DERIVATIVES AND RISK
MANAGEMENT
Op#on Valua#on I
(Lecture 2)
Ma/ Suominen
January 2018

To value [email protected] (C
0
) we first need a model of stock
prices:
STOCK PRICES AS A RANDOM WALK
In eﬃcient markets prices reﬂect all past public
[email protected]
ð
The expected returns depend only on beta
ð
Prices move when unexpected news come to the
market
Normal [email protected] (with a mean that depends on beta)
is a good [email protected] for the [email protected] of stock
returns.

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
60
70
80
90
100
110
0
50
100 150 200 250
50
60
70
80
90
100
110
120
130
140
150
0
50
100 150 200 250
50
60
70
80
90
100
110
120
130
140
150
0
50
100 150 200 250

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
.
S
0
u
S
d
S
ud
S
u
2
S
d
2
S
T
Δ
t
If we select
u
and
d
“correctly”, then as
Δ
t
→
0 this discrete
[email protected]
approaches a
[email protected] 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 [email protected] expires. In
the next lecture we will
extend the analysis to
[email protected] periods.

EXACT OPTION PRICING
Basic idea:
[email protected] (and other [email protected]) are priced by
no arbitrage.
Find a por\olio [email protected] the payoff of the [email protected]
A simple case:
Δ
t = 1
S
0
= 50,
u
= 1.1,
d
= 0.9
S
0
= 50
uS
0
= 55
dS
0
= 45
What is the price of a call [email protected], expiring at t = 1, with EX = 50 when r
f
= 5% ?
C
0
= ?
C
u
= max(S-‐EX;0) = 5
C
d
= max(S-‐EX;0) = 0

[email protected] Pricing by No-‐Arbitrage
•
Given our model of stock prices, we price [email protected] by
[email protected] a
replica#ng porMolio
.
•
[email protected] por\olio:
–
buy h
0
stocks
–
borrow B
0
So that:
55 h
0
– 1.05 B
0
= 5
45 h
0
– 1.05 B
0
= 0

[email protected] Pricing by No-‐Arbitrage
This leads to two [email protected] in two unknowns, h
0
and B
0
55 h
0
– 1.05 B
0
= 5
45 h
0
– 1.05 B
0
= 0
=> h
0
= 0.5
[Hedge [email protected]]
B
0
= 21.43
C
0
=
This is the basic idea in [email protected] pricing: The price of the [email protected] must equal
the value of the [email protected] por\olio.
h
0
S – B
0
= 0.5
x
50 – 21.43 = 3.57
By no-‐arbitrage, the current value of the [email protected] must equal the current value
of the [email protected] por\olio.

How to take advantage of mispricing?
•
What if the call [email protected] is trading at 3.80 in the market?
–
Buy low:
–
Sell high:
t = 0
t = 1
‘up’ state
‘down’ state
Buy
½
share
Borrow 21.43
Write call
Net
+21.43
-‐25
+3.80
+0.23
-‐ 22.5
+27.5
-‐5
0
-‐22.5
= (1+r)21.43
+22.5
0
0
= 3.57
= 5
= 0
[email protected]
por\olio

More generally:
Number of shares bought:
h
0
=
Δ
=
C
u
−
C
d
S
u
−
S
d
=

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