any time before maturity.

- Call option: right to buy.

- Put option: right to sell.

- At-the-money, In-the-money, out-of-the-money.

- Intrinsic value (For example call: max[0, S - X])


Trading Strategies

Single Option Holdings.

Long call

(bullish)

Long put

(bearish)

0X

ST

Payoff

0X

ST

Payoff


Trading Strategies

Single Option Holdings.

Short call

(bearish)

Short put

(bullish)

0X

ST

Payoff

0X

ST

Payoff


- Discussions: Comparing long call with short put, gain in shorting put is limited, and that in long

call is not. But long call involves upfront cash outflow, while short put involves

upfront cash inflow;

Comparing long put and short call, we see that shorting a call leads to unlimited

loss but upfront cash inflows; long put leads to limited loss, but requires upfront

cash outflow.

Unlike stocks, options cannot be bought on margin. Must be paid in full.

Margin for naked call: max [mc1, mc2], where

mc1 : a total of 100% of the option proceeds plus 20% of

the underlying share price less the amount if any

by which the option is out of the money

mc2: a total of 100% of the option proceeds plus 10% of

the underlying share price

Trading Strategies


- Discussions: (cont’d)

Margin for naked put: max [mp1, mp2], where

mp1 : a total of 100% of the option proceeds plus

20% of the underlying share price less the amount if

any by which the option is out of the money

mp2: a total of 100% of the option proceeds plus

10% of the exercise price

Example: Peter writes 5 naked call contracts for $3 per call. Stock price is

$45 and exercise price is $47. Then

mc1 = 500*[3 + 0.2*45 - (47 - 45)] = $5,000

mc2 = 500*[3 + 0.1*45] = $3,750

Therefore the initial margin is $5,000.

Trading Strategies


- Straddle: one call and one put

Upfront cost of 2 options, p + c (next page)

Use when you expect market to move a lot either way (Bre-X, 1997)

When market doesn’t move, you lose the premium.

If you expect market to be stable, sell straddle.

Nick Leeson (1995). Straddle on Nikkei 225 betting on no movements,

but market went down. Bought in to shore up Nikkei, then Kobe

earthquake did him in.

Trading Strategies

0X

ST

Payoff


- Straddle: (cont’d)

Payoff

Range of payoff from payoff from total Net

stock price call put payoffprofit

ST ≤ X 0 X - ST X - ST X - ST - c - p

ST ≥ X ST - X 0 ST - X ST - X - c – p

Example:

c = $4, p = $1.5, X = $40.

If ST = 45, then net profit is, 45 - 40 - 4 - 1.5 = - $0.5.

If ST = 55, then net profit is, 55 - 40 - 4 - 1.5 = $9.5

Trading Strategies


- Strip: 1 call and 2 puts

Upfront cost of 3 options, c + 2p Use when you expect market to move, and more likely to go down

(Good for those who think Guzman’s death was fishy). If market does not move, then lose a bundle.

Trading Strategies

0X

ST

Payoff


- Strip: (cont’d)

payoff


stock price call put payoff profit

ST ≤ X 0 2(X - ST) 2(X - ST) 2(X - ST) - c - 2p

ST ≥ X ST - X 0 ST - X ST - X - c - 2p

Example:

c = $4, p = $1.5, X = $40.

If ST = 45, then net profit is, 45 - 40 - 4 - 2(1.5) = - $2.

If ST = 35, then net profit is, 2(40 - 35) - 4 - 2(1.5) = $3.

Trading Strategies


- Strap: 2 calls and 1 put

Upfront cost of 3 options, 2c + p Similar to Strip, except that you think market will more likely move up

(good for those who think there really is gold). For both Strip and Strap you can make your position more aggressive

by choosing proper exercise prices. For example, if you really believe there is gold, then you can choose the options with very low exercise prices, X. This way you are guaranteed to make money even if the stock moves up only a little. But nothing is free. The two in-the-money calls will cost you more.

