Jeanblanc M., Yor M., Chesney M. Mathematical Methods for Financial Markets
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4.4 Parisian Options 257
4.4.4 American Parisian Options
American Parisian options are also considered. Grau [403] combined Monte
Carlo simulations and PDE solvers (see also Grau and Kallsen [404]) in
order to price European and American Parisian options. The PDE approaches
developed by Haber et al. [415] and Wilmott [846] can also be used in order
to value these options. In the same setting, where the risk-neutral dynamics
of the underlying are given by (4.4.1), Chesney and Gauthier [172] developed
a probabilistic approach for the pricing of American Parisian options. They
derived the following result for currency options:
Proposition 4.4.4.1 The price of an American Parisian down-and-out call
(ADOPa) can be decomposed as follows:
ADOPa (S
0
,L,D,T)=DOPa(S
0
,L,D,T)
+ δS
0
T
0
e
−αu
E
'
1
{W
u
≥
¯
b(u)}
1
{u<G
−,
D
(W )}
exp ((ν + σ) W
u
)
(
du
− rK
T
0
e
−αu
E
'
1
{W
u
≥
¯
b(u)}
1
{u<G
−,
D
(W )}
exp (νW
u
)
(
du
where
α = r +
ν
2
2
,ν=
1
σ
r − δ −
σ
2
2
,
¯
b(u)=
1
σ
ln
b
c
(T − u)
S
0
,
=
1
σ
ln
L
S
0
≤ 0
and where {b
c
(T − u),u ∈ [0,T]} is the exercise boundary (see Section 3.11
for the general definition). Here, the process W is a Brownian motion.
This decomposition can also be written as follows:
ADOPa (S
0
,L,D,T)=DOPa(S
0
,L,D,T)+δ
T
0
DOPa(S
0
,b
c
(T − u),u)du
+ δ
T
0
b
c
(T − u) −
r
δ
K
BinDOCPa (S
0
,b
c
(T − u),u) du
where DOPa(S
0
,b
c
(T − u),u) is the price of the European Parisian down-
and-out call option with maturity u,strikepriceb
c
(T − u), barrier L and
delay D, BinDOCPa (S
0
,b
c
(T − u),u) is the price of a Parisian binary call
(see Subsections 3.6.2 and 3.6.3 for the definitions of binary calls and binary
barrier options) which generates at maturity a pay-off of one monetary unit
if the underlying value is higher than the strike price and if the first instant
–when the underlying price spends consecutively more than D units of time
under the level L
1
– is greater than the maturity u. Otherwise, the payoff is
equal to zero.
258 4 Complements on Brownian Motion
Denote by ADOPa (S
0
,L,D) the price of a perpetual American Parisian
option. The following proposition is obtained:
Proposition 4.4.4.2 The price of a perpetual American Parisian down-and-
out call is given by:
ADOPa (S
0
,L,D)=δ
+∞
0
DOPa(S
0
,L
c
,u) du
+ δ
+∞
0
L
c
−
r
δ
K
BinDOCPa (S
0
,L
c
,u) du
or
ADOPa(S
0
,L,D)=
1 −
Ψ(−κ
√
D)
Ψ(κ
√
D)
e
2κ
1
σκ
S
0
L
c
γ
δL
c
γ −1
−
r
γ
K
with κ =
√
2r + ν
2
, γ =
−ν+
√
2r+ν
2
σ
and where the exercise boundary L
c
is
defined implicitly by:
L
c
− K =
1 −
Ψ(−κ
√
D)
Ψ(κ
√
D)
L
L
c
2
κ
σ
1
σκ
δL
c
γ − 1
−
r
γ
K
where the function Ψ is defined in equation (4.3.12).
Solutions when the excursion has already started and for the “in” barrier
case are also derived. The latter case is easier to analyze. Indeed, in this
setting, the option holder cannot do or decide anything before the option
is activated; once the option is activated then it does not have a barrier
anymore, but is just a plain vanilla American call. The exercise frontier for
an American Parisian “in” barrier option is therefore the exercise frontier of
the corresponding plain vanilla option, starting at the activation time.
5
Complements on Continuous Path Processes
In this chapter, we present the important notion of time change, which will
be crucial when studying applications to finance in a L´evy process setting.
We then introduce the operation of dual predictable projection, which will
be an important tool when working with the reduced form approach in the
default risk framework (of course, it has many other applications as will appear
clearly in subsequent chapters). We present important facts about general
homogeneous diffusions, in particular concerning their Green functions, scale
functions and speed measures. These three quantities are of great interest
when valuing options in a general setting. We study applications related to
last passage times. A section is devoted to enlargements of filtrations, an
important subject when dealing with insider trading.
