Jeanblanc M., Yor M., Chesney M. Mathematical Methods for Financial Markets
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540 9 General Processes: Mathematical Facts
We now assume that S is locally square integrable and that M
P
(S)isnot
empty, and we decompose the Q-local martingale S,whereQ ∈M
P
(S), into
the sum of its continuous and purely discontinuous parts:
S
t
= S
c
t
+ S
d
t
.
Theorem 9.5.2.4 Suppose that S enjoys the PRP under Q and that S is
continuous. Then,
(i) ΔS
t
= m
t
1
{ΔS
t
=0}
,wherem is predictable,
(ii) the random measures S
c
and S
d
are mutually singular.
Proof: The process [S
d
]
t
−S
d
t
is a Q-local martingale. From the PRP of
S under Q,wehave
[S
d
]
t
−S
d
t
=
t
0
m
s
dS
s
, (9.5.2)
for some predictable process m.SinceS is continuous, S
d
is continuous;
recall that [S
d
]
t
=
s≤t
(ΔS
s
)
2
, hence, we get ΔS
t
= m
t
1
{ΔS
t
=0}
.
Moreover, from the PRP assumption,
S
c
t
=
t
0
ϕ
(c)
s
dS
s
,S
d
t
=
t
0
ϕ
(d)
s
dS
s
for some predictable processes ϕ
(c)
and ϕ
(d)
. Then, since S
c
,S
d
=0,we
obtain
t
0
ϕ
(c)
s
ϕ
(d)
s
dS
s
=0
Therefore,
0=ϕ
(c)
s
ϕ
(d)
s
dS
s
,
or, equivalently
(ϕ
(c)
s
)
2
(ϕ
(d)
s
)
2
=0,dSa.s..
Then, using the fact that
dS
c
s
=(ϕ
(c)
s
)
2
dS
s
,dS
d
s
=(ϕ
(d)
s
)
2
dS
s
the property (ii) follows.
Note that under the hypotheses of the above theorem, S is a PCSM (see
Definition 9.4.1.1).
9.5 Existence and Uniqueness of the e.m.m. 541
Sufficient Condition for Existence
Theorem 9.5.2.5 Let S be an (F, Q)-local martingale and suppose that:
(i) every continuous F-martingale is a stochastic integral with respect to S
c
,
(ii) every discontinuous F-martingale is a stochastic integral with respect
to S
d
,
(iii) the random measures S
c
and S
d
have disjoint supports.
Then S enjoys the PRP.
Proof: Let M = M
c
+ M
d
be a martingale. It admits a representation of
the form
M
t
= m +
t
0
m
(c)
s
dS
c
s
+
t
0
m
(d)
s
dS
d
s
.
Let Γ
c
(resp. Γ
d
) be the support of the measure dS
c
t
(resp. dS
d
t
). Then
S
c
t
=
t
0
1
Γ
c
(s)dS
s
,S
d
t
=
t
0
1
Γ
d
(s)dS
s
.
Indeed
E
S
c
t
−
t
0
1
Γ
c
(s)dS
s
2
= E
S
c
t
− 2
t
0
1
Γ
c
(s)dS
c
s
+
t
0
1
Γ
c
(s)dS
s
= E
t
0
1
Γ
c
(s)(dS
s
−S
c
s
)
= E
t
0
1
Γ
c
(s)(dS
d
s
=0.
Hence
M
t
= M
c
t
+ M
d
t
= m +
t
0
m
(c)
s
1
Γ
c
(s)+m
(d)
s
1
Γ
d
(s)
dS
s
.
We first present a case where the PRP property is not satisfied, and an
example where it is.
Let B be a Brownian motion and Π the compensated martingale
associated with a Poisson process N of intensity λ = 1 in the same filtration
(hence B and Π are independent). We leave it to the reader to write the result
for a general fixed λ or even for a deterministic function λ(t).
