1
Finance School of Management
Chapter 15,Options &
Contingent Claims
Objective
?To show how the law of one price may
be used to derive prices of options
?To show how to infer implied
volatility from option
prices
2
Finance School of Management
Chapter 15 Contents
? How Options Work
? Investing with Options
? The Put-Call Parity Relationship
? Volatility & Option Prices
? Two-State Option Pricing
? Dynamic Replication & the Binomial Model
? The Black-Scholes Model
? Implied Volatility
3
Finance School of Management
Objectives
?To show how the Law of One Price can be
used to derive prices of options
?To show how to infer implied volatility
from option prices
4
Finance School of Management
Terms
– Underlying Asset,Call,Put,Strike (Exercise)
Price,Expiration (Maturity) Date,American /
European Option
– Out-of-the-money,in-the-money,at-the-money
– Tangible (Intrinsic) value,Time Value
5
Finance School of Management
T a b l e 1 5, 1 L i st o f I BM O p t i o n Pri ce s
( S o u r c e, W a l l S t r e e t J o u r n a l I n t e r a c t i v e E d i t i o n,M a y 2 9,1 9 9 8 )
I BM (I BM ) U n d e rl y i n g s t o c k p ri c e 1 2 0 1 / 1 6
C a l l, P u t,
S t r i k e E x pi r a t i on V ol um e La s t O pe n V ol um e La s t O pe n
I nt e r e s t I nt e r e s t
115 J u n 1372 7 4483 756 1 3 / 1 6 9692
115 O c t … … 2584 10 5 967
115 J a n … … 15 53 6 3 / 4 40
120 J u n 2377 3 1 / 2 8049 873 2 7 / 8 9849
120 O c t 121 9 5 / 1 6 2561 45 7 1 / 8 1993
120 J a n 91 1 2 1 / 2 8842 … … 5259
125 J u n 1564 1 1 / 2 9764 17 5 3 / 4 5900
125 O c t 91 7 1 / 2 2360 … … 731
125 J a n 87 1 0 1 / 2 124 … … 70
6
Finance School of Management
T a b l e 1 5, 2 L i s t o f I n d e x O p ti o n Pr i c e s
( S o u r c e, W a l l S t r e e t J o u r n a l I n t e r a c t i v e E d i t i o n,J u n e 6,1 9 9 8 )
S & P 5 0 0 I N D EX -A M C h i c a g o Ex c h a n g e
U nd e r l y i ng H i gh Low C l os e N e t From %
C ha ng e 3 1 - D e c C ha ng e
S & P 5 0 0 1 1 1 3, 8 8 1 0 8 4, 2 8 1 1 1 3, 8 6 1 9, 0 3 1 4 3, 4 3 1 4, 8
( S P X ) N e t O pe n
St r ik e V ol um e La s t C ha ng e I nt e r e s t
J u n 1 1 1 0 c a l l 2,0 8 1 1 7 1 / 4 8 1 / 2 1 5,7 5 4
J u n 1 1 1 0 p u t 1,0 7 7 10 - 1 1 1 7,1 0 4
J u l 1 1 1 0 c a l l 1,2 7 8 3 3 1 / 2 9 1 / 2 3,7 1 2
J u l 1 1 1 0 p u t 152 2 3 3 / 8 - 1 2 1 / 8 1,0 4 0
J u n 1 1 2 0 c a l l 80 12 7 1 6,5 8 5
J u n 1 1 2 0 p u t 211 17 - 1 1 9,9 4 7
J u l 1 1 2 0 c a l l 67 2 7 1 / 4 8 1 / 4 5,5 4 6
J u l 1 1 2 0 p u t 10 2 7 1 / 2 - 1 1 4,0 3 3
7
Finance School of Management
Terminal or Boundary Conditions for Call and Put Options
-20
0
20
40
60
80
100
120
0 20 40 60 80 100 120 140 160 180 200
Underlying Price
Do
lla
rs
Call Put
8
Finance School of Management
The Put-Call Parity Relation
? Two ways of creating a stock investment that is
insured against downside price risk
– Buying a share of stock and a put option (a protective-
put strategy)
– Buying a pure discount bond with a face value equal to
the option’s exercise price and simultaneously buying a
call option
9
Finance School of Management
Terminal Conditions of a Call and a Put Option with Strike = 100
Share Call Put Share_Put Bond Call_Bond
0 0 100 100 100 100
10 0 90 100 100 100
20 0 80 100 100 100
30 0 70 100 100 100
40 0 60 100 100 100
50 0 50 100 100 100
60 0 40 100 100 100
70 0 30 100 100 100
80 0 20 100 100 100
90 0 10 100 100 100
100 0 0 100 100 100
110 10 0 110 100 110
120 20 0 120 100 120
130 30 0 130 100 130
140 40 0 140 100 140
150 50 0 150 100 150
160 60 0 160 100 160
170 70 0 170 100 170
180 80 0 180 100 180
190 90 0 190 100 190
200 100 0 200 100 200
10
Finance School of Management
S t o c k,C a l l,P u t,B o n d
0
20
40
60
80
100
120
140
160
180
200
0 20 40 60 80 100 120 140 160 180 200
S t o c k P r i c e
S
t
o
c
k
,
C
a
l
l
,
P
u
t
,
B
o
n
d
,
P
u
t
+
S
t
o
c
k
,
C
a
l
l
+
B
o
n
d
C a l l
P u t
S h a r e _ P u t
B o n d
C a l l _ B o n d
S h a r e
11
Finance School of Management
Payoff Structure for Protective Put
Strategy
P os i t i on I f S T < E I f S T > E
S t oc k S T S T
P ut E - S T 0
S t oc k pl us pu t E S T
V alue of P os i t i on at M atur i t y D ate
12
Finance School of Management
Payoff Structure for a Pure Discount
Bond Plus a Call
P os i t i on I f S T < E I f S T > E
P ur e di s c ou nt bo nd w i t h f a c e E E
v a l u e o f E
C a l l 0 S T - E
P ur e di s c ou nt bo nd pl us c a l l E S T
V alue of P os i t i on at M atur i t y D ate
13
Finance School of Management
Put-Call Parity Equation
? ? S h a r eM a t u r i t yS t r i k eP u tr
S t r i k eM a t u r i t yS t r i k eC a l l
M a t u ri t y ???? ),(1),(
? ?
