Martingale invariance and utility maximization
Thorsten Rheinlander
Jena, June 2010
Thorsten Rheinlander ()
Martingale invariance
Jena, June 2010
1 / 27
Martingale invariance property
Consider two …ltrations F
G . F is immersed in G if every
F -martingale is a G -martingale.
Let S be some F -adapted locally bounded semi-martingale. Typical
choice: F = F S
Maximizing expected exponential utility from terminal wealth:
sup E
ϑ2Θ
exp
Z T
0
ϑt dSt
Is it possible for an agent to achieve higher expected utility by using
G -predictable strategies? What rôle does the immersion property play
here?
Thorsten Rheinlander (London School of Economics)
Martingale invariance
Jena, June 2010
2 / 27
Duality approach to utility maximization
Under mild assumptions, the following key duality holds (Delbaen et
al (2002)):
sup E
ϑ2Θ
exp
Z T
0
ϑt dSt
=
exp
inf H (Q, P )
Q 2Me
Here Θ denotes a set of admissible strategies, Me the set of all
equivalent martingale measures for S, and H (Q, P ) is the relative
entropy.
The link between the two problems is provided by the minimal
entropy martingale measure Q E and its density representation
dQ E
= exp c +
dP
Z T
0
η t dSt
.
Here c = H Q E , P , and η is the optimal strategy for the primal
problem. Note that in general η depends on the time horizon T .
Thorsten Rheinlander (London School of Economics)
Martingale invariance
Jena, June 2010
3 / 27
Proposition: Let S be a locally bounded F -adapted process such that the
entropy measure Q E (G) for S exists (and then Q E (F ) exists as well). If
E
E
E
F is immersed in G , then
h Z (Fi) = Z (G). Here Z (H) is the density
process, ZtE (H) = E
dQ E
dP
Ht , relative to …ltration H.
Proof: A martingale measure Q for S wrt. G induces a martingale
b wrt. F via d Qb = E dQ FT .
measure Q
dP
dP
b P
By the conditional Jensen’s inequality, we have H Q,
H (Q, P ).
E (G) has …nite relative entropy,
Applying this to Q E (G) we get that Q\
hence there is some martingale measure wrt. F which has …nite relative
entropy. Therefore Q E (F ) exists uniquely. As Z E (F ) S is a local
(P, F )-martingale, it is by immersion also a local (P, G)-martingale.
Hence Z E (F ) is the density process of a martingale measure wrt. G , with
terminal value dQ E (F ) /dP. As
H Q E (F ) , P
E (G), P
H Q\
H Q E (G) , P ,
the result follows by uniqueness of Q E (G ).
Thorsten Rheinlander (London School of Economics)
Martingale invariance
Jena, June 2010
4 / 27
Remarks. 1) In particular, if Z E (G) is not F -adapted, then F cannot be
immersed into G . This can be useful in case it is complicated to determine
the structure of all F -martingales.
2) It is in general not true that Q E (F ) = Q E (G) implies
Z E (F ) = Z E (G), as will be seen later on. In particular, the density of
Q E (G) can be FT -measurable, whereas the density process Z E (G) is not
F -adapted.
3) Summing up, if F is immersed in G , then no additional expected utility
can be gained by employing G -predictable strategies. However, if the
immersion property fails, one can in general not conclude that this leads to
a utility gain.
4) To apply these results, the main task is to calculate Z E (G), the density
process of the entropy measure wrt. G .
Thorsten Rheinlander (London School of Economics)
Martingale invariance
Jena, June 2010
5 / 27
Optimal martingale measure equation
We work with a …ltration G such that all G -martingales are
continuous. We know that the density of the minimal entropy
measure Q E can necessarily be written as
dQ E
= exp c +
dP
Z T
0
η t dSt
for some constant c and some G -predictable process η.
By the structure condition, there exists a local martingale M and a
G -predictable process λ such that
S
KT
Thorsten Rheinlander (London School of Economics)
= M+
=
Z T
0
Z
λ d [M ] ,
λ2t d [M ]t < ∞.
Martingale invariance
Jena, June 2010
6 / 27
Every martingale measure Q can be written as
dQ
dP
= E
= exp
= exp
Z
λ dM
Z
L
T
1
KT
2
Z
1
λt dSt + KT
2
λt dMt
LT
LT
1
[L]
2 T
1
,
[L]
2 T
where L is some local (P, G)-martingale strongly orthogonal to M.
As we look for a candidate measure with representation as above, we
would like to …nd such an L as well as a constant c and a
G -predictable process ψ such that we can decompose
1
KT = c +
2
Z T
0
ψt dSt + LT +
1
[L] .
2 T
We are looking for a solution (c, ψ, L) to this BSDE, and set
η = ψ λ.
