No short sale constraints, i.e., we can take any long and short positions in all traded
assets.
There is no bidask spread, i.e., we sell and we buy at the same price.
There is no counterparty risk.
Financial Engineering with Stochastic Calculus I
27 / 30
The binomial model
Oneperiod binomial model: Solution
Simplest example: Oneperiod binomial model
We assume that prices for the money market account
B
t
and stock
S
t
are quoted at
times
t
= 0
(today) and
t
= 1
(option maturity).
Suppose
B
0
=
B
1
= 1
,
S
0
= 100
,
and
S
1
is a random variable taking values
S
1
= 115
or
S
1
= 90
each with probability
p
=
1
2
.
Consider a call with strike
K
= 100
. Value at time
0
?
Naive response:
Expectation of the payoff
(
S
1

100)
+
? Gives
E
(
S
1

100)
+
=
1
2
(115

100)
+
+
1
2
(90

100)
+
= 7
.
5
Turns out to be
too expensive
: A suitable investment strategy can
generate the
option payoff
out of less capital!
Indeed, suppose at time
0
we invest into
η
shares of bond and
Δ
shares of stock, and
require that at time
1
η
+ Δ
·
115 = (115

100)
+
,
(2.1)
η
+ Δ
·
90 = (90

100)
+
.
(2.2)
We find
Δ = 0
.
6
,
η
=

54
. This requires
η
+ Δ
·
100 = 6
initial capital and does the
same job as the call.
Financial Engineering with Stochastic Calculus I
28 / 30
The binomial model
Oneperiod binomial model: Solution
The initial capital
C
0
= 6
is the
fair value
at time
0
of the option in the above example.
Any price
C
0
6
= 6
for the option would introduce an
arbitrage opportunity
into the
market:
If
C
0
>
6
, the seller could make a guaranteed profit of
C
0

6
by selling the option
and simultaneously employing the above investment strategy.
If
C
0
<
6
, the buyer could make a guaranteed profit of
6

C
0
by buying the option
and simultaneously employing the reverse strategy (
Δ =

0
.
6
,
η
= 54
).
Such arbitrage opportunities (possibility of a riskless profit without net investment of
capital) are unrealistic in most markets.
Absence of arbitrage
is a fundamental concept in financial market models and in the
theory of derivative pricing.
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 Fall '09
 J.WISSEL