Versión de HTML Básico
On Dufresne’s Translated Perpetuity and Some Black-Scholes Annuities
Analíti a
k
7
Revista de Análisis Estadístico
Journal of Statistical Analysis
and
∫
+
∞
0
M
2
s
(
M
2
s
+
2
a
M
s
+
b
)
2
ds
(law)
=
´ınf
{
t
≥
0,
R
(
3,
√
a
2
−
b
)
t
=
η
(
∞
)
}
,
with R
(
3,
√
a
2
−
b
)
0
=
η
(
b
/
x
)
.
where
η
(
x
) =
∫
x
0
dz
z
2
+
2
az
+
b
.
We give two proofs of this result:
i
)
a direct proof relying on a martingale approach and
on the weak absolute continuity formula between
Brownian motion and the three-dimensional Bessel
process,
ii
)
and a slightly more probabilistic proof relying on the
third Ray-Knight theorem and on a decomposition of
the three-dimensional Bessel paths at its last passage
times.
In the remainder of the paper, we study via a Feynman-Kac
approach the law of the pair :
(
∫
t
0
(
e
2
β
(
B
s
+
ν
s
)
−
1
)
+
ds
,
∫
t
0
(
e
2
b
(
B
s
+
ν
s
)
−
1
)
−
ds
)
where
x
+
=
m´ax
(
0,
x
)
and
x
−
=
m´ın
(
0,
x
)
. This study ans-
wers a problem raised in the monograph [14, Chapter 4],
where the authors compute the Laplace transform of the
Black-Scholes call perpetuity
∫
+
∞
0
(
e
x
+
B
u
+
ν
u
−
K
)
+
du
and leave as an open question the study of the analo-
gous annuity. We then discuss several special cases (among
which the Black-Scholes call annuity, the positive sojourn
time of Brownian motion with drift and Yor’s functional),
and recover the Laplace transform of the associated perpe-
tuities.
Note that annuities also appear in the computation of
Asian options, where the payoff is determined by the ave-
rage price of the underlying asset
(
S
t
,
t
≥
0
)
on the con-
sidered period, see [7]. For instance, the price of an Asian
Call option with exercise price
K
and maturity
T
is given
by
E
[ (
1
T
∫
T
0
S
t
dt
−
K
)
+
]
.
This is somewhat different from the price of a classic Euro-
pean Call option, where only the final value of the underl-
ying asset at time
T
is considered (see [3]) :
E
[
(
S
T
−
K
)
+
]
.
Of course, Asian options are harder to compute in practice
as they depend on the entire past history of the underlying
asset, but they make it possible to reduce the risk of price
manipulation near the maturity date.
The paper is finally concluded by a short appendix on Bes-
sel functions and Bessel processes (with drift).
2 A generalization of Dufresne’s
translated perpetuity
In this section, we compute the law of the perpetuities :
∫
+
∞
0
E
2
s
(
E
2
s
+
2
a
E
s
+
b
)
2
ds
and
∫
+
∞
0
M
2
s
(
M
2
s
+
2
a
M
s
+
b
)
2
ds
.
T
HEOREM
2.
Assume that the polynomial z
2
+
2
az
+
b does
not have positive roots. For
2
λ
+
a
2
−
b
≥
0
, we have:
E
x
[
exp
(
−
λ
∫
+
∞
0
E
2
s
(
E
2
s
+
2
a
E
s
+
b
)
2
ds
)]
=
√
x
2
+
2
ax
+
b
x
sinh
(
√
2
λ
+
a
2
−
b
∫
x
0
dz
z
2
+
2
az
+
b
)
sinh
(
√
2
λ
+
a
2
−
b
∫
+
∞
0
dz
z
2
+
2
az
+
b
)
and
E
x
[
exp
(
−
λ
∫
+
∞
0
M
2
s
(
M
2
s
+
2
a
M
s
+
b
)
2
ds
)]
=
√
b
+
2
ax
+
x
2
√
b
sinh
(
√
2
λ
+
a
2
−
b
∫
b
/
x
0
dz
z
2
+
2
az
+
b
)
sinh
(
√
2
λ
+
a
2
−
b
∫
+
∞
0
dz
z
2
+
2
az
+
b
)
.
The equality in law given in Theorem 1 follows directly
from this result, since when
a
2
−
b
≥
0, one recognizes in
the right-hand side the expression of the Laplace transform
of the first passage time of a three-dimensional Bessel pro-
cess with drift
√
a
2
−
b
. A short review of Bessel processes
with drift is given in Section A, where these Laplace trans-
forms are also inverted thanks to Jacobi’s theta function.
2.1 A martingale approach to Dufresne’s
translated perpetuity
Let
x
>
0 and assume that
E
0
=
x
. By Lamperti’s rela-
tion (see Theorem 7), there exists a three-dimensional Bes-
sel process
(
R
t
,
t
≥
0
)
started from
x
such that:
E
t
=
R
A
t
with
A
t
=
⟨E⟩
t
=
∫
t
0
(
E
s
)
2
ds
.
Analítika,
Revista de análisis estadístico
, 4 (2014), Vol. 7(1): 7-19
9