Cross hedging with stochastic correlation
Gregor Heyne
(joint work with Stefan Ankirchner)
Humboldt Universit ¨at Berlin, QPL
March 20th, 2009
Finance and Insurance, Jena
Motivation
Consider a call option on the DAX.
How to hedge?
Motivation
0 20 40 60 80 100 120 140 160 180 4000 4500 5000 5500 6000 6500 7000 7500 Days to maturity DAX − FDAX Dax FDaxRelated literature
Henderson, 2002, Valuation of claims on nontraded assets using
utility maximization;
Musiela, Zariphopoulou, 2004, An example of indifference prices
under exponential preferences;
Davis, 2006 , Optimal hedging with basis risk;
Motivation
40 60 80 100 120 140 160 180 0.85 0.9 0.95 1 Days to maturity CorrelationModelling correlation
Direct modelling with Jacobi process:
d
ρ
t
=
κ
(
ϑ
−
ρ
t
)
dt
+
α
p
(
1
−
(
ρ
t
)
2
W
ˆ
t
, with
α
,
κ
>
0,
ϑ
∈
(
−
1
,
1
)
Indirect modelling with OU process:
Set
ρ
t
=
b
−
1
(
U
t
)
, with
dU
t
=
a
(
ϑ
−
U
t
)
dt
+
σ
U
d
W
ˆ
t
,
where a
>
0,
ϑ
∈
R
,
σ
U
>
0 and b
: (
−
1
,
1
)
→
R
continuous
bijection.
The Model
We want to hedge the contingent claim H
=
h
(
I
T
)
by trading the asset
X .
dX
t
=
X
t
((
µ
X
−
r
)
dt
+
σ
X
dW
t
1
)
dI
t
=
I
t
((
µ
I
−
r
)
dt
+
σ
I
(
ρ
t
dW
t
1
+
q
1
−
ρ
2
t
dW
t
2
))
d
ρ
t
=
a
(
ρ
t
)
dt
+
g
(
ρ
t
)
d
W
ˆ
t
FS decomposition via BSDEs
El Karoui, Peng, Quenez, 1997:
The FS decomposition of H
=
h
(
I
T
)
is given by
h
(
I
T
) =
Y
0
+
Z
T
0
Z
s
1
σ
X
X
s
dX
s
+
Z
T
0
Z
2
s
dW
s
2
+
Z
T
0
Z
3
s
dW
s
3
,
where Y and Z are determined by
Y
t
=
h
(
I
T
)
−
Z
T
t
Z
s
dW
s
−
Z
T
t
Z
s
1
µ
X
−
r
σ
X
ds
,
for 0
≤
t
≤
T .
Explicit solution of linear BSDE
The solution of
Y
t
=
h
(
I
T
)
−
Z
T
t
Z
s
dW
s
−
Z
T
t
Z
1
s
µ
X
−
r
σ
X
ds
,
is given by
Y
t
= ˜
IE
[
h
(
I
T
)
|
F
t
]
and
Z
t
=
σ
(
t
,
I
t
,
ρ
t
)
∗
∂
x
ψ
(
t
,
I
t
,
ρ
t
)
∂
v
ψ
(
t
,
I
t
,
ρ
t
)
,
where
ψ
(
t
,
x
,
v
) = ˜
IE
[
h
(
I
T
t,x,v
)]
.
Technical details
ψ
(
t
,
x
,
v
) = ˜
IE
[
h
(
I
T
t,x,v
)]
1
Under which conditions is
ψ
(
t
,
x
,
v
)
differentiable with respect to
x and v ?
2
Under which conditions are I
t,x,v
and
ρ
t,v
differentiable with
respect to x and v ?
