Aalto University
Derivatives
LECTURE 2
Matti Suominen

2
STOCK PRICES AS A RANDOM WALK
•
In efficient markets prices reflect all past public information.
ð༏
The expected returns depend only on beta
ð༏
Prices move when unexpected news come to the market
•
Normal distribution (with a mean that depends on beta) is a useful
approximation for the distribution of stock returns.

3
Examples of stock price paths: real and simulated
50
60
70
80
90
100
110
0
50
100
150
200
250
50
60
70
80
90
100
110
0
50
100
150
200
250
50
70
90
110
130
150
170
0
50
100
150
200
250
50
70
90
110
130
150
0
50
100
150
200
250
50
70
90
110
130
150
0
50
100
150
200
250
50
60
70
80
90
100
110
0
50
100
150
200
250

4
BINOMIAL MODEL OF STOCK PRICES
We model stock prices using a binomial tree:
Each period the stock price can go up by a factor
u
or down by a factor
of
d
.
If we select
u
and
d
“correctly”, then as
Δ
t
→
0 this discrete
approximation approaches a continuous model of stock prices where
the stock returns are normally distributed.
To begin with, we will assume that there is only one period before the
option expires. In the next lecture we will extend the analysis to
multiple periods.
u
²
S
uS
S
udS
dS
d
²
S
Δ
t
T

5
EXACT OPTION PRICING
Basic idea:
Options (and other derivatives) are priced by
no arbitrage.
Find a portfolio replicating the payoff of the option.
A simple case:
Δ
t = 1
S
0
= 50,
u
= 1.1,
d
= 0.9
50
What is the price of a call option, expiring at t = 1, with EX = 50 when
r
f
=
5% ?
1.1 x 50 = 55
0.9 x 50 = 45

6
Assume traders can borrow and lend at the risk free rate.
Find a portfolio composed of the underlying stock (buy
h
0
stocks)
combined with riskless lending/borrowing (borrow
B
0
) that replicates
the payoff of the call.
t = 0
t = 1
Value of call at t = 1
Value of replicating
portfolio at t = 1
55
C
u
=
50
45
C
d
=
h
0
and B
0
must satisfy the following two conditions:
*
*
Value of call:
This is the basic idea in derivatives pricing: The price of the derivative
must equal the value of the replicating portfolio.

7
More generally:
Number of shares bought:
Ratio
Hedge
or
Delta
Option
45
55
0
5
0
=
−
−
=
−
−
=
d
u
d
u
S
S
C
C
h
Amount borrowed:
f
d
d
r
C
S
h
B
+
−
=
1
0
0
Value of the call:
C
0
= h
0
S
0
- B
0
Note that call price does
not
depend on the probability that the stock
price goes up; nor on the traders’ risk preferences !!!

8
VALUE OF PUT OPTION
With similar procedure, consider a put option with EX = 50 with
expiration date at t = 1:
t = 0
t = 1
Value of put at t = 1
Value of replicating
portfolio at t = 1
55
P
u
= max{EX-S
u
,0} = 0
55h
0
+ 1.05B
0
50
45
P
d
= max{EX-S
d
,0} = 5
45h
0
+ 1.05B
0
Hedge ratio:
5
.
0

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- Winter '12
- Options, share price