Trading Strategies

0X

ST

Payoff


- Strap: (cont’d)

Payoff



ST ≤ X 0 X - ST X - ST X - ST - 2c - p

ST ≥ X 2(ST - X) 0 2(ST - X) 2(ST - X) - 2c - p

Example:

c = $4, p = $1.5, X = $40.

If ST = 45, then Net profit is, 2(45 - 40) - 2(4) - (1.5) = 40.5.

If ST = 35, then net profit is, 40 - 35 - 2(4) - 1.5 = -$4.5

Trading Strategies


- Strangle: 1 call and 1 put with call exercise price higher than put’s

Upfront cost of 2 options, c + p (next page) Similar to Straddle, but you think that the market will move a lot either

way. Less expensive than a Straddle. Sell Strangle if you don’t expect market to move a lot. Reason for being less expensive: If at X1, then call with X2 is less

expensive; if at X2, then put with X1 is less expensive.

Trading Strategies

0X1

ST

Payoff

X2


Trading Strategies- Strangle: (cont’d)

payoff



ST ≤ X1 0 X1 - ST X1 - ST X1 - ST - c - p

X1≤ST≤X2 0 0 0 -c - p

ST ≥ X2 ST - X2 0 ST - X2 ST - X2 - c - p

Example:

c = $3, p = $1.2, X1 = $25, X2 = 30.

If ST = 34, then net profit is, 34 - 30 - 3 - 1.2 = -$0.2.

If ST = 27, then net profit is, - 3 - 1.2 = -$4.2.

If ST = 20, then net profit is, 25- 20 - 3 - 1.2 = $0.8.


- Bull Spread: (long call with X1) + (short call with X2), with X2 > X1

c(X2) < c(X1), upfront cash outflow, - c1 + c2

Use when you expect market to go up. Different from single call, because it is less expensive; different from

selling put, because loss is limited to a small amount.

Trading Strategies

0X1

ST

Payoff

X2


Trading Strategies- Bull Spread: (cont’d)

payoff


stock price call one call two payoff profit

ST ≤ X1 0 0 0 -c1 + c2

X1≤ST≤X2 ST - X1 0 ST - X1 ST - X1 - c1 + c2

ST ≥ X2 ST - X1 -(ST - X2) X2 - X1 X2 - X1 - c1 + c2

Example:

c1 = $5, c2 = $1.5, X1 = $26, X2 = 30.

If ST = 34, then net profit is, 30 - 26 - 5 + 1.5 = $0.5.

If ST = 27, then net profit is, 27 - 26 - 5 + 1.5 = -$2.5.

If ST = 20, then net profit is, - 5 + 1.5 = - $3.5.


- Bear Spread: (short call with X1) + (long call with X2), with X2 > X1

Since c(X1) > c(X2), upfront cash inflow, c1 - c2

Use when you expect market to go down.

Both gain and loss are limited, just like bull spread.

Trading Strategies

0X1

ST

Payoff

X2


- Bear Spread: (cont’d)

payoff (exactly the opposite of bull spread)


stock price call one call two payoff profit

ST ≤ X1 0 0 0 c1 - c2

X1≤ST≤X2 -(ST - X1) 0 -(ST - X1) -(ST - X1) + c1 - c2

ST ≥ X2 -(ST - X1) (ST - X2) -(X2 - X1) -(X2 - X1) + c1 - c2

Example:

c1 = $5, c2 = $1.5, X1 = $26, X2 = 30.

If ST = 34, then net profit is, -(30 - 26) + 5 - 1.5 = -$0.5.

If ST = 27, then net profit is, -(27 - 26) + 5 - 1.5 = $2.5.

If ST = 20, then net profit is, 5 - 1.5 = $3.5.

Trading Strategies


- Butterfly Spread: (long call with X1) + (long call with X3) +

(short 2 calls with X2 = (X1 + X3) / 2)

Requires a small initial investment, c1 + c3 - 2c2 (next page) Use when you expect market to be stable. Different from selling Straddle, because the loss is limited to a small

amount in both directions of market movements. Exercise: what if (long put with X1) + (long put with X3) +

(short 2 puts with X2 = (X1 + X3) / 2)?