Differentiability of I and
ρ
Denote
dv
d
ρ
s
t,v
and
dv
d
I
t,x,v
s
by
ρ
¯
t,v
s
and
¯
I
t,x,v
s
. Then
¯
ρ
t,v
s
=
1
+
Z
s
t
a
′
(
ρ
t,v
u
)¯
ρ
t,v
u
du
+
Z
s
t
g
′
(
ρ
t,v
u
)¯
ρ
t,v
u
d
W
ˆ
u
,
¯
I
t,x,v
s
=
Z
s
t
¯
I
t,x,v
u
((
µ
I
−
r
)
du
+
σ
I
(
ρ
u
t,v
dW
u
1
+
q
1
−
(
ρ
u
t,v
)
2
dW
u
2
))
+
Z
s
t
I
u
t,x,v
σ
I
(¯
ρ
u
t,v
dW
u
1
−
ρ
t,v
u
q
1
−
(
ρ
u
t,v
)
2
¯
ρ
t,v
u
dW
u
2
)
.
Differentiability of I and
ρ
Set
τ
=
inf
{
s
≥
0
:
ρ
s
∈ {−
1
,
1
}}
.
(H1)
τ
>
T
,
IP
−
a
.
s
.
Moreover
Z
T
t
(¯
ρ
s
t,v
)
2
1
−
(
ρ
s
t
,v
)
2
du
<
∞
,
IP
−
a
.
s
.
Differentiability of
ψ
(
t
,
x
,
v
) = ˜
IE
[
h
(
I
T
t
,
x
,
v
)]
(H2) There exists p
>
1 such that for every v0
∈
(
−
1
,
1
)
there exists an
open intervall U
⊂
(
−
1
,
1
)
of v
0
such that
sup
v
∈
U
IE
[
Z
T
t
(¯
ρ
s
t,v
)
2
1
−
(
ρ
s
t,v
)
2
p
ds
]
<
∞
.
Lemma
Let h be Lipschitz such that the weak derivative h
′
is
Lebesgue-almost everywhere continuous. Under the Conditions (H1)
and (H2) the partial derivative
∂
v
ψ
(
t
,
x
,
v
)
exists and is given by
Representation of Z
We want to justify
Z
s
t,x,v
=
σ
(
s
,
I
t
,x,v
s
,
ρ
t,v
s
)
∗
∂
x
ψ
(
s
,
I
s
t,x,v
,
ρ
t,v
s
)
∂
v
ψ
(
s
,
I
s
t,x,v
,
ρ
t,v
s
)
!
.
Problem:
ψ
(
s
,
x
,
v
) = ˜
IE
[
h
(
I
T
t,x,v
)]
is only locally Lipschitz.
We use
dI
t
n
=
I
t
n
(
µ
I
−
r
)
dt
+
σ
I
I
n
t
(
1
−
1
n
)
ρ
t
dW
1
t
+
r
1
−
(
1
−
1
n
)
2
ρ
2
t
dW
2
t
!
and
ψ
n
(
t
,
x
,
v
) = ˜
IE
[
h
n
(
I
T
n,t,x,v
)]
in approximation proof.
Representation of Z
Lemma
Assume that (H1) and (H2) hold, and that a and g are continuously
differentiable with bounded derivatives. Let h be Lipschitz such that
the weak derivative h
′
is Lebesgue-almost everywhere continuous.
Then
Z
s
t,x,v
=
σ
(
s
,
I
t
,x,v
s
,
ρ
t,v
s
)
∗
∂
x
ψ
(
s
,
I
s
t,x,v
,
ρ
t,v
s
)
∂
v
ψ
(
s
,
I
s
t,x,v
,
ρ
t,v
s
)
!
.
(1)
Representation of Z
Lemma
Assume that (H1) and (H2) hold, and that a and g are continuously
differentiable with bounded derivatives. Let h be Lipschitz such that
the weak derivative h
′
is Lebesgue-almost everywhere continuous.
Then
Z
s
t,x,v
=
σ
(
s
,
I
t
,x,v
s
,
ρ
t,v
s
)
∗
∂
x
ψ
(
s
,
I
s
t,x,v
,
ρ
t,v
s
)
∂
v
ψ
(
s
,
I
s
t,x,v
,
ρ
t,v
s
)
!
.