SP
r
E
C
T
??
?
?
1
14
Finance School of Management
Synthetic Securities
?The put-call parity relationship may be
solved for any of the four security variables
to create synthetic securities,
? C=S+P-B
? S=C-P+B
? P=C-S+B
? B=S+P-C
15
Finance School of Management
Converting a Put into a Call
?S = $100,E = $100,T = 1 year,r = 8%,P = $10,
C = 100 – 100/1.08 + 10 = $17.41
?If C = $18,the arbitrageur would sell calls at a price of
$18,and synthesize a synthetic call at a cost of $17.41,and
pocket the $0.59 difference between the proceed and the
cost
? ?
P
r
E
SC
T
?
?
??
1
16
Finance School of Management
Put-Call Arbitrage
I m m e d i a t e
P o s i t i o n C a s h F l o w
S e l l a c a l l $18 0 - ( S
T - $ 1 0 0 )
B u y a s t o c k ( $ 1 0 0 ) S
T S T
B o r r o w t h e p r e s e n t v a l u e o f $ 1 0 0 $ 9 2, 5 9 ( $ 1 0 0 ) ( $ 1 0 0 )
B u y a p u t ( $ 1 0 ) $ 1 0 0 - S
T
0
N e t c a s h f l o w s $ 0, 5 9 0 0
C a s h F l o w a t M a t u r i t y D a t e
I f S T < $ 1 0 0 I f S T > $ 1 0 0
B u y R e p l i c a t i n g P o r t f o l i o ( S y n t h e t i c C a l l )
17
Finance School of Management
Options and Forwards
? We saw in the last chapter that the discounted
value of the forward was equal to the current spot
? The relationship becomes
? ? TT r
FP
r
EC
)1(1 ?
??
?
?
Tr
EFPC
)1( ?
???
18
Finance School of Management
Implications for European Options
? If (F > E) then (C > P)
? If (F = E) then (C = P)
? If (F < E) then (C < P)
? E is the common exercise price
? F is the forward price of underlying share
? C is the call price
? P is the put price
19
Finance School of Management
C a l l a n d P u t a s a F u n c t i o n o f F o r w a r d
0
2
4
6
8
10
12
14
16
90 92 94 96 98 100 102 104 106 108 110
F o r w a r d
P
u
t
,
C
a
l
l
V
a
l
u
e
s
c a l l
put
a s y _ c a l l _ 1
a s y _ p u t _ 1
Strike = Forward
Call = Put
20
Finance School of Management
Put and Call as Function of Share Price
-10
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120 130 140 150
Share Price
Pu
t a
nd
C
all
Pr
ice
s
call
put
asy_call_1
asy_call_2
asy_put_1
asy_put_2
100/(1+r
)
21
Finance School of Management
P u t a n d C a l l a s F u n c t i o n o f S h a r e P r i c e
0
5
10
15
20
80 85 90 95 100 105 110 115 120
S h a r e P r i c e
P
u
t
a
n
d
C
a
l
l
P
r
i
c
e
s
c a l l
put
a s y _ c a l l _ 1
a s y _ c a l l _ 2
a s y _ p u t _ 1
a s y _ p u t _ 2
PV Strike Strike
22
Finance School of Management
V o l a t i l i t y a n d O p t i o n P r i c e s,P 0 = $ 1 0 0,S t r i k e = $ 1 0 0
S t o c k P r i c e C a l l P a y o f f P u t P a y o f f
L o w V o l a t i l i t y C a s e
R i s e 120 20 0
F a l l 80 0 20
E x p e c t a t i o n 100 10 10
H i g h V o l a t i l i t y C a s e
R i s e 140 40 0
F a l l 60 0 40
E x p e c t a t i o n 100 20 20
23
Finance School of Management
Two-State Option Pricing,Simplification
– The stock price can take only one of two possible
values at the expiration date of the option
– The exercise price is equal to the forward price of
the underlying stock
? The option’s price depends only on the volatility and
the time to maturity,the interest rate is assumed to be
zero
? The put and call have the same price
24
Finance School of Management
Two-State Option Pricing,An Example
?S = $100,E = $100,T = 1 year,d = 0,r = 0
?The stock price can either rise or fall by 20%
during the year
25
Finance School of Management
Binary Model,Call
?Implementation,
– the synthetic call,C,is created by
? buying a fraction x of shares,of the stock,S,and
simultaneously selling short risk-free bonds with a
market value y
? the fraction x is called the hedge ratio
yxSC ??
