options - examples

MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

OPTIONS - EXAMPLES


COVERED STRATEGIES: Take a position in the option and the underlying stock.

SPREAD STRATEGIES: Take a position in 2 or more options of the same type (A spread).

COMBINATION STRATEGIES: Take a position in a mixture of calls and puts.

SOME STRATEGIES


STANDARD SYMBOLS:◦ C = current call price, P = current put price◦ S0 = current stock price, ST = stock price at

time T◦ T = time to maturity◦ X = exercise price (or K in some books)◦P = profit from strategy

STAKES:◦ NC = number of calls◦ NP = number of puts◦ NS = number of shares of stock

TYPES OF STRATEGIES


These symbols imply the following:◦ NC or NP or NS > 0 implies buying (going long)◦ NC or NP or NS < 0 implies selling (going short)

Recall the PROFIT EQUATIONS◦ Profit equation for calls held to expiration P = NC[Max(0,ST - X) – Cexp(rT)]

For buyer of one call (NC = 1) this implies P = Max(0,ST - X) - Cexp(rT) For seller of one call (NC = -1) this implies P = -Max(0,ST - X) + Cexp(rT)

Types of Strategies


The Profit Equations (continued) Profit equation for puts held to expiration P = NP[Max(0,X - ST) - Pexp(rT)] For buyer of one put (NP = 1) this implies

P = Max(0,X - ST) - Pexp(rT) For seller of one put (NP = -1) this implies P = -Max(0,X - ST) + Pexp(rT)

Types of Strategies


The Profit Equations (continued)◦Profit equation for stock P = NS[ST - S0] For buyer of one share (NS = 1) this

implies P = ST - S0

For short seller of one share (NS = -1) this implies

P = -ST + S0

Types of Strategies


Positions in an Option & the Underlying

Profit

STK

Profit

ST

K

Profit

ST

K

Profit

STK

(a) (b)

(c)

(d)


Bull Spread Using Calls

Bull Spread Using Calls: Buying a call option on a stock with a particular strike price and selling a call option on the same stock with a higher strike price.

Payoff from a Bull Spread:

Stock price Range

Payoff from Long Call Option

Payoff from Short Call Option

Total Payoff

ST ≥ K2

K1 < ST < K2

ST ≤ K1

ST - K1

ST - K1

0

K2 - ST 00

K2 - K1

ST ≥ K2

0



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

Payoff from a Bull Spread:

Stock price Range



Total Payoff

ST ≥ $35$30 < ST < $35ST ≤ $30

ST - $30 - $3ST - $30 -$30 - $3

$35 - ST +$10+$10+$1

$3ST - $32-$2



K1 K2

Profit

ST


Bull Spread Using Puts

K1 K2

Profit

ST


Bear Spread Using Puts-buying one put with a strike price of K2 and selling one put with a strike price of K1

K1 K2

Profit

ST


Bear Spread Using Calls

Stock price Range



Total Payoff

ST ≥ K2

K1 < ST < K2

ST ≤ K1

ST - K2

00

K1 - ST K1 - ST

0

-(K2 - K1) -(ST ≥ K1)0

Bear Spread: Buying a call option on a stock with a particular strike price and selling a call option on the same stock with a lower strike price.



Stock price Range



Total Payoff

ST ≥ $35$30 < ST < $35ST ≤ $30

ST - $3500

$30 - ST $30 - ST

0

-($35 - $30) -(ST ≥ $30)0

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



K1 K2

Profit

ST


A combination of a bull call spread and a bear put spread

If all options are European a box spread is worth the present value of the difference between the strike prices

If they are American this is not necessarily so.