Trading Strategies

0X1

ST

Payoff

X2 X3


Trading Strategies- Butterfly Spread: (cont’d)

Payoff

Range of payoff from payoff from payoff from Net

stock price call one call two call three profit

ST ≤ X1 0 0 0 - c1 - c3 + 2c2

X1≤ST≤X2 ST - X1 0 0 ST - X1 - c1 - c3 + 2c2

X2≤ST≤X3 ST - X1 -2(ST - X2)0 X3 -ST - c1 - c3 + 2c2

ST ≥ X3 ST - X1 -2(ST - X2)ST - X3 - c1 - c3 + 2c2

(Note: 2X2 - X1 = X3, and 2X2 - X1 - X3 = 0)

Example:

X1 = $26, X3 = 32, X2 = (26+32)/2 = 29,

c1 = $5, c2 = $2.5, c3 = $1.5,

If ST = 36, then net profit is, - 5 - 1.5 + 2(2.5) = -$1.5.

If ST = 27, then net profit is, 27-26 - 5 - 1.5 + 2(2.5)= -$0.5.

If ST = 20, then net profit is, - 5 - 1.5 + 2(2.5) = - $1.5.


- Stock price and exercise price:

Call: S increases ==> C increases,

X increases ==> C decreases

Put: S increases ==> P decreases,

X increases ==> P increases

- Time to Maturity

Both American calls and puts become more valuable as time to maturity increases. This is because owners of longer maturity options enjoy all the rights available to a shorter maturity option owner, plus more.

For European call and put options, longer time to maturity may not be beneficial because sometimes it is better to exercise early. For example, when the stock price is close to zero, it is better to exercise a put early so that interests can be earned on the exercise price. (For calls, when dividends are high, longer maturities may hurt too.)

Factors Affecting Option Prices


- Interest Rate

For both American and European,

r increases ==> c increases

r increases ==> p decreases

Interest rate affects option price in two ways. When interest rate is high, the overall growth in the stock price is high. But the discount rate also becomes higher. For put options, both effects work against the put price, hence, unambiguously, put price goes down as interest rate goes up. For call options, it can be shown that the growth effect dominates the discount effect, hence r increases ==> c increases.

- Volatility

For both call and put options, higher volatility benefits prices. This is because of the non-symmetric nature of the payoff.



- DividendsSince stock values go down after dividend payments, dividends will reduce call option value and increase put option value.


Variable Europeancall

Europeanput

Americancall

Americanput

stock price + - + -

strike price - + - +

time to maturity ? ? + +

volatility + + + +

interest rate + - + -

dividends - + - +


- NotationS: current stock price

X: exercise priceT: time to maturity

C: value of America call

c: value of European call

P: value of America put

p: value of European put

- Call options S ≥ C ≥ c ≥ max [ 0, S - Xe-rT ] Proof :

S ≥ C ≥ c is obvious what about c ≥ max [ 0, S - Xe-rT ] ?

Bounds for Option Prices


- Call options Proof : (cont’d)

Portfolio A: call and T-bill with face value X, and maturity T

Portfolio B: stock

VA = c + Xe-rT

VB = S

At Maturity: VA

T ST XVB

T ST ST

VAT ≥ VB

T

Therefore, VA ≥ VB, or

c + Xe-rT ≥ S, i.e.

c ≥ S - Xe-rT

Since c > 0, thereforec ≥ max [ 0, S - Xe-rT ].

Q.E.D.


S X S XT T


- Call options What if c < S - Xe-rT ?

