Aalto University
2013
Derivatives
LECTURE 3
Matti Suominen
2
MULTIPERIOD CASE
Extend basic case (EX = 50, r
f
= 5%) by 1 period:
t = 0
t = 1
t=2
t = 0
t = 1
t=2
60.5
C
uu
= 10.5
55
C
u
= 7.5
50
49.5
C
0
= 5.36
C
ud
= 0
45
C
d
= 0
40.5
C
dd
= 0
At time t=1, how much money is needed to purchase the
duplicating portfolio?
•
If S
u
= 55
h
u
= 0.955
B
u
= 45.0
ð
C
u
= 7.5
•
If S
d
= 45
C
d
= 0
How about at time 0?
•
S
0
= 50
h
0
= 0.75
B
0
= 32.14
C
0
= 5.36
3
Dynamic Hedging:
# of shares
€
amount
in shares
€
amount
borrowed
Value of
portfolio
At t=0, S
0
=50
At t=1, if S
1
=55
before
rebalancing
after rebalancing
rebalancing trades:
Buy:
Borrow:
At t=1, if S
1
=45
before
rebalancing
after rebalancing
rebalancing trades:
Sell:
Borrow:
4
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
ð
5.36
=
1.05
0.25x0
+
0.75x7.5
=
r
+
1
p)C

(1
+
pC
=
C
f
d
u
0
5
ALLOWING FOR MORE FREQUENT PRICE CHANGES:
Keep T fixed and reduce
Δ
t:
u
²
S
uS
S
udS
dS
d
²
S
Δ
t
T
How to choose u and d?
Take
u
=
e
t
σ
Δ
and
d
= 1/
u,
where
σ
is the empirically observed volatility of the return on
the stock.
6
What is the final distribution of stock prices?
As
Δ
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
σ
2
T.
This implies that the stock prices are distributed
lognormally
:
7
What is the limiting value of the call option?
BlackScholes Option Pricing Formula
Let:
σ
= 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
Δ
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 (
/
)
σ
σ
d
2
=
d
1

σ
T