26
Finance School of Management
Binary Model,Creating the
Synthetic Call I m m e d i a t e
P o s i t i o n C a s h F l o w
C a l l o p t i o n -$ C $20 0
B u y x s h a r e s o f s t o c k - $ 1 0 0 x $120 x $80 x
B o r r o w y y - y - y
T o t a l r e p l i c a t i n g p o r t f o l i o - $ 1 0 0 x + y $20 0
C a s h F l o w a t M a t u r i t y D a t e
I f S 1 = $ 1 2 0 I f S 1 = $ 8 0
S y n t h e t i c C a l l
27
Finance School of Management
Binary Model,Call
?Specification,
– We have an equation,and given the value of the
terminal share price,we know the terminal
option value for two cases,
– By inspection,the solution is x=1/2,y = 40
yx
yx
??
??
800
12020
28
Finance School of Management
Binary Model,Call
?Solution,
– We now substitute the value of the parameters
x=1/2,y = 40 into the equation
– to obtain,
yxSC ??
10$401 0 0
2
1 ???C
29
Finance School of Management
Binary Model,Put
?Implementation,
– the synthetic put,P,is created by
? sell short a fraction x of shares,of the stock,S,and
simultaneously buy risk free bonds with a market
value y
? the fraction x is called the hedge ratio
yxSP ???
30
Finance School of Management
Binary Model,Creating the
Synthetic Put I m m e d i a t e
P o s i t i o n C a s h F l o w
P u t o p t i o n -$ P 0 $20
S e l l s h o r t x s h a r e s o f s t o c k $100 x - $ 1 2 0 x - $ 8 0 x
I n v e s t y i n t h e r i s k - f r e e a s s e t - y y y
T o t a l r e p l i c a t i n g p o r t f o l i o $100 x - y 0 $20
C a s h F l o w a t M a t u r i t y D a t e
I f S 1 = $ 1 2 0 I f S 1 = $ 8 0
S y n t h e t i c P u t
31
Finance School of Management
Binary Model,Put
?Specification,
– We have an equation,and given the value of
the terminal share price,we know the terminal
option value for two cases,
– By inspection,the solution is x = 1/2,y = 60
yx
yx
??
??
800
12020
32
Finance School of Management
Binary Model,Put
?Solution,
– We now substitute the value of the parameters
x=1/2,y = 60 into the equation
– to obtain,
yxSP ???
10$601 0 0
2
1 ????P
33
Finance School of Management
Decision Tree for Dynamic Replication
of a Call Option
F
$80
E
$100
D
$120
C
$90
A
$100
B
$110
34
Finance School of Management
Decision Tree for Dynamic Replication
of a Call Option
E
$100
D
$120
B
$110
0
C11
$20
–The terminal option value for two cases,
120x – y = 20
100x – y = 0
–By inspection,the solution is x=1,y = $100
–Thus,C11 = 1*$110 - $100 = $10
35
Finance School of Management
Decision Tree for Dynamic Replication
of a Call Option
F
$80
E
$100
C
$90
0
C12
0
–The terminal option value for two cases,
90x – y = 0
80x – y = 0
–By inspection,the solution is x=0,y = 0
–Thus,C12 = 0*$90 - $0 = $0
36
Finance School of Management
Decision Tree for Dynamic Replication
of a Call Option
C
$90
B
$110
A
$100
0
C0
$10
–The terminal option value for two cases,
110x – y = 10
90x – y = 0
–By inspection,the solution is x=1/2,y = $45
–Thus,C0 = (1/2)*$100 - $45 = $5
37
Finance School of Management
Decision Tree for Dynamic Replication
of a Call Option
F
$80
E
$100
D
$120
C
$90
A
$100
B
$110
Buy 1/2 share of stock
Borrow $45
Total investment $5
Buy another half share of stock
Increase borrowing to $100
Sell stock and pay off debt
Sell shares $120
Pay off debt -$100
Total $20
Sell shares $100
Pay off debt -$100
Total 0
38
Finance School of Management
Decision Tree for Dynamic Replication
of a Call Option
< - - - - - - - - - 0 M o n t h s - - - - - - - - - - > < - - - - - - - - - - - - - - - - - - 6 M o n t h s - - - - - - - - - - - - - - - - > 1 2 M o n t h s
S t o c k P r i c e x y C a l l P r i c e x y C a l l P r i c e
$ 1 2 0, 0 0 $ 2 0, 0 0
$ 1 1 0, 0 0 $ 1 0, 0 0 1 0 0, 0 0 % - $ 1 0 0, 0 0
$ 1 0 0, 0 0 5 0, 0 0 % - $ 4 5, 0 0 $ 0, 0 0
$ 9 0, 0 0 $ 0, 0 0 0, 0 0 % $ 0, 0 0
$ 8 0, 0 0 $ 0, 0 0
($120*100%) + (-$100) = $20
39
Finance School of Management
The Black-Scholes Model,The Limiting
Case of Binomial Model
? One can continuously and costlessly adjust the
replicating portfolio over time
? As the decision intervals in the binomial model
become shorter,the resulting option price from
the binomial model approaches the Black-Scholes
option price
40
Finance School of Management
The Black-Scholes Model,Notation
? C = price of call
? P = price of put
? S = price of stock
? E = exercise price
? T = time to maturity
? ln(.) = natural
logarithm
? e = 2.71828..,
? N(.) = cum,norm,dist’n
? The following are annual,
compounded
continuously,
? r = domestic risk free rate
of interest
? d = foreign risk free rate
or constant dividend yield
? σ = volatility
41
Finance School of Management
The Black-Scholes Model,Equations
? ? ? ?