E.g., S = 100, X = 100, r = 0.1, T = 1 yr, c = 8.50

Then, S - Xe-rT = 100 - 100e - 0.1 = 9.516 > c

Arbitrage: Short a stock, purchase a T-bill and a call. Proceeds from the transaction is:

V0 = S - Xe-rT - c = $1.016

At maturity:

ST ≥ X = 100 ST ≤ X = 100

T-bill 100 100

stock short -ST -ST

call ST - 100 0 .

Total 0 100 - ST > 0

End result: pocket $1.016 today, and possibly

100 - ST > 0 at maturity.

Can not last long: c ↑ and S ↓, until

c ≥ max [ 0, S - Xe-rT ]



- Put OptionsAmerican: X ≥ P ≥ max [0, X - S]

European: Xe-rT ≥ p ≥ max [0, Xe-rT - S]

proof :

X ≥ P ≥ max [0, X - S] is obvious Xe -rT ≥ p

The best scenario for a European put is when the current stock price is zero, and remains at zero. In this case, the put will be for sure worth X at maturity, which is Xe -rT in today’s terms. This is the maximum value of the put. Hence, p ≤ Xe -rT.



- Put Options proof : (cont’d)

p ≥ max [0, Xe -rT - S]

Portfolio A: long put and stock

Portfolio B: T-bill with face value X

VA = p + S

VB = Xe -rT

At maturity:

VAT ST X

VBT X X

Since VAT ≥ VB

T,

we have VA ≥ VB, that is,

p + S ≥ Xe -rT , or

p ≥ Xe -rT - S

Since p > 0 , therefore

p ≥ max[0, Xe -rT - S] Q.E.D.


S X S XT T


- Put OptionsExample: p = $2.5, X = $50, S = 45, r = 5%, t = 0.5

Then, p = 2.5 < Xe -rT - S = 3.765

Arbitrage strategy: buy stock and put, short T-bill (i.e. borrow PV of X)

Current position: V0 = Xe-rT - p - S = $1.265

At maturity:

ST ≥ X = 50 ST ≤ X = 50

T-bill (loan) -50 -50

stock ST ST

put 0 50 – ST Total ST - 50 > 0

0


ST - 50 > 0 at maturity.

Can not last long: p ↑ and S ↑, until

p ≥ max[0, Xe -rT - S]



- A call option’s price is convex in its exercise price, i.e.

If X3 = αX1 + (1 - α)X2 (0 < α < 1)

Then c(X3) ≤ αc(X1) + (1 - α)c(X2), and

C(X3) ≤ αC(X1) + (1 - α)C(X2)

- A put option’s price is convex in its exercise price, i.e.

If X3 = αX1 + (1 - α)X2 (0 < α < 1)

Then p(X3) ≤ αp(X1) + (1 - α)p(X2), and

P(X3) ≤ αP(X1) + (1 - α)P(X2)



- Example of convexity

three options with the same time to maturity.

S = 40, X1 = 40, X2 = 50 c1 = 4, c2 = 3.

X3 = (0.2)(40) + (0.8)(50) = 48,

then the value of the option with exercise price of 48 cannot be more than

(0.2)(4) + (0.8)(3) = 3.2

Otherwise, there is arbitrage opportunity.

Suppose c3 = $3.6.

Since c(X3) = 3.6 > αc(X1) + (1 - α)c(X2) = 3.2,

Short c3, buy 0.2 units of c1 and 0.8 units of c2

Current position: V0 = 3.6 - (0.2)(4) - (0.8)(3)=$0.4



Bounds for Option Prices- Example of convexity (Cont’d)

At maturity:

ST ≥ 50 48 ≤ ST ≤ 50

c3 -(ST - 48) -(ST - 48)

0.2c1 0.2(ST - 40) 0.2(ST - 40)

0.8c2 0.8(ST - 50) 0

Total 0 0.8(50 - ST) > 0

40 ≤ ST ≤ 48 ST ≤ 40

c3 0 0

0.2c1 0.2(ST - 40) 0

0.8c2 0 0

Total 0.2(ST - 40) > 0 0


more at maturity. Cannot last long.