The books of Borodin and Salminen [109], Itˆo[462], Itˆo and McKean [465],
Karlin and Taylor [515], Karatzas and Shreve [513], Kallenberg [505], Knight
[528], Øksendal [684], [RY] and Rogers and Williams [741, 742] are highly
recommended. See also the review of Varadhan [826].
An excellent reference for the study of first hitting times of a fixed level
for a diffusion is the book of Borodin and Salminen [109] where many results
can be found. The general theory of stochastic processes is presented in
Dellacherie [240], Dellacherie and Meyer [242, 244] and Dellacherie, Meyer
and Maisonneuve [241]. Some results about the general theory of processes
can also be found in Chapter 9.
5.1 Time Changes
5.1.1 Inverse of an Increasing Process
In this paragraph, we deal with processes on a probability space but do not
make any reference to a given filtration. Let us recall that by definition (see
Subsection 1.1.10) an increasing process is equal to 0 at time 0; it is right
M. Jeanblanc, M. Yor, M. Chesney, Mathematical Methods for Financial
Markets, Springer Finance, DOI 10.1007/978-1-84628-737-4
5,
c
Springer-Verlag London Limited 2009
259
260 5 Complements on Continuous Path Processes
continuous and of course increasing. Let A be an increasing process and let C
be the right inverse of A, that is the increasing process defined by:
C
u
=inf{t : A
t
>u} (5.1.1)
where inf{∅} = ∞. We shall use C
u
or C(u) for the value at time u of the
process C. The process C is right-continuous and satisfies
C
u−
=inf{t : A
t
≥ u}
and {C
u
>t} = {A
t
<u}.WealsohaveA
C
s
≥ s and A
t
=inf{u : C
u
>t}.
(See [RY], Chapter 0, section 4 for details.) Moreover, if A is continuous and
strictly increasing, C is continuous and C(A
t
)=t.
Proposition 5.1.1.1 Time changing in integrals can be effected as follows:
if f is a positive Borel function
[0,∞[
f(s) dA
s
=
∞
0
f(C
u
) 1
{C
u
<∞}
du .
Proof: For f = 1
[0,v]
, the formula reads
A
v
=
∞
0
1
{C
u
≤v}
du
and is a consequence of the definition of C. The general formula follows from
the monotone class theorem.
5.1.2 Time Changes and Stopping Times
In this section, F is a right-continuous filtration, and A is a right-continuous
adapted increasing process with right inverse C.Fromtheidentity
{C
u
≤ t} = {A
t
≥ u},
we see that (C
u
,u≥ 0) is a family of F-stopping times. This leads us to define
a time change C as a family (C
u
,u ≥ 0) of stopping times such that the
map u → C
u
is a.s. increasing and right continuous. We denote by F
C
the
filtration F
C
=(F
C
t
,t ≥ 0). For every t the r.v. A
t
is an F
C
-stopping time
(indeed {A
t
<u} = {C
u
>t}).
Example 5.1.2.1 We have studied a very special case of time change while
dealing with Ornstein-Uhlenbeck processes in Section 2.6. These processes are
obtained from a Brownian motion by means of a deterministic time change.
5.1 Time Changes 261
Example 5.1.2.2 Let W be a Brownian motion and let
T
t
=inf{s ≥ 0:W
s
>t} =inf
s ≥ 0: max
u≤s
W
u
>t
be the right-continuous inverse of M
t
=max
u≤t
W
u
. The process (T
t
,t ≥ 0)
is increasing, and right-continuous (see Subsection 3.1.2). See Section 11.8
for applications.
Exercise 5.1.2.3 Let (B, W) be a two-dimensional Brownian motion and
define
T
t
=inf{s ≥ 0:W
s
>t}.
Prove that (Y
t
= B
T
t
,t ≥ 0) is a Cauchy process, i.e., a process with
independent and stationary increments, such that Y
t
has a Cauchy law with
characteristic function exp(−t|u|).
Hint: E(e
iuB
T
t
)=
e
−
1
2
u
2
T
t
(ω)
P(dω)=E(e
−
1
2
u
2
T
t
)=e
−t|u|
.
5.1.3 Brownian Motion and Time Changes
Proposition 5.1.3.1 (Dubins-Schwarz’s Theorem.) A continuous mar-
tingale M such that
M
∞
= ∞
is a time-changed Brownian motion. In other words, there exists a Brownian
motion W such that M
t
= W
M
t
.
