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≤
At time
t
T
the option is said to be
in the money
,if
S
t
>K
, the option is
out of
the money
,if
S
t
<K
, and the option is said to be
at the money
,if
S
t
≈
K
.
Modelling Assumptions
Most market models for stocks assume the existence
of a
riskless bank account
with
riskless interest rate r
0. We will also consider
stochastic interest rate models where this is not the case. However, unless explic-
itly stated otherwise, we assume that money can be deposited and borrowed from
this bank account with continuously compounded, known interest rate
r
. Therefore,
1 currency unit in this account at
t
≥
0 will give
e
rt
=
currency units at time
t
, and
0, we will have to pay back
e
rt
currency
units at time
t
.Wealsoassumea
frictionless market
, i.e. there are no transaction
costs, and we assume further that there is no default risk, all market participants are
rational, and the market is efficient, i.e. there is no arbitrage.
if 1 currency unit is borrowed at time
t
=
1.2 Stochastic Processes
We refer to the texts Mao [120] and Øksendal [131] for an introduction to stochastic
processes and stochastic differential equations. Much more general stochastic pro-
cesses in the Markovian and non-Markovian setup are treated in the monographs
Gihman and Skorohod [71-73] as well as Jacod and Shiryaev [97].
Prices of the so-called risky assets can be modelled by stochastic processes in
continuous time
t
where the maturity
T>
0isthe
time horizon
. To describe
stochastic price processes, we require a probability space
(Ω,
F
,
P
)
. Here,
Ω
is the
set of elementary events,
∈[
0
,T
]
F
is a
σ
-algebra which contains all events (i.e. subsets
of
Ω
) of interest and
P :
F
→[
0
,
1
]
assigns a probability of any event
A
∈
F
.
We shall always assume the probability space to be complete, i.e. if
B
⊂
A
with
A
∈
F
and
P[
A
]=
0, then
B
∈
F
. We equip
(Ω,
F
,
P
)
with a
filtration
,i.e.afam-
ily
F ={
F
t
:
0
≤
t
≤
T
}
of
σ
-algebras which are monotonic with respect to
t
in
the sense that for 0
≤
s
≤
t
≤
T
holds that
F
s
⊆
F
t
⊆
F
T
⊂
F
. In financial mod-
elling, the
σ
-algebra
represents the information available in the model up to
time
t
. We assume that the filtered probability space
(Ω,
F
,
P
,
F
)
satisfies the
usual
assumptions
,i.e.
F
t
∈ F
(i)
F
is
P
-complete,
(ii)
F
0
contains all
P
-null subsets of
Ω
and
F
t
=
s>t
F
s
.
(iii) The filtration
F
is right-continuous:
Definition 1.2.1
(Stochastic processes) A
stochastic process X
={
X
t
:
0
≤
t
≤
T
}
is a family of random variables defined on a probability space
(Ω,
F
,
P
,
F
)
,
parametrised by the time variable
t
.For
ω
∈
Ω
, the function
X
t
(ω)
of
t
is called
a sample path
of
X
. The process is
F
-
adapted
if
X
t
is
F
t
measurable (denoted by
X
t
∈
F
t
) for each
t
.
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