Box Spread


Butterfly Spread Using Calls Butterfly Spread: buying a call option with a relative

low strike price, K1,, buying a call option with a relative high strike price. K3, and selling two call options with a strike price halfway in between, K2.Stock price Range

Payoff from First Long Call Option

Payoff from Second Long Call Option

Payoff from Short Calls

Total Payoff

ST ≥ K3

K2 < ST < K3

K2 < ST < K3

ST ≤ K1

ST - K1 ST - K1

ST - K1

0

ST - K3 000

-2(ST - K2) -2(ST - K2) 00

0K3 - ST ST - K1

0


Butterfly Spread Using Calls Example: Call option prices on a $61 stock are: $10 for a $55 strike, $7

for a $60 strike, and $5 for a $65 strike. The investor could create a butterfly spread by buying one call with $55 strike price, buying a call with a $65 strike price, and selling two calls with a $60 strike price.

Stock price Range

Payoff from First Long Call Option

Payoff from Second Long Call Option

Payoff from Short Calls

Total Payoff

ST ≥ $65$60 < ST <$65$55 < ST <$60ST ≤ $55

ST - $55ST - $55ST - $550

ST - $65000

-2(ST - $60) -2(ST - $60)00

0$65 - ST ST -$550


Butterfly Spread Using Calls

K1 K3

Profit

STK2


Butterfly Spread Using Puts

K1 K3

Profit

STK2


Calendar Spread Using Calls

Profit

ST

K


Calendar Spread Using Puts

Profit

ST

K


A Straddle Combination

Stock price Range

Payoff from Call Payoff from Put Total Payoff

ST ≥ KST < K

ST – K 0

0K - ST

ST - K K - ST

Straddle: Buying a call and a put with the same strike price and expirationDate.



Stock price Range

Payoff from Call Payoff from Put Total Payoff

ST ≥ $70ST < $70

ST – $70 -$40 - $4

0 -$3$70 - ST - $3

ST - $77$63 - ST

Example: An investor buying a call and a put with a strike price of $70 and an expiration date in 3 months. Suppose the call costs $4 and the put $3.



Profit

STK


Strip & StrapStrip: combining one long call with two long putsStrap: combining two long calls with one long put

Profit

K ST

Profit

K ST

Strip Strap


A Strangle Combinationbuying one call with a strike price of K2 and buying one put with a strike price of K1

K1 K2

Profit

ST


BINOMIAL MODELS - EXAMPLES

A stock price is currently $20 In three months it will be either $22 or $18

Stock Price = $22

Stock Price = $18

Stock price = $20


Stock Price = $22Option Price = $1

Stock Price = $18Option Price = $0

Stock price = $20Option Price=?

A Call Option

A 3-month call option on the stock has a strike price of 21.


Consider the Portfolio: long D shares short 1 call option

Portfolio is riskless when 22D – 1 = 18D or D = 0.25

22D – 1

18D

Setting Up a Riskless Portfolio


Valuing the Portfolio(Risk-Free Rate is 12%)

The riskless portfolio is: long 0.25 sharesshort 1 call option

The value of the portfolio in 3 months is 22 ´ 0.25 – 1 = 4.50

The value of the portfolio today is 4.5e – 0.12´0.25 = 4.3670


Valuing the Option The portfolio that is

long 0.25 sharesshort 1 option

is worth 4.367 The value of the shares is

5.000 (= 0.25 ´ 20 ) The value of the option is therefore

0.633 (= 5.000 – 4.367 )


Generalization

A derivative lasts for time T and is dependent on a stock

S(1+a)=Su

ƒu

S(1-a)=Sd

ƒd

Sƒ


Generalization(continued) Consider the portfolio that is long D shares and

short 1 derivative

The portfolio is riskless when SuD – ƒu = Sd D – ƒd or

du

du

SSfƒ

D

SuD – ƒu

SdD – ƒd


Generalization(continued)

Value of the portfolio at time T is Su D – ƒu

Value of the portfolio today is (Su D – ƒu )e–rT

Another expression for the portfolio value today is S D – f

Hence ƒ = S D – (Su D – ƒu )e–rT


Generalization(continued)

