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4.5 Extensions of the Black-Scholes Model
We end this chapter by considering two extensions of the Black-Scholes model
d
S
t
=
rS
t
d
t
+
σS
t
d
W
t
,S
0
=
s
≥
0
.
In the
constant elasticity of variance
(CEV) model,
σS
t
is replaced by
σS
t
for some
0
<ρ<
1. Another possible extension is to replace the constant volatility
σ
by a
deterministic function
σ(s)
which leads to the so-called
local volatility
models.
4.5.1 CEV Model
Under a unique equivalent martingale measure, the stock price dynamics are given
by
σS
t
d
S
t
=
rS
t
d
t
+
d
W
t
,
0
=
s
≥
0
,
(4.17)
where
ρ
is the
elasticity of variance
. We assume 0
<ρ<
1. Under this condition,
the point 0 is an attainable state. As soon as
S
0, we keep
S
equal to zero, with
the resulting process still satisfying (
4.17
). Note that (
4.17
)isoftheform(1.2)but
with
σ(t,s)
=
σs
ρ
, non-Lipschitz. The transformation to log-price,
x
log
s
, will
not allow removing the factor
s
ρ
. A formal application of Theorem
4.1.4
yields that
the value
V(t,s)
of a European vanilla with payoff
g
is the solution of
=
=
CEV
ρ
∂
t
V
+
A
V
−
rV
=
0in
J
× R
≥
0
,
(4.18)
=
with the terminal condition
V(T,s)
g(s)
and generator
1
2
σ
2
s
2
ρ
∂
ss
f(s)
CEV
ρ
(
A
f )(s)
=
+
rs∂
s
f(s).
Note that for
ρ
1, the CEV generator is the same as the BS generator. In (
4.18
),
we change to time-to-maturity
t
=
→
T
−
t
and localize to a bounded domain
G
:=
(
0
,R)
,
R>
0. Thus, we consider
CEV
v
∂
t
v
−
A
+
rv
=
0
in
J
×
G,
(4.19)
v
=
0
on
J
×{
R
}
,
v(
0
,s)
=
g(s)
in
G.
For the variational formulation of (
4.19
), we multiply the first equation in (
4.19
)by
a test function
w
and integrate from
s
R
.Using
s
2
ρ
∂
ss
v
∂
s
(s
2
ρ
∂
s
v)
=
0to
s
=
=
−
2
ρs
2
ρ
−
1
∂
s
v
, we find, upon integration by parts,
2
ρ
R
R
R
s
2
ρ
∂
s
vw
s
=
R
s
2
ρ
∂
ss
vw
d
s
s
2
ρ
∂
s
v∂
s
w
d
s
s
2
ρ
−
1
∂
s
vw
d
s
=−
−
+
0
.
s
=
0
0
0
C
0
The boundary terms vanish for
w
∈
(G)
.
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