- Relative pricing

If X2 > X1, then

for European calls, c(X1) - c(X2) ≤ (X2 - X1)e -rT

for European puts, p(X2) - p(X1) ≤ (X2 - X1)e -rT

Example: X1 = 40, X2 = 42, c1 = 4,

T = 1.0 yr, r = 0.1,

then,

4 - c2 ≤ e -(0.1)(1.0) (42 - 40) = 1.81

===> c2 ≥ 2.19.

Therefore, the value of the second call cannot be smaller than

2.19.

Exercise: show how to take advantage of arbitrage if

c2 = $2.



- European options with dividends

Let D be the present value of dividends before option maturity call options, c ≥ max [ 0, S - D - Xe -rT] put options, p ≥ max [ 0, D + Xe -rT - S]

Proof (for call):

Portfolio A: One European call plus T-bill of face value De rT + X.

Portfolio B: One share.

At maturity:

VAT ST + De rT De rT + X

VBT ST + De rT ST + De rT


S X S XT T



- European options with dividends

Proof (for call): (cont’d)

Since VAT ≥ VB

T

Therefore, VA ≥ VB, i.e.

c + D + Xe -rT ≥ S, or

c ≥ S - D - Xe –rT

Since c > 0, therefore

c ≥ max [ 0, S - D - Xe -rT]

Proof (for put):

Similar. Do it as an exercise.


- Put - Call Parity. European options with no dividends:

c + Xe -rT = p + S

Proof : Portfolio A: Call + T-bill with face value of X

Portfolio B: Put and stock

At maturity:

VAT ST X

VBT ST X

Therefore: VA = VB

European options with dividends

c + D + Xe -rT = p + S

D = PV of dividends.

Proof : Similar. Do it as an exercise.


S X S XT T


- Without Dividends Call Options

Since C ≥ c, and c ≥ max[0, S - Xe -rT] early exercise is not desirable, because

S - X < S - Xe -rT. To see it another way, call options provide insurance against the

price going down. If exercise early and price goes down subsequently, then suffer the loss. Also, by postponing exercising, you save the interest earnings on the exercise price.

Put Options An American put may be exercised early. Therefore P ≥ p.

Also, P ≥ X - S. Consider an extreme example, where the stock is worth zero now

due to bankruptcy. If exercise now, you get X. But if wait until maturity, you still get X. And you lose the interest earnings on X.

Early Exercise of American Options


- With Dividends

Call Options

Early exercise of calls is likely because of the drop in stock price on

ex-dividend date.

If the price drop is much bigger than the interest earnings on the

exercise price, then early exercise may be optimal.

Ex-dividend date: t1 t2 t3...........tn, with

Dividends: D1 D2 D3.........Dn.



(cont’d from previous slide)

At time tn:

If exercise: S(tn) - X.

If wait, then option worth at least

S(tn) - Dn - Xe -r(T - tn)

condition for not early exercise is

S(tn) - Dn - Xe -r(T - tn) > S(tn) - X

or,

Dn < X (1 - e -r(T - tn))

i.e. not early exercise if the size of dividend is not as big as

the interest earnings on the exercise price.

In general, at time ti, condition for not early exercise is:


)( )( ii ttri eXD 11


Example: American Call, T - t = 6 months. Two dividends before maturity:

D1 = $1.2 to be paid one month from now, and D2 = $2.5 to be

paid four months from now. X = $100, r = 0.1

At t2 = 4 months:

X [1 - e -r(T - t2)] = 100[1 - e -0.1 * 2/12] = $1.65

Since D2 = 2.5 > 1.65

there is a possibility that call may be exercised at t2 = 4 months.

At t1 = one month,

X[1 - e -r(t2 - t1)] = 100 [1 - e -0.1 x 3/12] = $2.47

Since D1 = 1.2 < 2.47,

There is no early exercise possibility at t = one month.

Put Options

As in the case of no dividends, early exercise is still possible. But dividend payments make early exercise less likely.


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Trading Strategy Options