(1) Let S
t
= B
t
+ Π
t
. This process does not satisfy the PRP. Note that
(i) in Theorem 9.5.2.4 is satisfied, but (ii) is not: indeed, S
c
t
= S
d
t
= t.
(2) Let dX
t
= f(t)dB
t
+g(t)dΠ
t
where f and g are deterministic functions
such that fg = 0. The filtration F
X
is the filtration generated by the processes
542 9 General Processes: Mathematical Facts
(
t
0
f(s)dB
s
,t ≥ 0) and (
t
0
g(s)dΠ
s
,t ≥ 0) and is included in F
B
∨ F
N
.Let
us set d
X
t
= f(t)dB
t
+ g(t)dN
t
.ThesetS which contains all the r.v.’s
Φ
∞
=exp
∞
0
ϕ
s
d
X
s
−
1
2
∞
0
ϕ
2
s
f
2
(s)ds −
∞
0
ds
e
ϕ
s
g(s)
− 1
where ϕ belongs to the set Δ of deterministic functions, is total in L
2
(F
X
∞
, P).
Indeed, if Y ∈ L
2
(F
X
∞
, P) is orthogonal to Φ
∞
,thenY is orthogonal to
exp(
∞
0
ϕ
s
d
X
s
) for every ϕ ∈ Δ, hence to Y itself.
The process X enjoys the PRP w.r.t. F
X
. Indeed,
exp
t
0
ϕ
s
d
X
s
−
1
2
t
0
ϕ
2
s
f
2
(s)ds −
t
0
ds
e
ϕ
s
g(s)
− 1
=exp
t
0
ϕ
s
f(s)dB
s
−
1
2
t
0
ϕ
2
s
f
2
(s)ds
× exp
t
0
ϕ
s
g(s)dN
s
− λ
t
0
ds
e
ϕ
s
g(s)
− 1
= E
(1)
t
E
(2)
t
and the two martingales E
(i)
are orthogonal. It follows that
E
(1)
t
E
(2)
t
=1+
t
0
E
(1)
s
E
(2)
s
−
ϕ
s
f(s)dB
s
+
t
0
E
(1)
s
E
(2)
s
−
ϕ
s
g(s)dΠ
s
=1+
t
0
E
(1)
s
E
(2)
s
−
ϕ
s
dX
s
,
since fg =0.
9.5.3 Uniqueness Property
Theorem 9.5.3.1 Let S be a continuous F-adapted process. Suppose that
M
P
(S) is not empty and let Q ∈M
P
(S). The following two properties are
equivalent:
(i) PRP holds under Q with respect to S,
(ii) The set M
P
(S) is a singleton.
Proof: (i) implies (ii): Let Q and Q
∗
belong to M
P
(S)andletq be the
Radon-Nikod´ym density of Q
∗
w.r.t. Q: Q
∗
|
F
t
= q
t
Q|
F
t
. The canonical
decomposition of S as a Q
∗
-semi-martingale is
S
t
=
S
t
+
t
0
dS, q
s
q
s
−
where
S is a Q
∗
-local martingale. Since S is a Q
∗
-local martingale, S, q =0,
hence the Q-martingale q is orthogonal to S under Q, which is not possible
unless q = 1, because S enjoys the PRP under Q, hence Q
∗
= Q.
9.5 Existence and Uniqueness of the e.m.m. 543
(ii) implies (i). If (i) does not hold, there exists a locally bounded non-
trivial martingale orthogonal to S,sayq
0
(see Jacod and Yor [472] and Protter
[727] for details). By stopping we may assume that q
0
is uniformly bounded,
sayby1/2, and q
0
0
= 0 and we consider q
t
=1+q
0
t
. The probability Q
∗
defined
by Q
∗
|
F
t
= q
t
Q|
F
t
belongs to M
P
(S) and is different from Q.
A Particular Case
We recall a result in a simple case, when the coefficient of the Brownian
motion, i.e., the volatility, may vanish (see Steele [806]).