Sketch of the Proof: Let A = M and define the process W as
W
u
= M
C
u
where C is the inverse of A. One can then show that W is a
continuous local martingale, with bracket W
u
= M
C
u
= u. Therefore,
W is a Brownian motion, and replacing u by A
t
in W
u
= M
C
u
, one obtains
M
t
= W
A
t
.
Comments 5.1.3.2 (a) This theorem was proved in Dubins and Schwarz
[268]. It admits a partial extension due to Knight [527] to the multidimensional
case: if M is a d-dimensional martingale such that M
i
,M
j
=0,i = j
and M
i
∞
= ∞, ∀i, then the process W =(M
i
C
i
(t)
,i ≤ d, t ≥ 0) is a d-
dimensional Brownian motion w.r.t. its natural filtration, where the process
C
i
is the inverse of M
i
. See, e.g., Rogers and Williams [741]. The assumption
M
∞
= ∞ can be relaxed (See [RY], Chapter V, Theorem 1.10).
(b) Let us mention another two-dimensional extension of Dubins and
Schwarz’s theorem for complex valued local martingales which generalize
complex Brownian motion. Getoor and Sharpe [390] introduced the notion
of a continuous conformal local martingale as a process Z = X + iY , valued
in C, the complex plane, where X and Y are real valued continuous local
martingales and Z
2
is a local martingale. A prototype is the complex-valued
262 5 Complements on Continuous Path Processes
Brownian motion. If Z is a continuous conformal local martingale, then, from
Z
2
t
= X
2
t
−Y
2
t
+2iX
t
Y
t
, we deduce that X
t
= Y
t
and X, Y
t
= 0. Hence,
applying Knight’s result to the two-dimensional local martingale (X, Y ), there
exists a complex-valued Brownian motion B such that Z = B
X
. In fact, in
this case, B can be shown to be a Brownian motion w.r.t. (F
α
u
,u≥ 0), where
α
u
=inf{t : X
t
>u}.If(Z
t
,t ≥ 0) denotes now a C-valued Brownian
motion, and f : C → C is holomorphic, then (f(Z
t
),t ≥ 0) is a conformal
martingale. The C-extension of the Dubins-Schwarz-Knight theorem may then
be written as:
f(Z
t
)=
Z
R
t
0
|f
(Z
u
)|
2
du
,t≥ 0 (5.1.2)
where f
is the C-derivative of f,and
Z denotes another C-valued Brownian
motion. This is an extremely powerful result due to L´evy, which expresses the
conformal invariance of C-valued Brownian motion. It is easily shown, as a
consequence, using the exponential function that, if Z
0
= a,then(Z
t
,t ≥ 0)
shall never visit b = a (of course, almost surely). As a consequence, (5.1.2)
may be extended to any meromorphic function from C to itself, when P (Z
0
∈
S) = 0 with S the set of singular points of f.
(c) See Jacod [468], Chapter 10 for a detailed study of time changes, and
El Karoui and Weidenfeld [311]andLeJan[569].
Exercise 5.1.3.3 Let f be a non-constant holomorphic function on C and
Z = X + iY a complex Brownian motion. Prove that there exists another
complex Brownian motion B such that f(Z
t
)=f(Z
0
)+B(
t
0
|f
(Z
s
)|
2
dX
s
)
(see [RY], Chapter 5). As an example, exp(Z
t
)=1+B
R
t
0
ds exp(2X
s
)
.
We now come back to a study of real-valued continuous local martingales.
Lemma 5.1.3.4 Let M be a continuous local martingale with M
∞
= ∞,
W the Brownian motion such that M
t
= W
M
t
and C the right-inverse of
M.IfH is an adapted process such that for any t,
t
0
H
2
s
dM
s
=
M
t
0
H
2
C
u
du
< ∞ ,
then
t
0
H
s
dM
s
=
M
t
0
H
C
u
dW
u
,
C
t
0
H
s
dM
s
=
t
0
H
C
u
dW
u
.
Lemma 5.1.3.5 Let X
t
=
C
t
0
H
s
dW
s
,whereC is a time change with respect
to F, differentiable with respect to time. Assume that C
t
=0for any t.Then,
dX
t
= H
C
t
C
t
dB
t
,
where B is an F
C
-Brownian motion.
5.1 Time Changes 263
Proof: From the previous lemma
C
t
0
H
s
dW
s
=
t
0
H
C
u
dW
C
u
,
hence, dX
t
= H
C
t
dW
C
t
. The process (W
C
u
,u≥ 0) is a local martingale with
bracket C
u
. The process
B
t
=
t
0
1
C
u
dW
C
u
is a Brownian motion.
