DERIVATIVES AND RISK
MANAGEMENT
Option Valuation II
(Lecture 3)
Matti Suominen
January 2018
TwoPeriod Binomial Tree Model
•
S
0
=50,
u
=1.1,
d
=0.9, (1+r
f
)=1.05, EX=50, T=2
S
0
=
S
u
= 55
S
d
= 45
S
uu
= 60.5 = 50 x (1.1)
2
S
ud
= 49.5
S
dd
= 40.5
C
0
=
C
u
=
C
d
=
C
uu
=
C
ud
=
C
dd
=
•
Stock price tree:
•
Call price tree:
max(10.5;0) =10.5
0
0
TwoPeriod Binomial Tree Model
•
Start at t=2 and work backwards, solving oneperiod trees,
using either
–
Replicating portfolio method, or
–
Riskneutral pricing.
•
Using replicating portfolio method
First, at
S
u
node
(time t=1)
(replicating portfolio: h
u
shares, borrow B
u
)
50
.
7
00
.
45
55
955
.
0
B
S
h
C
00
.
45
05
.
1
0
5
.
49
955
.
0
)
r
1
(
C
S
h
B
955
.
0
5
.
49
5
.
60
0
5
.
10
S
S
C
C
h
u
u
u
u
f
ud
ud
u
u
ud
uu
ud
uu
u
=

´
=

=
=

´
=
+

=
=


=


=
TwoPeriod Binomial Tree Model
•
This we already saw but just to check, at
S
d
node
(time t=1)
(replicating portfolio: h
d
shares, borrow B
d
)
•
Now finally at
S
0
node
(time t=0)
(replicating portfolio: h
0
shares, borrow B
0
)
36
.
5
14
.
32
50
75
.
0
B
S
h
C
14
.
32
05
.
1
0
45
75
.
0
)
r
1
(
C
S
h
B
75
.
0
45
55
0
5
.
7
S
S
C
C
h
0
0
0
0
f
d
d
0
0
d
u
d
u
0
=

´
=

=
=

´
=
+

=
=


=


=
0
C
,
0
B
,
0
h
d
d
d
=
=
=
What if call
trades at 7,36?
Arbitrage!
# of shares
€ amount
in shares
€ amount
borrowed
Value of
portfolio
At t=0, S
0
=50
0.75
37.5
(=0.75x50)
32.14
5.36
At t=1, if S
1
=55
before
rebalancing
0.75
41,25
33,75
7.5
after rebalancing
0.955
52.5
45
7.5
rebalancing trades:
Buy:
0,9550,75 shares @ 55
11,25
Borrow:
(45  33,75)
11,25
NET
0
At t=1, if S
1
=45
before
rebalancing
0,75
33,75
33,75
0
after rebalancing
0
0
0
0
Dynamic Hedging:
rebalancing trades:
Sell:
0,75 shares @ 45
33,75
Pay back loan:
33,75
 33,75
NET
0
Note that “risk neutral pricing” still works.
We just move down the “option tree”:
0.75
=
0.9

1.1
0.9

1.05
=
d

u
d

)
r
+
(1
=
p
f
7.5
=
1.05
0.25x0
+
0.75x10.5
=
r
+
1
p)C

(1
+
pC
=
C
f
ud
uu
u
0
=
C
d
5.36
=
1.05
0.25x0
+
0.75x7.5
=
r
+
1
p)C

(1
+
pC
=
C
f
d
u
0
ALLOWING FOR MORE FREQUENT PRICE CHANGES:
Keep T fixed and reduce Δt:
u
²
S
uS
S
udS
dS
d
²
S
D
t
T
How to choose u and d?
Take
u
=
e
t
s
D
and
d
= 1/
u,
where
σ
is the empirically observed volatility of the return on the stock.
Annual volatility
Time during which
prices change once as
fraction of year
What is the final distribution of stock prices?
As
D
t
®
0 this approaches a continuous model
of stock prices, where the stock return (between
time t = 0 and t = T) is distributed
normally
with some mean
μ
T (that depends on the true
probability of stock price going up) and
variance
s
2
T.
And
the
stock
prices
are
distributed
lognormally
:
What is the limiting value of the call option?
BlackScholes Option Pricing Formula
Let:
s
= volatility of returns
r
c
= interest rate (continuously compounded)
T
= time to expiration
N(
d
) = Pr {z
£
d
}, where z is distributed according to a
standard normal distribution.
As
D
t
®
0 the duplicating portfolio at time 0 approaches:
C
0
= h
0
S
0
 B
0
where:
h
0
= N(
d
1
)
B
0
= N(
d
2
) PV(EX)
d
S
EX
r T
T
T
c
1
0
2
=
+
+
ln (
/
)
s
s
d
2
=
d
1

s
T
Example:
Call option on a stock:
S
0
= 50
EX
= 45
s
= 18% (annual)
T
= 3 months = .25
r
A
= 6% (annual)
(
)
d
r T
c
1
50 45
18
25
18
25
2
=
+
+
ln
/
.
.
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 Summer '14
 sam
 Derivatives, Valuation