Proposition 9.5.3.2 Let W be the Wiener measure, X the canonical process
X
t
(ω)=ω(t) and Z
t
=
t
0
Φ(s, ω)dX
s
where Φ ∈ L
2
loc
(X). There exists a
unique probability Q equivalent to W such that Z is a (Q, F)-local martingale
if and only if
{(s, ω):Φ(s, ω)=0} (9.5.3)
is ds ×dW negligible. (Of course, then, this unique probability coincides with
W.)
Proof: Suppose that (9.5.3) does not hold. Then, for any λ = 0 the process
L
λ
t
=exp
λ
t
0
1
{Φ(s,ω)=0}
dX
s
−
1
2
λ
2
t
0
1
{Φ(s,ω)=0}
ds
is a martingale which is not identically equal to 1. Let Q
λ
|
F
t
= L
λ
t
W|
F
t
;
the process X
λ
t
= X
t
− λ
t
0
1
{Φ(s,ω)=0}
ds is a (Q
λ
, F)-local martingale and
Z
t
=
t
0
Φ(s, ω)dX
λ
s
, hence there is an infinity of e.m.m’s.
The converse study is easy.
9.5.4 Examples
Lemma 9.5.4.1 Let M and N be two martingales in their own respective
filtrations.
(i) If M has the F
M
-PRP, if N has the F
N
-PRP, and if M and N are
F
M
∨ F
N
-martingales, then they are orthogonal in F
M
∨ F
N
if and only if
they are independent.
(ii) If M has the F
M
-PRP and N the F
N
-PRP, and if M and N are
independent, then the pair (M,N) enjoys the F
M
∨ F
N
-PRP.
Proof: (i) Take Φ ∈ L
2
(F
M
∞
)andΨ ∈ L
2
(F
N
∞
). Then,
Φ = E(Φ)+
∞
0
ϕ
s
dM
s
,Ψ= E(Ψ )+
∞
0
ψ
s
dN
s
.
544 9 General Processes: Mathematical Facts
Now, we consider these stochastic integrals as written in F
M
∨ F
N
and from
the orthogonality hypothesis, we deduce
E(ΦΨ)=E(Φ)E(Ψ) .
(ii) Under the hypotheses, we have, with the previous notation
ΦΨ = E(ΦΨ)+
∞
0
Ψ
s
−
ϕ
s
dM
s
+
∞
0
Φ
s−
ψ
s
dN
s
where
Ψ
s
= E(Ψ|F
N
s
)=E(Ψ|F
N
s
∨F
M
∞
) ,
Φ
s
= E(Φ|F
M
s
)=E(Φ|F
M
s
∨F
N
∞
) .
The general representation result follows from the totality of the products ΦΨ
in L
2
(F
M
∞
F
N
∞
).
While using this result, it is important to prove that M and N are
martingales in the ”large” filtration F
M
∨F
N
. One may recall that it is possible
to construct a Poisson process adapted to a Brownian filtration F
B
, i.e., a
Poisson process N such that F
N
⊂ F
B
(see Jeulin [494]), so that a fortiori it
is not true that a Poisson process and a Brownian motion constructed on the
same probability space are independent.
Theorem 9.5.4.2 Suppose that the SDE
dS
t
= b(S
t
)dt + σ(S
t
)dB
t
admits a unique weak solution. Then, any F
S
-local martingale can be written
as M
t
= m +
t
0
m
s
dS
mar
s
where S
mar
is the martingale part of S, i.e.,
S
mar
t
= S
t
− S
0
−
t
0
b(S
u
)du =
t
0
σ(S
u
)dB
u
.
Proof: See Hunt and Kennedy [455] and Jacod and Yor [472].
9.6 Self-financing Strategies and Integration by Parts
We present a simple, but useful application of the integration by parts formula
for the characterization of self-financing strategies in finance.