? ? ? ?
21
21
1
2
2
2
1
2
1
ln
2
1
ln
dNEedNSeP
dNEedNSeC
Td
T
Tdr
E
S
d
T
Tdr
E
S
d
rTdT
rTdT
?????
??
??
?
?
?
?
?
?
????
?
?
?
?
?
?
?
?
?
?
?
?
????
?
?
?
?
?
?
??
??
?
?
?
?
?
42
Finance School of Management
The Black-Scholes Model,Equations
(Forward Form)
? ?
? ?
? ?
? ?
? ?? ?
? ?
? ?
? ?? ?EdNSedNeP
EdNSedNeC
T
T
E
Se
d
T
T
E
Se
d
TdrrT
TdrrT
Tdr
Tdr
21
21
2
2
2
1
2
1
ln
2
1
ln
?????
??
??
?
?
?
?
?
?
??
?
?
?
?
?
?
??
??
?
?
?
?
?
?
43
Finance School of Management
The Black-Scholes Model,Equations
(Simplified)
? ?
? ? ? ?? ?
? ? ? ?? ?
TST
S
PC
dNdNSPC
d
PdNdNSeC
TdTd
SeE
dT
Tdr
??
?
??
3 9 8 8 6.0
2
0 If
2
1;
2
1
If
21
21
21
???
???
?
???
???
?
?
?
44
Finance School of Management
Increases in,Call Put
Stock Price,S Increase Decrease
Exercise Price,E Decrease Increase
Volatility,sigma Increase Increase
Time to Expiration,T Ambiguous Ambiguous
Interest Rate,r Increase Decrease
Cash Dividends,d Decrease Increase
Determinants of Option Prices
45
Finance School of Management
V a l u e o f a C a l l a n d P u t O p t i o n s w i t h S t r i k e =
C u r r e n t S t o c k P r i c e
0
1
2
3
4
5
6
7
8
9
10
11
0, 00, 10, 20, 30, 40, 50, 60, 70, 80, 91, 0
T i m e - t o - M a t u r i t y
C
a
l
l
a
n
d
P
u
t
P
r
i
c
e
c a l l put
46
Finance School of Management
C a l l a n d P u t P r i c e s a s a F u n c t i o n o f V o l a t i l i t y
0
1
2
3
4
5
6
0, 0 0 0, 0 2 0, 0 4 0, 0 6 0, 0 8 0, 1 0 0, 1 2 0, 1 4 0, 1 6 0, 1 8 0, 2 0
V o l a t i l i t y
C
a
l
l
a
n
d
P
u
t
P
r
i
c
e
s
c a l l put
47
Finance School of Management
Implied Volatility
? The value of σ that makes the observaed market
price of the option equal to its Black-Scholes
formula value
? Approximation,
TS
C ?
?
2
?
48
Finance School of Management
C o m p u t i n g I m p l i e d V o l a t i l i t y
v o l a t i l i t y 0, 3 1 5 4
c a l l 1 0, 0 0 0 0
s t r i k e 1 0 0, 0 0 0 0
s h a r e 1 0 5, 0 0 0 0
r a t e _ d o m 0, 0 5 0 0
r a t e _ f o r 0, 0 0 0 0
m a t u r i t y 0, 2 5 0 0
f a c t o r 0, 0 2 4 9
d_1 0, 4 6 7 5
d_2 0, 3 0 9 8
n_d_1 0, 6 7 9 9
n_d_2 0, 6 2 1 7
c a l l _ p a r t _ 1 7 1, 3 9 3 4
c a l l _ p a r t _ 2 - 6 1, 3 9 3 4
e r r o r 0, 0 0 0 0
Insert any number to
start
Formula for option value
minus the actual call
value
49
Finance School of Management
C o m p u t i n g I m p l i e d V o l a t i l i t y
v o l a t i l i t y 0, 3 1 5 3 7 8 1 2 7 1 0 1 8 5 2
c a l l 10
s t r i k e 100
s h a r e 105
r a t e _ d o m 0, 0 5
r a t e _ f o r 0
m a t u r i t y 0, 2 5
f a c t o r = ( r a t e _ d o m - r a t e _ f o r + ( v o l a t i l i t y ^ 2 ) / 2 ) * m a t u r i t y
d_1 = ( L N ( s h a r e / s t r i k e ) + f a c t o r ) / ( v o l a t i l i t y * S Q R T ( m a t u r i t y ) )
d_2 = d _ 1 - v o l a t i l i t y * S Q R T ( m a t u r i t y )
n_d_1 = N O R M S D I S T ( d _ 1 )
n_d_2 = N O R M S D I S T ( d _ 2 )
c a l l _ p a r t _ 1 = n _ d _ 1 * s h a r e * E X P ( - r a t e _ f o r * m a t u r i t y )
c a l l _ p a r t _ 2 = - n _ d _ 2 * s t r i k e * E X P ( - r a t e _ d o m * m a t u r i t y )
e r r o r = c a l l _ p a r t _ 1 + c a l l _ p a r t _ 2 - c a l l
Finance School of Management
Chapter 15,Options &
Contingent Claims
Objective
?To show how the law of one price may
be used to derive prices of options
?To show how to infer implied
volatility from option
prices
2
Finance School of Management
Chapter 15 Contents
? How Options Work
? Investing with Options
? The Put-Call Parity Relationship
? Volatility & Option Prices
? Two-State Option Pricing
? Dynamic Replication & the Binomial Model
? The Black-Scholes Model
? Implied Volatility
3
Finance School of Management
Objectives
?To show how the Law of One Price can be
used to derive prices of options
?To show how to infer implied volatility
from option prices
4
Finance School of Management
Terms
– Underlying Asset,Call,Put,Strike (Exercise)