Remarks 5.1.3.6 (a) Up to an enlargement of probability space, one can
generalize the previous lemma to the case where the condition C
t
=0does
not hold, but where we keep the assumption that C is differentiable. (The
proof is left to the reader.)
(b) A time-changed local martingale is not necessarily a local martingale
with respect to the time-changed filtration. As seen in Example 5.1.2.2,ifT
t
is the first hitting time of the level t for the Brownian motion B, the process
t → T
t
is increasing and is a time change. However, B
T
t
= t is not a local
martingale. This illustrates, although very roughly, Monroe’s theorem (see
Remark 5.1.3.6) which states that any semi-martingale (even discontinuous)
is a time changed Brownian motion. [655, 656]
However, if X is a continuous F-local martingale and C a continuous
time change, then (X
C
t
)
t≥0
is a continuous F
C
-local martingale. (See [RY],
Chapter V, Section 1).
Comments 5.1.3.7 (a) Changes of time are extensively used for finance
purposes in the papers of Geman, Madan and Yor [379, 385, 380, 381].
(b) The “pli cachet´e” of Doeblin [255] may have been one of the first papers
studying time changes.
(c) Further extensions to Markov processes are found in Volkonski [831].
See also McKean’s paper [637] for other aspects of this major idea and
important applications to Bessel processes in Chapter 6.
Exercise 5.1.3.8 Let Y be the solution of
dY
t
=(cY
t
+ kY
2
t
)dt +
Y
t
dW
t
Prove that Y
t
= Z(
t
0
Y
s
ds)wheredZ (u)=(c + kZ(u))du + d
W
u
.
Exercise 5.1.3.9 Let Z be a complex BM Z
t
= X
t
+ iY
t
. Consider the two
martingales |Z
t
|
2
− 2t and
t
0
(X
s
dY
s
− Y
s
dX
s
). Prove that
1
2
|Z
t
|
2
− 2t
+ i
t
0
(X
s
dY
s
− Y
s
dX
s
)
264 5 Complements on Continuous Path Processes
is a conformal martingale which can be represented as
Z
u
= β
u
+ iγ
u
time-
changed by
t
0
|Z
s
|
2
ds with β and γ two independent BM’s. Prove that
σ(β
u
,u≥ 0) = σ(|Z
t
|,t≥ 0), hence γ and |Z| are independent.
5.2 Dual Predictable Projections
In this section, after recalling some basic facts about optional and predictable
projections, we introduce the concept of a dual predictable projection
1
,which
leads to the fundamental notion of predictable compensators. We recommend
the survey paper of Nikeghbali [674].
Recall that a process is said to be optional if it is measurable with
respect to the σ-algebra on R
+
×Ω generated by c`adl`ag F-adapted processes,
considered as mappings on R
+
× Ω,whereasapredictable process is
measurable with respect to the σ-algebra on R
+
× Ω generated by c`ag F-
adapted processes (see Subsection 9.1.3 for comments).
5.2.1 Definitions
Let X be a bounded (or positive) process, and F a given filtration. The
optional projection of X is the unique optional process
(o)
X which satisfies:
for any F-stopping time τ
E(X
τ
1
{τ<∞}
)=E(
(o)
X
τ
1
{τ<∞}
) . (5.2.1)
For any F-stopping time τ,letΓ ∈F
τ
and apply the equality (5.2.1)tothe
stopping time τ
Γ
= τ1
Γ
+ ∞1
Γ
c
. We get the re-inforced identity:
E(X
τ
1
{τ<∞}
|F
τ
)=
(o)
X
τ
1
{τ<∞}
.
In particular, if A is an increasing process, then, for s ≤ t:
E(
(o)
A
t
−
(o)
A
s
|F
s
)=E(A
t
− A
s
|F
s
) ≥ 0 . (5.2.2)
Note that, for any t, E(X
t
|F
t
)=
(o)
X
t
. However, E(X
t
|F
t
) is defined almost
surely for any t; thus uncountably many null sets are involved, hence, a priori,
E(X
t
|F
t
) is not a well-defined process whereas
(o)
X takes care of this difficulty.
Likewise, the predictable projection of X is the unique predictable
process
(p)
X such that for any F-predictable stopping time τ
E(X
τ
1
{τ<∞}
)=E(
(p)
X
τ
1
{τ<∞}
) . (5.2.3)
As above, this identity reinforces as
E(X
τ
1
{τ<∞}
|F
τ−
)=
(p)
X
τ
1
{τ<∞}
,
for any F-predictable stopping time τ (see Subsection 1.2.3 for the definition
of F
τ−
).
1
See Dellacherie [240] for the notion of dual optional projection.