9.6 Self-financing Strategies and Integration by Parts 545
9.6.1 The Model
Assume that there is a financial market where S
1
t
,S
2
t
,...,S
k
t
represent values
at time t of k assets. Here, S
1
,S
2
,...,S
k
are semi-martingales on some
probability space (Ω,F, F, P), satisfying the usual conditions. We assume that
S
1
is stricly positive. We emphasize that a priori, there is no riskless asset
beingtradedinthemarket.Letπ =(π
1
,π
2
,...,π
k
) be a trading strategy; in
particular, the processes π
i
are F-predictable. The component π
i
t
represents
the number of units of the i-th asset held in the portfolio at time t. Then the
wealth V
t
(π) of the trading strategy π =(π
1
,π
2
,...,π
k
) equals
V
t
(π)=
k
i=1
π
i
t
S
i
t
(9.6.1)
and we say that π is a self-financing strategy if
V
t
(π)=V
0
(π)+
k
i=1
t
0
π
i
u
dS
i
u
. (9.6.2)
By combining the last two formulae, we obtain
dV
t
(π)=
V
t
(π) −
k
i=2
π
i
t
S
i
t
(S
1
t
)
−1
dS
1
t
+
k
i=2
π
i
t
dS
i
t
.
The latter representation shows that the wealth process only depends on k −1
components of π.
9.6.2 Self-financing Strategies and Change of Num´eraire
Choosing S
1
as a num´eraire, and denoting
V
1
t
(π)=V
t
(π)(S
1
t
)
−1
,S
i,1
t
= S
i
t
(S
1
t
)
−1
,
we get the following well-known result which proves that the self-financing
property does not depend on the choice of the num´eraire,
Lemma 9.6.2.1 (i) Let π =(π
1
,π
2
,...,π
k
) be a self-financing strategy.
Then we have
V
1
t
(π)=V
1
0
(π)+
k
i=2
t
0
π
i
u
dS
i,1
u
, ∀t ∈ [0,T]. (9.6.3)
(ii) Conversely, let X be an F
T
-measurable random variable, and assume
that there exist x ∈ R and predictable processes π
i
,i=2,...,k such that
X = S
1
T
x +
k
i=2
t
0
π
i
u
dS
i,1
u
.
Then there exists a predictable process π
1
such that the strategy π is self-
financing and replicates X.
546 9 General Processes: Mathematical Facts
Proof: We give the proof for k =2.
(i) Let V =(V
t
(π),t ≥ 0) be the value of a self-financing strategy π,
and V
1
= V/S
1
the value of this strategy in the num´eraire S
1
.Fromthe
integration by parts formula
d(V
1
t
)=V
t
−
d(S
1
t
)
−1
+(S
1
t
−
)
−1
dV
t
+ d[(S
1
)
−1
,V]
t
.
From the self-financing condition
d(V
1
t
)=π
1
t
S
1
t
−
d(S
1
t
)
−1
+ π
2
t
S
2
t
−
d(S
1
t
)
−1
+(S
1
t
−
)
−1
π
1
t
dS
1
t
+ π
2
t
(S
1
t
−
)
−1
dS
2
t
+ π
1
t
d[(S
1
)
−1
,S
1
]
t
+ π
2
t
d[(S
1
)
−1
,S
2
]
t
= π
1
t
S
1
t
−
d(S
1
)
−1
t
+(S
1
)
−1
t
−
dS
1
t
+ d[(S
1
)
−1
,S
1
]
t
+ π
2
t
S
2
t
−
d(S
1
t
)
−1
+(S
1
t
−
)
−1
dS
2
t
−
+ d[(S
1
)
−1
,S
2
]
t
.
We now note that
S
1
t
−
d(S
1
t
)
−1
+(S
1
t
−
)
−1
dS
1
t
+ d[(S
1
)
−1
, (S
1
)]
t
= d(S
1
(S
1
)
−1
)
t
=0
and
S
2
t
−
d(S
1
t
)
−1
+(S
1
t
−
)
−1
dS
2
t
+ d[(S
1
)
−1
,S
2
]
t
= d((S
1
t
)
−1
S
2
t
)
hence,
dV
1
t
= π
2
t
dS
2,1
t
.