Price,Expiration (Maturity) Date,American /
European Option
– Out-of-the-money,in-the-money,at-the-money
– Tangible (Intrinsic) value,Time Value
5
Finance School of Management
T a b l e 1 5, 1 L i st o f I BM O p t i o n Pri ce s
( S o u r c e, W a l l S t r e e t J o u r n a l I n t e r a c t i v e E d i t i o n,M a y 2 9,1 9 9 8 )
I BM (I BM ) U n d e rl y i n g s t o c k p ri c e 1 2 0 1 / 1 6
C a l l, P u t,
S t r i k e E x pi r a t i on V ol um e La s t O pe n V ol um e La s t O pe n
I nt e r e s t I nt e r e s t
115 J u n 1372 7 4483 756 1 3 / 1 6 9692
115 O c t … … 2584 10 5 967
115 J a n … … 15 53 6 3 / 4 40
120 J u n 2377 3 1 / 2 8049 873 2 7 / 8 9849
120 O c t 121 9 5 / 1 6 2561 45 7 1 / 8 1993
120 J a n 91 1 2 1 / 2 8842 … … 5259
125 J u n 1564 1 1 / 2 9764 17 5 3 / 4 5900
125 O c t 91 7 1 / 2 2360 … … 731
125 J a n 87 1 0 1 / 2 124 … … 70
6
Finance School of Management
T a b l e 1 5, 2 L i s t o f I n d e x O p ti o n Pr i c e s
( S o u r c e, W a l l S t r e e t J o u r n a l I n t e r a c t i v e E d i t i o n,J u n e 6,1 9 9 8 )
S & P 5 0 0 I N D EX -A M C h i c a g o Ex c h a n g e
U nd e r l y i ng H i gh Low C l os e N e t From %
C ha ng e 3 1 - D e c C ha ng e
S & P 5 0 0 1 1 1 3, 8 8 1 0 8 4, 2 8 1 1 1 3, 8 6 1 9, 0 3 1 4 3, 4 3 1 4, 8
( S P X ) N e t O pe n
St r ik e V ol um e La s t C ha ng e I nt e r e s t
J u n 1 1 1 0 c a l l 2,0 8 1 1 7 1 / 4 8 1 / 2 1 5,7 5 4
J u n 1 1 1 0 p u t 1,0 7 7 10 - 1 1 1 7,1 0 4
J u l 1 1 1 0 c a l l 1,2 7 8 3 3 1 / 2 9 1 / 2 3,7 1 2
J u l 1 1 1 0 p u t 152 2 3 3 / 8 - 1 2 1 / 8 1,0 4 0
J u n 1 1 2 0 c a l l 80 12 7 1 6,5 8 5
J u n 1 1 2 0 p u t 211 17 - 1 1 9,9 4 7
J u l 1 1 2 0 c a l l 67 2 7 1 / 4 8 1 / 4 5,5 4 6
J u l 1 1 2 0 p u t 10 2 7 1 / 2 - 1 1 4,0 3 3
7
Finance School of Management
Terminal or Boundary Conditions for Call and Put Options
-20
0
20
40
60
80
100
120
0 20 40 60 80 100 120 140 160 180 200
Underlying Price
Do
lla
rs
Call Put
8
Finance School of Management
The Put-Call Parity Relation
? Two ways of creating a stock investment that is
insured against downside price risk
– Buying a share of stock and a put option (a protective-
put strategy)
– Buying a pure discount bond with a face value equal to
the option’s exercise price and simultaneously buying a
call option
9
Finance School of Management
Terminal Conditions of a Call and a Put Option with Strike = 100
Share Call Put Share_Put Bond Call_Bond
0 0 100 100 100 100
10 0 90 100 100 100
20 0 80 100 100 100
30 0 70 100 100 100
40 0 60 100 100 100
50 0 50 100 100 100
60 0 40 100 100 100
70 0 30 100 100 100
80 0 20 100 100 100
90 0 10 100 100 100
100 0 0 100 100 100
110 10 0 110 100 110
120 20 0 120 100 120
130 30 0 130 100 130
140 40 0 140 100 140
150 50 0 150 100 150
160 60 0 160 100 160
170 70 0 170 100 170
180 80 0 180 100 180
190 90 0 190 100 190
200 100 0 200 100 200
10
Finance School of Management
S t o c k,C a l l,P u t,B o n d
0
20
40
60
80
100
120
140
160
180
200
0 20 40 60 80 100 120 140 160 180 200
S t o c k P r i c e
S
t
o
c
k
,
C
a
l
l
,
P
u
t
,
B
o
n
d
,
P
u
t
+
S
t
o
c
k
,
C
a
l
l
+
B
o
n
d
C a l l
P u t
S h a r e _ P u t
B o n d
C a l l _ B o n d
S h a r e
11
Finance School of Management
Payoff Structure for Protective Put
Strategy
P os i t i on I f S T < E I f S T > E
S t oc k S T S T
P ut E - S T 0
S t oc k pl us pu t E S T
V alue of P os i t i on at M atur i t y D ate
12
Finance School of Management
Payoff Structure for a Pure Discount
Bond Plus a Call
P os i t i on I f S T < E I f S T > E
P ur e di s c ou nt bo nd w i t h f a c e E E
v a l u e o f E
C a l l 0 S T - E
P ur e di s c ou nt bo nd pl us c a l l E S T
V alue of P os i t i on at M atur i t y D ate
13
Finance School of Management
Put-Call Parity Equation
? ? S h a r eM a t u r i t yS t r i k eP u tr
S t r i k eM a t u r i t yS t r i k eC a l l
M a t u ri t y ???? ),(1),(
? ?
