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OPTIONS - EXAMPLES. SOME STRATEGIES. 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. - PowerPoint PPT Presentation
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MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
OPTIONS - EXAMPLES
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
Positions in an Option & the Underlying
Profit
STK
Profit
ST
K
Profit
ST
K
Profit
STK
(a) (b)
(c)
(d)
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
Bull Spread Using Calls
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
Payoff from Long Call Option
Payoff from Short Call Option
Total Payoff
ST ≥ $35$30 < ST < $35ST ≤ $30
ST - $30 - $3ST - $30 -$30 - $3
$35 - ST +$10+$10+$1
$3ST - $32-$2
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
Bull Spread Using Calls
K1 K2
Profit
ST
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
Bull Spread Using Puts
K1 K2
Profit
ST
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
Bear Spread Using Calls
Stock price Range
Payoff from Long Call Option
Payoff from Short Call Option
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.
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
Bear Spread Using Calls
Stock price Range
Payoff from Long Call Option
Payoff from Short Call Option
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.
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
Bear Spread Using Calls
K1 K2
Profit
ST
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
Butterfly Spread Using Calls
K1 K3
Profit
STK2
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
Butterfly Spread Using Puts
K1 K3
Profit
STK2
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
Calendar Spread Using Calls
Profit
ST
K
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
Calendar Spread Using Puts
Profit
ST
K
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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.
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
A Straddle Combination
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.
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
A Straddle Combination
Profit
STK
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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.
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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 )
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
Generalization
A derivative lasts for time T and is dependent on a stock
S(1+a)=Su
ƒu
S(1-a)=Sd
ƒd
Sƒ
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
Generalization(continued)
Substituting for D we obtain ƒ = [ p ƒu + (1 – p )ƒd ]e–rT
where
aaep
rT
21
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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 )
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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 )
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
A Two-Step Example
Each time step is 3 months K=21, r=12%
20
22
18
24.2
19.8
16.2
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
Behaviorof Stock Prices
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
Discrete time; discrete variable Discrete time; continuous variable Continuous time; discrete variable Continuous time; continuous variable
Categorization of Stochastic Processes
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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)
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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 . . .
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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)
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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 ( , ) ( , )
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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. .
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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