We now prove part (ii). We define
V
1
t
= x +
k
i=2
t
0
π
i
u
dS
i,1
u
, (9.6.4)
and we set
π
1
t
= V
1
t
−
k
i=2
π
i
t
S
i,1
t
=(S
1
t
)
−1
V
t
−
k
i=2
π
i
t
S
i
t
,
where V
t
= V
1
t
S
1
t
. From the definition of V ,wehavedV
1
t
=
k
i=2
π
i
t
dS
i,1
t
and
thus
dV
t
= d(V
1
t
S
1
t
)=V
1
t
−
dS
1
t
+ S
1
t
−
dV
1
t
+ d[S
1
,V
1
]
t
= V
1
t
−
dS
1
t
+
k
i=2
π
i
t
S
1
t
−
dS
i,1
t
+ d[S
1
,S
i,1
]
t
.
From the obvious equality
dS
i
t
= d(S
i,1
t
S
1
t
)=S
i,1
t
−
dS
1
t
+ S
1
t
−
dS
i,1
t
+ d[S
1
,S
i,1
]
t
,
it follows that
9.7 Valuation in an Incomplete Market 547
dV
t
= V
1
t
−
dS
1
t
+
k
i=2
π
i
t
dS
i
t
− S
i,1
t
−
dS
1
t
=
V
1
t
−
−
k
i=2
π
i
t
S
i,1
t
−
dS
1
t
+
k
i=2
π
i
t
dS
i
t
.
The needed equality dV
t
=
k
i=1
π
i
t
dS
i
t
holds if
π
1
t
= V
1
t
−
k
i=2
π
i
t
S
i,1
t
= V
1
t
−
−
k
i=2
π
i
t
S
i,1
t
−
,
i.e., if ΔV
1
t
=
k
i=2
π
i
t
ΔS
i,1
t
, which is the case from the definition (9.6.4)of
V
1
t
. Note also that the process π
1
is indeed predictable.
For more examples, in particular for a study of constrained strategies, see
Bielecki et al [92, 93].
9.7 Valuation in an Incomplete Market
We study a market in which both a savings account with constant interest
rate r,andd risky assets (S
i
,i =1,...,d) are traded. We shall denote for
simplicity by π
t
S
t
the amount of money invested in the risky assets (that
is, we set π
t
S
t
=
d
i=1
π
i
t
S
i
t
)and
S
t
= S
t
e
−rt
is the discounted price. We
assume here that the market is incomplete and arbitrage free. More precisely,
we assume that the set M
P
(
S) of e.m.m’s is not empty and not reduced to a
singleton. By definition of an incomplete market, if H is a contingent claim,
it is in general not possible to construct a hedging strategy for H, i.e., a
self-financing portfolio with terminal value equal to H (note that contingent
claims of the form H = e
rT
x +
T
0
π
s
d
S
s
such as for example c + κS
T
,
are hedgeable). We recall that some extra conditions are required to avoid
doubling strategies, the most common one being that the value of the strategy
is bounded below.
The set
{E
Q
(He
−rT
), Q ∈M
P
(
S)}
is called the set of viable prices. If the asset H is traded at price E
Q
(He
−rT
)
for Q ∈M
P
(
S), this does not induce arbitrage opportunities. If the asset H
is traded at a price which is not in the set of viable prices, this induces
an arbitrage opportunity. We present briefly some methods of evaluating
contingent claims in an incomplete market.
One way is to find a hedging strategy which replicates as best as possible
the contingent claim. The dual approach is to choose a particular equivalent
548 9 General Processes: Mathematical Facts
martingale measure, e.g., the F¨ollmer-Schweizer minimal probability measure
or the minimal entropy measure. In that setting, one gives an arbitrary value
to the market price of risk.