SP
r
E
C
T
??
?
?
1
14
Finance School of Management
Synthetic Securities
?The put-call parity relationship may be
solved for any of the four security variables
to create synthetic securities,
? C=S+P-B
? S=C-P+B
? P=C-S+B
? B=S+P-C
15
Finance School of Management
Converting a Put into a Call
?S = $100,E = $100,T = 1 year,r = 8%,P = $10,
C = 100 – 100/1.08 + 10 = $17.41
?If C = $18,the arbitrageur would sell calls at a price of
$18,and synthesize a synthetic call at a cost of $17.41,and
pocket the $0.59 difference between the proceed and the
cost
? ?
P
r
E
SC
T
?
?
??
1
16
Finance School of Management
Put-Call Arbitrage
I m m e d i a t e
P o s i t i o n C a s h F l o w
S e l l a c a l l $18 0 - ( S
T - $ 1 0 0 )
B u y a s t o c k ( $ 1 0 0 ) S
T S T
B o r r o w t h e p r e s e n t v a l u e o f $ 1 0 0 $ 9 2, 5 9 ( $ 1 0 0 ) ( $ 1 0 0 )
B u y a p u t ( $ 1 0 ) $ 1 0 0 - S
T
0
N e t c a s h f l o w s $ 0, 5 9 0 0
C a s h F l o w a t M a t u r i t y D a t e
I f S T < $ 1 0 0 I f S T > $ 1 0 0
B u y R e p l i c a t i n g P o r t f o l i o ( S y n t h e t i c C a l l )
17
Finance School of Management
Options and Forwards
? We saw in the last chapter that the discounted
value of the forward was equal to the current spot
? The relationship becomes
? ? TT r
FP
r
EC
)1(1 ?
??
?
?
Tr
EFPC
)1( ?
???
18
Finance School of Management
Implications for European Options
? If (F > E) then (C > P)
? If (F = E) then (C = P)
? If (F < E) then (C < P)
? E is the common exercise price
? F is the forward price of underlying share
? C is the call price
? P is the put price
19
Finance School of Management
C a l l a n d P u t a s a F u n c t i o n o f F o r w a r d
0
2
4
6
8
10
12
14
16
90 92 94 96 98 100 102 104 106 108 110
F o r w a r d
P
u
t
,
C
a
l
l
V
a
l
u
e
s
c a l l
put
a s y _ c a l l _ 1
a s y _ p u t _ 1
Strike = Forward
Call = Put
20
Finance School of Management
Put and Call as Function of Share Price
-10
0
10
20
30
40
50
60
50 60 70 80 90 100 110 120 130 140 150
Share Price
Pu
t a
nd
C
all
Pr
ice
s
call
put
asy_call_1
asy_call_2
asy_put_1
asy_put_2
100/(1+r
)
21
Finance School of Management
P u t a n d C a l l a s F u n c t i o n o f S h a r e P r i c e
0
5
10
15
20
80 85 90 95 100 105 110 115 120
S h a r e P r i c e
P
u
t
a
n
d
C
a
l
l
P
r
i
c
e
s
c a l l
put
a s y _ c a l l _ 1
a s y _ c a l l _ 2
a s y _ p u t _ 1
a s y _ p u t _ 2
PV Strike Strike
22
Finance School of Management
V o l a t i l i t y a n d O p t i o n P r i c e s,P 0 = $ 1 0 0,S t r i k e = $ 1 0 0
S t o c k P r i c e C a l l P a y o f f P u t P a y o f f
L o w V o l a t i l i t y C a s e
R i s e 120 20 0
F a l l 80 0 20
E x p e c t a t i o n 100 10 10
H i g h V o l a t i l i t y C a s e
R i s e 140 40 0
F a l l 60 0 40
E x p e c t a t i o n 100 20 20
23
Finance School of Management
Two-State Option Pricing,Simplification
– The stock price can take only one of two possible
values at the expiration date of the option
– The exercise price is equal to the forward price of
the underlying stock
? The option’s price depends only on the volatility and
the time to maturity,the interest rate is assumed to be
zero
? The put and call have the same price
24
Finance School of Management
Two-State Option Pricing,An Example
?S = $100,E = $100,T = 1 year,d = 0,r = 0
?The stock price can either rise or fall by 20%
during the year
25
Finance School of Management
Binary Model,Call
?Implementation,
– the synthetic call,C,is created by
? buying a fraction x of shares,of the stock,S,and
simultaneously selling short risk-free bonds with a
market value y
? the fraction x is called the hedge ratio
yxSC ??