Substituting for D we obtain ƒ = [ p ƒu + (1 – p )ƒd ]e–rT

where

aaep

rT

21


Risk-Neutral Valuation

ƒ = [ p ƒu + (1 – p )ƒd ]e-rT

The variables p and (1 – p ) can be interpreted as the risk-neutral probabilities of up and down movements

The value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate Su

ƒu

Sd

ƒd

Sƒ

p

(1 – p )


Original Example Revisited

Since p is a risk-neutral probability 20e0.12 ´0.25 = 22p + 18(1 – p ); p = 0.6523

Su = 22 ƒu = 1

Sd = 18 ƒd = 0

S ƒ

p

(1 – p )


Valuing the Option

The value of the option is e–0.12´0.25 [0.6523´1 + 0.3477´0] = 0.633

Su = 22 ƒu = 1

Sd = 18 ƒd = 0

Sƒ

0.6523

0.3477


Estimating pOne way of matching the volatility is to set

where s is the volatility and Dt is the length of

the time step. This is the approach used by Cox, Ross, and Rubinstein

12

1

Dt

rT

eaa

aep

s


A Two-Step Example

Each time step is 3 months K=21, r=12%

20

22

18

24.2

19.8

16.2


Valuing a Call Option

Value at node B = e–0.12´0.25(0.6523´3.2 + 0.3477´0) = 2.0257

Value at node A = e–0.12´0.25(0.6523´2.0257 + 0.3477´0)

= 1.2823

201.2823

22

18

24.23.2

19.80.0

16.20.0

2.0257

0.0

A

B

C

D

E

F


A Put Option Example; K=52

K = 52, Dt = 1yrr = 5%

504.1923

60

40

720

484

3220

1.4147

9.4636

A

B

C

D

E

F


Behaviorof Stock Prices


Discrete time; discrete variable Discrete time; continuous variable Continuous time; discrete variable Continuous time; continuous variable

Categorization of Stochastic Processes


We can use any of the four types of stochastic processes to model stock prices

The continuous time, continuous variable process proves to be the most useful for the purposes of valuing derivative securities

Modeling Stock Prices


In a Markov process future movements in a variable depend only on where we are, not the history of how we got where we are

We will assume that stock prices follow Markov processes

Markov Processes


The assertion is that it is impossible to produce consistently superior returns with a trading rule based on the past history of stock prices. In other words technical analysis does not work.

A Markov process for stock prices is clearly consistent with weak-form market efficiency

Weak-Form Market Efficiency


A stock price is currently at $40 At the end of 1 year it is considered

that it will have a probability distribution of (40,10) where f(m,s) is a normal distribution with mean m and standard deviation s.

Example of a Discrete Time Continuous Variable Model


What is the probability distribution of the stock price at the end of 2 years?

½ years? ¼ years? Dt years? Taking limits we have defined a

continuous variable, continuous time process

Questions


In Markov processes changes in successive periods of time are independent

This means that variances are additive Standard deviations are not additive

Variances & Standard Deviations


In our example it is correct to say that the variance is 100 per year.

It is strictly speaking not correct to say that the standard deviation is 10 per year.

Variances & Standard Deviations (continued)


We consider a variable z whose value changes continuously

The change in a small interval of time Dt is Dz

The variable follows a Wiener process if: 1.

2. The values of Dz for any 2 different (non-overlapping) periods of time are independent

A Wiener Process

(0,1) N from drawing random a is where tz DD


Mean of [z (T ) – z (0)] is 0 Variance of [z (T ) – z (0)] is T Standard deviation of [z (T ) – z (0)] is

Properties of a Wiener Process

T


What does an expression involving dz and dt mean?

It should be interpreted as meaning that the corresponding expression involving Dz and Dt is true in the limit as Dt tends to zero

In this respect, stochastic calculus is analogous to ordinary calculus

Taking Limits . . .