Another method is to relate the price of the contingent claim to a utility
function approach. We now present briefly these three approaches.
9.7.1 Replication Criteria
Super-replication
The super-replication price of H is the smallest initial wealth v such that
there exists a self-financing strategy π which super-hedges the contingent
claim, i.e., in the case r =0,v satisfies
v =inf{w : ∃π, w +
T
0
π
s
dS
s
≥ H }.
(See Subsection 10.5.3 for an example). The super-replication price v is
also called the selling price. It is proved in El Karoui and Quenez [307]that
this super-replication price is the supremum of the viable prices, i.e.,
sup
Q∈M
P
(
e
S
)
E
Q
(e
−rT
H) .
See also Kramkov [543]. However, from the practitioner’s viewpoint, this price
is too large (see Subsection 10.5.2 for an example). In the case of stochastic
volatility (see Frey and Sin [360]) the selling price of a European option is
often infinite, except if the volatility is bounded (see El Karoui et al. [301]).
This method can be applied in models where transaction costs are taken into
account, with the same drawback: for example, the super-replication price for
a call is the price of the underlying asset (see Soner et al. [796], Hubalek and
Schachermayer [449]).
Quadratic Hedging
F¨ollmer and Sonderman [352] suggest minimizing the quadratic error under
the historical probability measure, i.e., finding v and π which minimize
E
P
((H − V
v,π
T
)
2
)
over initial wealth v and self-financing strategies π,whereV
v,π
T
= x+
T
0
π
s
dS
s
is the terminal wealth associated to the strategy π (we assumed that the
interest rate is null). The solution is the L
2
-projection of H on the vector
space x +
T
0
π
s
dS
s
. (One needs to find some conditions which ensure that
this space is closed, see Delbaen et al. [232].)
9.7 Valuation in an Incomplete Market 549
It can be proved that there exists, at least in markets with continuous
asset prices, a Radon-Nikod´ym density L
qh
which does not depend on the
choice of H such that v = E
P
(e
−rT
HL
qh
). Despite this explicit solution to
the original question, again practitioners dislike this criterion, which gives the
same weight for losses and gains of V relative to H. There are also asymmetric
criteria, but then the mathematical theory is more involved and results are
less explicit. We refer to the book of F¨ollmer and Schied [350] for a complete
study.
In the case of weather derivatives, or in a default risk setting, an interesting
question is the measurability criteria for the strategies. Indeed, even if the
market is incomplete, it is possible to include the information about the
weather or on the default in the choice of the portfolio (see Bielecki et al.
[89] in the case of credit risk).
9.7.2 Choice of an Equivalent Martingale Measure
This method consists in the choice of an equivalent martingale measure (or a
state price density) in an appropriate way. One possibility is to minimize
E
P
(f(L
T
)) over the set of Radon-Nikod´ym densities for a given convex
function f. However, the solution depends on the choice of num´eraire. We
now present two different, but classical, choices of convex functions.
Minimal Entropy Measure: f(x)=x ln x
Let P and Q be two equivalent probability measures. The relative entropy of
Q w.r.t. P corresponds to the choice f (x)=x ln x and is
H(Q|P)=E
Q
ln
dQ
dP
= E
P
dQ
dP
ln
dQ
dP
. (9.7.1)
Let S denote the value of the asset and M
P
S
the set of e.m.m’s. Any
probability Q
∗
such that
i) Q
∗
∈M
P
S
ii) ∀Q ∈M
P
S
,H(Q
∗
|P) ≤ H(Q|P)
is called a minimal entropy measure. See Choulli and Stricker [182], Delbaen
et al. [231], Frittelli [362], Frittelli et al. [364], Hobson [441] and Miyahara
[654] for a complete study. In Rouge and El Karoui [310], the authors provide
a general framework for pricing contingent claims, using a BSDE approach.