5.2 Dual Predictable Projections 265
Example 5.2.1.1 Let τ and ϑ be two stopping times such that ϑ ≤ τ and
Z a bounded r.v.. Let X = Z1
]] ϑ,τ ]]
. Then,
(o)
X = U1
]] ϑ,τ ]]
,
(p)
X = V 1
]] ϑ,τ ]]
where U (resp. V ) is the right-continuous (resp. left-continuous) version of
the martingale (E(Z|F
t
),t≥ 0).
Let τ and ϑ be two stopping times such that ϑ ≤ τ and X a positive
process. If A is an increasing optional process, then, since 1
]] ϑ,τ ]]
(t)is
predictable
E
τ
ϑ
X
t
dA
t
= E
τ
ϑ
(o)
X
t
dA
t
.
If A is an increasing predictable process, then
E
τ
ϑ
X
t
dA
t
= E
τ
ϑ
(p)
X
t
dA
t
.
The notion of interest in this section is that of dual predictable
projection, which we define as follows:
Proposition 5.2.1.2 Let (A
t
,t≥ 0) be an integrable increasing process (not
necessarily F-adapted). There exists a unique F-predictable increasing process
(A
(p)
t
,t≥ 0), called the dual predictable projection of A such that
E
∞
0
H
s
dA
s
= E
∞
0
H
s
dA
(p)
s
for any positive F-predictable process H.
In the particular case where A
t
=
t
0
a
s
ds, one has
A
(p)
t
=
t
0
(p)
a
s
ds (5.2.4)
Proof: See Dellacherie [240], Chapter V, Dellacherie and Meyer [244],
Chapter 6 paragraph (73), page 148, or Protter [727] Chapter 3, Section 5.
This definition extends to the difference between two integrable (for sim-
plicity) increasing processes. The terminology “dual predictable projection”
referstothefactthatitistherandommeasured
t
A
t
(ω) which is relevant when
performing that operation. If X is bounded and A has integrable variation
(not necessarily adapted), then
E((XA
(p)
)
∞
)=E((
(p)
XA)
∞
) .
This is equivalent to: for s<t,
E(A
t
− A
s
|F
s
)=E(A
(p)
t
− A
(p)
s
|F
s
) . (5.2.5)
If A is adapted (not necessarily predictable), then (A
t
− A
(p)
t
,t ≥ 0) is a
martingale. In that case, A
(p)
t
is also called the predictable compensator of A.
266 5 Complements on Continuous Path Processes
More generally, from Proposition 5.2.1.2 and (5.2.5), the process
(o)
A−A
(p)
is a martingale.
Proposition 5.2.1.3 If A is increasing, the process
(o)
A is a sub-martingale
and A
(p)
is the predictable increasing process in the Doob-Meyer decomposition
of the sub-martingale
(o)
A.
Example 5.2.1.4 Let W be a Brownian motion, M
t
=sup
s≤t
W
s
its running
maximum, and R
t
=2M
t
−W
t
. Then, from Pitman’s Theorem 5.7.2.1 and
its Corollary 5.7.2.2, for any positive Borel function f ,
E(f(M
t
)|F
R
t
)=
1
0
dxf(R
t
x) ,
hence, E(2M
t
|F
R
t
)=R
t
and the predictable projection of 2M
t
is R
t
.Onthe
other hand, from Pitman’s theorem
R
t
= β
t
+
t
0
ds
R
s
,
where β is a Brownian motion, therefore, the dual predictable projection of
2M
t
on F
R
t
is
t
0
ds
R
s
. Note that the difference between these two projections
is the (Brownian) martingale β.
In a general setting, the predictable projection of an increasing process A
is a sub-martingale whereas the dual predictable projection is an increasing
process. The predictable projection and the dual predictable projection of an
increasing process A are equal if and only if (
(p)
A
t
,t≥ 0) is increasing.
It will also be convenient to introduce the following terminology:
Definition 5.2.1.5 If ϑ is a random time, we call the predictable compen-
sator associated with ϑ the dual predictable projection A
ϑ
of the increasing
process 1
{ϑ≤t}
. This dual predictable projection A
ϑ
satisfies
E(k
ϑ
)=E
∞
0
k
s
dA
ϑ
s
(5.2.6)
for any positive, predictable process k.
5.2.2 Examples
In the sequel, we present examples of computation of dual predictable
projections. We end up with Az´ema’s lemma, providing the law of the
predictable compensator associated with the last passage at 0 of a BM before
T , evaluated at a (stopping) time T . See also Knight [529]and Sections 5.6
and 7.4.