26
Finance School of Management
Binary Model,Creating the
Synthetic Call I m m e d i a t e
P o s i t i o n C a s h F l o w
C a l l o p t i o n -$ C $20 0
B u y x s h a r e s o f s t o c k - $ 1 0 0 x $120 x $80 x
B o r r o w y y - y - y
T o t a l r e p l i c a t i n g p o r t f o l i o - $ 1 0 0 x + y $20 0
C a s h F l o w a t M a t u r i t y D a t e
I f S 1 = $ 1 2 0 I f S 1 = $ 8 0
S y n t h e t i c C a l l
27
Finance School of Management
Binary Model,Call
?Specification,
– We have an equation,and given the value of the
terminal share price,we know the terminal
option value for two cases,
– By inspection,the solution is x=1/2,y = 40
yx
yx
??
??
800
12020
28
Finance School of Management
Binary Model,Call
?Solution,
– We now substitute the value of the parameters
x=1/2,y = 40 into the equation
– to obtain,
yxSC ??
10$401 0 0
2
1 ???C
29
Finance School of Management
Binary Model,Put
?Implementation,
– the synthetic put,P,is created by
? sell short a fraction x of shares,of the stock,S,and
simultaneously buy risk free bonds with a market
value y
? the fraction x is called the hedge ratio
yxSP ???
30
Finance School of Management
Binary Model,Creating the
Synthetic Put I m m e d i a t e
P o s i t i o n C a s h F l o w
P u t o p t i o n -$ P 0 $20
S e l l s h o r t x s h a r e s o f s t o c k $100 x - $ 1 2 0 x - $ 8 0 x
I n v e s t y i n t h e r i s k - f r e e a s s e t - y y y
T o t a l r e p l i c a t i n g p o r t f o l i o $100 x - y 0 $20
C a s h F l o w a t M a t u r i t y D a t e
I f S 1 = $ 1 2 0 I f S 1 = $ 8 0
S y n t h e t i c P u t
31
Finance School of Management
Binary Model,Put
?Specification,
– We have an equation,and given the value of
the terminal share price,we know the terminal
option value for two cases,
– By inspection,the solution is x = 1/2,y = 60
yx
yx
??
??
800
12020
32
Finance School of Management
Binary Model,Put
?Solution,
– We now substitute the value of the parameters
x=1/2,y = 60 into the equation
– to obtain,
yxSP ???
10$601 0 0
2
1 ????P
33
Finance School of Management
Decision Tree for Dynamic Replication
of a Call Option
F
$80
E
$100
D
$120
C
$90
A
$100
B
$110
34
Finance School of Management
Decision Tree for Dynamic Replication
of a Call Option
E
$100
D
$120
B
$110
0
C11
$20
–The terminal option value for two cases,
120x – y = 20
100x – y = 0
–By inspection,the solution is x=1,y = $100
–Thus,C11 = 1*$110 - $100 = $10
35
Finance School of Management
Decision Tree for Dynamic Replication
of a Call Option
F
$80
E
$100
C
$90
0
C12
0
–The terminal option value for two cases,
90x – y = 0
80x – y = 0
–By inspection,the solution is x=0,y = 0
–Thus,C12 = 0*$90 - $0 = $0
36
Finance School of Management
Decision Tree for Dynamic Replication
of a Call Option
C
$90
B
$110
A
$100
0
C0
$10
–The terminal option value for two cases,
110x – y = 10
90x – y = 0
–By inspection,the solution is x=1/2,y = $45
–Thus,C0 = (1/2)*$100 - $45 = $5
37
Finance School of Management
Decision Tree for Dynamic Replication
of a Call Option
F
$80
E
$100
D
$120
C
$90
A
$100
B
$110
Buy 1/2 share of stock
Borrow $45
Total investment $5
Buy another half share of stock
Increase borrowing to $100
Sell stock and pay off debt
Sell shares $120
Pay off debt -$100
Total $20
Sell shares $100
Pay off debt -$100
Total 0
38
Finance School of Management
Decision Tree for Dynamic Replication
of a Call Option
< - - - - - - - - - 0 M o n t h s - - - - - - - - - - > < - - - - - - - - - - - - - - - - - - 6 M o n t h s - - - - - - - - - - - - - - - - > 1 2 M o n t h s
S t o c k P r i c e x y C a l l P r i c e x y C a l l P r i c e
$ 1 2 0, 0 0 $ 2 0, 0 0
$ 1 1 0, 0 0 $ 1 0, 0 0 1 0 0, 0 0 % - $ 1 0 0, 0 0
$ 1 0 0, 0 0 5 0, 0 0 % - $ 4 5, 0 0 $ 0, 0 0
$ 9 0, 0 0 $ 0, 0 0 0, 0 0 % $ 0, 0 0
$ 8 0, 0 0 $ 0, 0 0
($120*100%) + (-$100) = $20
39
Finance School of Management
The Black-Scholes Model,The Limiting
Case of Binomial Model
? One can continuously and costlessly adjust the
replicating portfolio over time
? As the decision intervals in the binomial model
become shorter,the resulting option price from
the binomial model approaches the Black-Scholes
option price
40
Finance School of Management
The Black-Scholes Model,Notation
? C = price of call
? P = price of put
? S = price of stock
? E = exercise price
? T = time to maturity
? ln(.) = natural
logarithm
? e = 2.71828..,
? N(.) = cum,norm,dist’n
? The following are annual,
compounded
continuously,
? r = domestic risk free rate
of interest
? d = foreign risk free rate
or constant dividend yield
? σ = volatility
41
Finance School of Management
The Black-Scholes Model,Equations
? ? ? ?
