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.
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010

STANDARD SYMBOLS:

STAKES:
◦ 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
◦ NC = number of calls
◦ NP = number of puts
◦ NS = number of shares of stock
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)
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)
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
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
Profit
Profit
K
K
ST
ST
(a)
(b)
Profit
Profit
K
ST
(c)
K
(d)
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
ST
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
0
0
K2 - K1
ST ≥ K2
0
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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 < $35
ST ≤ $30
ST - $30 - $3
ST - $30 -$3
0 - $3
$35 - ST +$1
0+$1
0+$1
$3
ST - $32
-$2
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
Profit
ST
K1
K2
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
Profit
K1
K2
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
ST
Profit
K1
K2
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
ST
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
Payoff from
Long Call
Option
Payoff from
Short Call
Option
Total Payoff
ST ≥ K2
K1 < ST < K2
ST ≤ K1
ST - K2
0
0
K1 - ST
K1 - ST
0
-(K2 - K1)
-(ST ≥ K1)
0
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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.
Stock price
Range
Payoff from
Long Call
Option
Payoff from
Short Call
Option
Total Payoff
ST ≥ $35
$30 < ST < $35
ST ≤ $30
ST - $35
0
0
$30 - ST
$30 - ST
0
-($35 - $30)
-(ST ≥ $30)
0
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
Profi
t
K1
K2
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
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.
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010

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
0
0
0
-2(ST - K2)
-2(ST - K2)
0
0
0
K3 - ST
ST - K1
0
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

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 <$60
ST ≤ $55
ST - $55
ST - $55
ST - $55
0
ST - $65
0
0
0
-2(ST - $60)
-2(ST - $60)
0
0
0
$65 - ST
ST -$55
0
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
Profit
K1
K2
K3
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
ST
Profit
K1
K2
K3
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
ST
Profit
ST
K
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
Profit
ST
K
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
Straddle: Buying a call and a put with the same strike price and expiration
Date.
Stock price
Range
Payoff from Call
Payoff from Put
Total Payoff
ST ≥ K
ST < K
ST – K
0
0
K - ST
ST - K
K - ST
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
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.
Stock price
Range
Payoff from Call
Payoff from Put
Total Payoff
ST ≥ $70
ST < $70
ST – $70 -$4
0 - $4
0 -$3
$70 - ST - $3
ST - $77
$63 - ST
MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010
Profit
K
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
ST
Profit
Profit
K
Strip
ST
K
Strap
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
ST
Profit
K1
K2
ST
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010


A stock price is currently $20
In three months it will be either $22 or $18
Stock Price = $22
Stock price = $20
Stock Price = $18
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
A 3-month call option on the stock has a strike
price of 21.
Stock Price = $22
Option Price = $1
Stock price = $20
Option Price=?
Stock Price = $18
Option Price = $0
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010

Consider the Portfolio:
long D shares
short 1 call option
22D – 1
18D

Portfolio is riskless when 22D – 1 = 18D or
D = 0.25
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010



The riskless portfolio is:
long 0.25 shares
short 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



The portfolio that is
long 0.25 shares
short 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
A derivative lasts for time T and is
dependent on a stock
S
ƒ
S(1+a)=Su
ƒu
S(1-a)=Sd
ƒd
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010

Consider the portfolio that is long D shares and
short 1 derivative
SuD – ƒu
SdD – ƒd

The portfolio is riskless when SuD – ƒu = Sd D – ƒd or
ƒu  f d
D
Su  Sd
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010




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

Substituting for D we obtain
ƒ = [ p ƒu + (1 – p )ƒd ]e–rT
where
p
e
rT
1 a
2a
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010



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 riskfree rate
Su
S
ƒ
ƒu
Sd
ƒd
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
Su = 22
ƒu = 1
S
ƒ

Sd = 18
ƒd = 0
Since p is a risk-neutral
probability 20e0.12 0.25 = 22p +
18(1 – p ); p = 0.6523
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
Su = 22
ƒu = 1
S
ƒ
Sd = 18
ƒd = 0
The value of the option is
e–0.120.25 [0.65231 + 0.34770]
= 0.633
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
One way of matching the volatility is to set
e rT  1  a
p
2a
s
ae
Dt
1
where s is the volatility and Dt is the length of
the time step. This is the approach used by
Cox, Ross, and Rubinstein
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
24.2
22
19.8
20
18


Each time step is 3 months
K=21, r=12%
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
16.2
D
22
20
1.2823
A
B
2.0257
18
E
F

19.8
0.0
C
0.0

24.2
3.2
16.2
0.0
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
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
K = 52, Dt = 1yr
r = 5%
D
60
50
4.1923
A
B
1.4147
40
72
0
48
4
E
C
9.4636
F
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
32
20
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010




Discrete time; discrete variable
Discrete time; continuous variable
Continuous time; discrete variable
Continuous time; continuous variable
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
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
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
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.
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

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
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.
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.
Dz   Dt where  is a random drawing from N(0,1)
2. The values of Dz for any 2 different (nonoverlapping) periods of time are independent
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
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
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
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
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
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
Dx  a Dt  b  Dt



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
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
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
Dx  a ( x , t ) Dt  b( x , t ) Dt
is only true in the limit as Dt tends to zero
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
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
dS  mSdt  sSdz
where m is the expected return s is
the volatility.
The discrete time equivalent is
DS  mSDt  sS Dt
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
DS  0.0014 S  0.02 S
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
Period
Stock Price at
Random
Start of Period Sample for 
Change in Stock
Price, 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
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010

A Taylor’s series expansion of G (x , t)
gives
G
G
G 2
DG 
Dx 
Dt  ½ 2 Dx
x
t
x
2G
2G 2

Dx Dt  ½ 2 Dt 
xt
t
2
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
In ordinary calculus we get
G
G
DG 
Dx
Dt
x
t
In stochastic calculus we get
G
G
 2G
2
DG 
D x
Dt ½
D
x
x
t
 x2
because Dx has a component which is of order Dt
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
Suppose
dx  a( x, t )dt  b( x, t )dz
so that
Dx = a Dt + b  Dt
Then ignoring terms of higher order than Dt
G
G
2G 2 2
DG 
Dx 
Dt  ½
b  Dt
2
x
t
x
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
Since   N (0,1), E ( )  0
E ( )  [ E ( )]  1
2
2
E ( )  1
2
It follows that E ( 2 Dt )  Dt
The variance of Dt is proportion al to Dt 2 and can
be ignored. Hence
G
G
1 G 2
DG 
Dx 
Dt 
b Dt
2
x
t
2 x
2
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
Taking limits
G
G
2G 2
dG 
dx 
dt  ½ 2 b dt
x
t
x
Substituting
dx  a dt  b dz
We obtain
 G
G
 2G 2 
G
dG  
a
 ½ 2 b  dt 
b dz
t
x
x
 x

This is Ito's Lemma
MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
The stock price process is
d S  m S dt  s S d z
For a function G of S & t
 G
G
 2G 2 2 
G
dG  
mS 
½
s S dz
2 s S  dt 
t
S
S
S

MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010
1. The forward price of a stock for a contract
maturing at time T
G  S er ( T  t )
dG  (m  r )G dt  sG dz
2. G  ln S
 s2 
dG   m   dt  s dz
2

MILJAN KNEŽEVIĆ, BANJA LUKA AUGUST 2010