(1)
By an argument of Imkeller, Reveillac, Richter one can show (1)
without the use of Malliavin calculus! This allows us to drop the
bounded derivatives assumption.
Main results
Theorem
Suppose that the coefficients a and g in the dynamics of
ρ
are
continuously differentiable. Assume furthermore that Conditions (H1)
and (H2) are satisfied. Let h be Lipschitz such that the weak
derivative h
′
is Lebesgue-almost everywhere continuous. Then, there
exists a locally risk minimizing strategy
ϕ
= (
ξ
,
η
)
for the derivative
h
(
I
T
)
.
ξ
is given by
ξ
t
=
ξ
(
t
,
I
t
,
X
t
,
ρ
t
)
, where, with
P denoting the
˜
minimal martingale measure of X ,
ξ
(
t
,
x
,
y
,
v
) =
v
x
σ
I
y
σ
X
˜
IE
[
h
′
(
I
T
t,x,v
)
I
T
t,1,v
] +
g
(
v
)
γ
y
σ
X
˜
IE
[
h
′
(
I
t
T
,x,v
)¯
I
T
t,x,v
]
.
Main results
Problem:
(H2) There exists p
>
1 such that for every v
0
∈
(
−
1
,
1
)
there exists an
open intervall U
⊂
(
−
1
,
1
)
of v
0
such that
sup
v
∈
U
IE
[
Z
T
t
(¯
ρ
s
t,v
)
2
1
−
(
ρ
s
t,v
)
2
p
ds
]
<
∞
.
Very technical, unintuitive assumption. BUT:
Theorem
Let a and g be bounded with bounded derivatives. We assume that
g
(
−
1
) =
g
(
1
) =
0, and that g does not have any roots in
(
−
1
,
1
)
. If
lim sup
x
↑
1
2a
(
x
)(
1
−
x
)
g
2
(
x
)
<
0
and
lim inf
x
↓−
1
2a
(
x
)(
1
+
x
)
g
2
(
x
)
>
0
,
Examples
Examples
d
ρ
t
=
κ
(
ϑ
−
ρ
t
)
dt
+
α
(
1
−
(
ρ
t
)
2
) ˆ
W
t
, with
α
,
κ
>
0,
ϑ
∈
(
−
1
,
1
)
Set
ρ
t
=
b
−
1
(
U
t
)
, with
dU
t
=
a
(
ϑ
−
U
t
)
dt
+
σ
U
d
W
ˆ
t
,
where a
>
0,
ϑ
∈
R
,
σ
U
>
0 and b
(
x
) =
√
x
1
−
x
2.
Examples
d
ρ
t
=
κ
(
ϑ
−
ρ
t
)
dt
+
α
(
1
−
(
ρ
t
)
2
) ˆ
W
t
, with
α
,
κ
>
0,
ϑ
∈
(
−
1
,
1
)
Set
ρ
t
=
b
−
1
(
U
t
)
, with
dU
t
=
a
(
ϑ
−
U
t
)
dt
+
σ
U
d
W
ˆ
t
,
where a
>
0,
ϑ
∈
R
,
σ
U
>
0 and b
(
x
) =
√
x
1
−
x
2.
d
ρ
t
=
κ
(
ϑ
−
ρ
t
)
dt
+
α
p
(
1
−
(
ρ
t
)
2
W
ˆ
t
, with
κ
≥
α
2
1
±
ϑ
and
−
1
<
ϑ
2
±
r
ϑ
2
4
+
α
2
2
κ
<
1
,
also fulfills Conditions (H1) and (H2)!
Conclusions
Correlation is random.
Explicit hedge formula under the assumption
(H2) There exists p
>
1 such that for every v0
∈
(
−
1
,
1
)
there
exists an open intervall U
⊂
(
−
1
,
1
)
of v0
such that
sup
v
∈
U
IE
[
Z
T
t
(¯
ρ
s
t,v
)
2
1
−
(
ρ
s
t,v
)
2
p
ds
]
<
∞
.
(H2) is fulfilled by a large class of models for correlation
dynamics (including Jacobi and OU processes).