? ? ? ?
21
21
1
2
2
2
1
2
1
ln
2
1
ln
dNEedNSeP
dNEedNSeC
Td
T
Tdr
E
S
d
T
Tdr
E
S
d
rTdT
rTdT
?????
??
??
?
?
?
?
?
?
????
?
?
?
?
?
?
?
?
?
?
?
?
????
?
?
?
?
?
?
??
??
?
?
?
?
?
42
Finance School of Management
The Black-Scholes Model,Equations
(Forward Form)
? ?
? ?
? ?
? ?
? ?? ?
? ?
? ?
? ?? ?EdNSedNeP
EdNSedNeC
T
T
E
Se
d
T
T
E
Se
d
TdrrT
TdrrT
Tdr
Tdr
21
21
2
2
2
1
2
1
ln
2
1
ln
?????
??
??
?
?
?
?
?
?
??
?
?
?
?
?
?
??
??
?
?
?
?
?
?
43
Finance School of Management
The Black-Scholes Model,Equations
(Simplified)
? ?
? ? ? ?? ?
? ? ? ?? ?
TST
S
PC
dNdNSPC
d
PdNdNSeC
TdTd
SeE
dT
Tdr
??
?
??
3 9 8 8 6.0
2
0 If
2
1;
2
1
If
21
21
21
???
???
?
???
???
?
?
?
44
Finance School of Management
Increases in,Call Put
Stock Price,S Increase Decrease
Exercise Price,E Decrease Increase
Volatility,sigma Increase Increase
Time to Expiration,T Ambiguous Ambiguous
Interest Rate,r Increase Decrease
Cash Dividends,d Decrease Increase
Determinants of Option Prices
45
Finance School of Management
V a l u e o f a C a l l a n d P u t O p t i o n s w i t h S t r i k e =
C u r r e n t S t o c k P r i c e
0
1
2
3
4
5
6
7
8
9
10
11
0, 00, 10, 20, 30, 40, 50, 60, 70, 80, 91, 0
T i m e - t o - M a t u r i t y
C
a
l
l
a
n
d
P
u
t
P
r
i
c
e
c a l l put
46
Finance School of Management
C a l l a n d P u t P r i c e s a s a F u n c t i o n o f V o l a t i l i t y
0
1
2
3
4
5
6
0, 0 0 0, 0 2 0, 0 4 0, 0 6 0, 0 8 0, 1 0 0, 1 2 0, 1 4 0, 1 6 0, 1 8 0, 2 0
V o l a t i l i t y
C
a
l
l
a
n
d
P
u
t
P
r
i
c
e
s
c a l l put
47
Finance School of Management
Implied Volatility
? The value of σ that makes the observaed market
price of the option equal to its Black-Scholes
formula value
? Approximation,
TS
C ?
?
2
?
48
Finance School of Management
C o m p u t i n g I m p l i e d V o l a t i l i t y
v o l a t i l i t y 0, 3 1 5 4
c a l l 1 0, 0 0 0 0
s t r i k e 1 0 0, 0 0 0 0
s h a r e 1 0 5, 0 0 0 0
r a t e _ d o m 0, 0 5 0 0
r a t e _ f o r 0, 0 0 0 0
m a t u r i t y 0, 2 5 0 0
f a c t o r 0, 0 2 4 9
d_1 0, 4 6 7 5
d_2 0, 3 0 9 8
n_d_1 0, 6 7 9 9
n_d_2 0, 6 2 1 7
c a l l _ p a r t _ 1 7 1, 3 9 3 4
c a l l _ p a r t _ 2 - 6 1, 3 9 3 4
e r r o r 0, 0 0 0 0
Insert any number to
start
Formula for option value
minus the actual call
value
49
Finance School of Management
C o m p u t i n g I m p l i e d V o l a t i l i t y
v o l a t i l i t y 0, 3 1 5 3 7 8 1 2 7 1 0 1 8 5 2
c a l l 10
s t r i k e 100
s h a r e 105
r a t e _ d o m 0, 0 5
r a t e _ f o r 0
m a t u r i t y 0, 2 5
f a c t o r = ( r a t e _ d o m - r a t e _ f o r + ( v o l a t i l i t y ^ 2 ) / 2 ) * m a t u r i t y
d_1 = ( L N ( s h a r e / s t r i k e ) + f a c t o r ) / ( v o l a t i l i t y * S Q R T ( m a t u r i t y ) )
d_2 = d _ 1 - v o l a t i l i t y * S Q R T ( m a t u r i t y )
n_d_1 = N O R M S D I S T ( d _ 1 )
n_d_2 = N O R M S D I S T ( d _ 2 )
c a l l _ p a r t _ 1 = n _ d _ 1 * s h a r e * E X P ( - r a t e _ f o r * m a t u r i t y )
c a l l _ p a r t _ 2 = - n _ d _ 2 * s t r i k e * E X P ( - r a t e _ d o m * m a t u r i t y )
e r r o r = c a l l _ p a r t _ 1 + c a l l _ p a r t _ 2 - c a l l