A Wiener process has a drift rate (ie average change per unit time) of 0 and a variance rate of 1

In a generalized Wiener process the drift rate & the variance rate can be set equal to any chosen constants

Generalized Wiener Processes


The variable x follows a generalized Wiener process with a drift rate of a & a variance rate of b2 if

dx=adt+bdz

Generalized Wiener Processes(continued)


Mean change in x in time T is aT Variance of change in x in time T is b2T Standard deviation of change in x in

time T is

Generalized Wiener Processes(continued)

D D Dx a t b t

b T


A stock price starts at 40 & has a probability distribution of (40,10) at the end of the year

If we assume the stochastic process is Markov with no drift then the process is

dS = 10dz If the stock price were expected to grow by

$8 on average during the year, so that the year-end distribution is (48,10), the process is

dS = 8dt + 10dz

The Example Revisited


In an Ito process the drift rate and the variance rate are functions of time

dx=a(x,t)dt+b(x,t)dz The discrete time equivalent

is only true in the limit as Dt tends to zero

Ito Process

D D Dx a x t t b x t t ( , ) ( , )


For a stock price we can conjecture that its expected proportional change in a short period of time remains constant not its expected absolute change in a short period of time

We can also conjecture that our uncertainty as to the size of future stock price movements is proportional to the level of the stock price

Why a Generalized Wiener Processis not Appropriate for Stocks


where m is the expected return s is the volatility.

The discrete time equivalent is

An Ito Process for Stock Prices

dS Sdt Sdz m s

D D DS S t S t m s


We can sample random paths for the stock price by sampling values for

Suppose m= 0.14, s= 0.20, and Dt = 0.01, then

Monte Carlo Simulation

DS S S 0 0014 0 02. .


Monte Carlo Simulation – One Path

PeriodStock Price atStart of Period

RandomSample for

Change in StockPrice, DS

0 20.000 0.52 0.236

1 20.236 1.44 0.611

2 20.847 -0.86 -0.329

3 20.518 1.46 0.628

4 21.146 -0.69 -0.262


If we know the stochastic process followed by x, Ito’s lemma tells us the stochastic process followed by some function G (x, t )

Since a derivative security is a function of the price of the underlying & time, Ito’s lemma plays an important part in the analysis of derivative securities

Ito’s Lemma


A Taylor’s series expansion of G (x , t) gives

Taylor Series Expansion

D D D D

D D D

G Gx

x Gt

t Gx

x

Gx t

x t Gt

t

½

½

2

22

2 2

22


Ignoring Terms of Higher Order Than Dt

In ordinary calculus we get

stic calculus we get

because has a component which is of order

In stocha

½

D D D

D D D D

D D

G Gx

x Gt

t

G Gx

x Gt

t Gx

x

x t

2

22


Substituting for DxSuppose ( , ) ( , )so that

= + Then ignoring terms of higher order than

½

dx a x t dt b x t dz

x a t b tt

G Gx

x Gt

t Gx

b t

D D DD

D D D D

2

22 2


The 2Dt Term

tbxGt

tGx

xGG

tt

ttE

E

EE

EN

DDDD

DD

DD

22

2

2

2

2

22

21

Hence ignored. be can and toalproportion is of varianceThe

)( that followsIt

1)(

1)]([)(

0)(,)1,0( Since


Taking Limits

Taking limits ½

Substituting

We obtain ½

This is Ito's Lemma

dG Gx

dx Gt

dt Gx

b dt

dx a dt b dz

dG Gx

a Gt

Gx

b dt Gx

b dz

2

22

2

22


Application of Ito’s Lemmato a Stock Price Process

The stock price process is For a function of &

½

d S S dt S d zG S t

dGGS

SGt

GS

S dtGS

S dz

m s

m

s

s2

22 2


Examples1. The forward price of a stock for a contract maturing at time

e

2.

T

G SdG r G dt G dz

G S

dG dt dz

r T t

( )

( )

ln

m s

ms

s2

2

options - examples

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