Thorsten Rheinlander (London School of Economics)
Martingale invariance
Jena, June 2010
7 / 27
We take as candidate
dQ
= exp c +
dP
Z T
0
η t dSt
=E
Z
λ dM
L
.
T
Veri…cation: we have to show that the stochastic exponential is a true
martingale, that H Q, P has …nite relative entropy, and that
Z T
0
η 2t d [S ]t 2 Lexp (P ).
Here Lexp (P ) is the Orlicz space generated by the Young function
exp ( ).
Thorsten Rheinlander (London School of Economics)
Martingale invariance
Jena, June 2010
8 / 27
We want to show that the following stochastic exponential is a
martingale:
Z
Z =E
λ dM
L
This could be done e.g. by a criterion due to Liptser & Shiryaev
(1977), however, here we give another approach yielding …nite relative
entropy of the resulting measure as well.
De…ne a fundamental sequence of stopping times as
Tn = min f t j Kt _ [L]t
ng ^ T
By Novikov’s criterion, the ZT n are then densities of probability
measures Qn . We have
ZT = lim ZT n
n
Thorsten Rheinlander (London School of Economics)
Martingale invariance
a.s.
Jena, June 2010
9 / 27
Proposition: The following assertions are equivalent.
h
i
(i ) supn EQ n KT n + [L]T n < ∞
(ii ) supn H (Qn , P ) < ∞
(iii ) ZT is the density of a prob. meas. Q, and H (Q, P ) < ∞.
Proof:
(i ) () (ii ) This follows by a calculation based on Girsanov’s theorem.
(ii ) =) (iii ) The sequence (ZT n ) is u.i. by de la Vallée-Poussins criterion.
(iii ) =) (ii ) Follows from Barron’s inequality.
Thorsten Rheinlander (London School of Economics)
Martingale invariance
Jena, June 2010
10 / 27
Stein & Stein model
The price process S is modelled as strong solution to the SDE
dSt = µVt2 St dt + σVt St dBt
and the volatility process V is given as
dVt = (m
αVt ) dt + β dWt .
Here µ, σ, m, α, β are positive non-zero constants and B, W are two
correlated Brownian motions with correlation coe¢ cient ρ 2 ( 1, +1).
Thorsten Rheinlander (London School of Economics)
Martingale invariance
Jena, June 2010
11 / 27
Solving the optimal martingale measure equation and following the
veri…cation procedure yields that the entropy measure Q E for S
relative to the …ltration G generated by the BM (B, W ) is given by
dQ E
= exp c
dP
Z T
0
1
ρρ
1
φt + ψt Vt
1
St
1
dSt
.
Here φ, ψ are bounded deterministic functions which can be obtained
as system of two di¤erential equations involving one Riccatti-type
equation. In case the mean-reversion level m is non-zero, ψ will be
non-zero.
Thorsten Rheinlander (London School of Economics)
Martingale invariance
Jena, June 2010
12 / 27
We choose F = F S , G = F B ,W . In case ρ 2 ( 1, +1), G is the
progressive enlargement of F S by sgn (V ) which is not F S -adapted.
As consequence, if the mean-reversion level m is non-zero and
ρ 2 ( 1, +1), then F is not immersed in G .
An agent who observes sgn (V ) can take advantage of this additional
information when maximizing her expected exponential utility from
terminal wealth, provided ρ, m 6= 0. In the uncorrelated case ρ = 0,
however, F is not immersed in G if m 6= 0, but since η is then equal
to 1/S, the optimal investment strategy does not use the additional
information and there is no bene…t for the insider. Note that in this
case we have Q E (G) = Q E (F ), but Z E (G) 6= Z E (F ).
It seems to be di¢ cult to calculate Z E wrt. F S in case m 6= 0
because of the rather complicated structure of F S .
Thorsten Rheinlander (London School of Economics)
Martingale invariance
Jena, June 2010
13 / 27
Models with default
Let (Ω, F , P ) be a probability space, supporting a random variable
modeling the time of default τ > 0.
Ht = Iτ t is the counting process associated with τ, H the …ltration
generated by H. Let F be a Brownian …ltration, generated by some
BM W .
We assume that there is a deterministic intensity function µ such that
Z t
P (τ > t ) = exp
0
µs ds
.
Let G = H _ F . We assume that F is immersed into G . In
particular, W stays a Brownian motion in the larger …ltration G .
The G -martingale M associated with the one-jump process is given as
Mt = Ht
Thorsten Rheinlander (London School of Economics)
Z τ ^t
0
Martingale invariance
µs ds.
Jena, June 2010
14 / 27
e of a defaultable security under P as
We model the price process S
et /S
et
dS
= at dt + bt dWt + ct dMt
The functions a, b, and c are deterministic and bounded on [0, T ].
Finding a pricing measure by calibration is sometimes di¢ cult in this
framework. Therefore, we will choose martingale measures according
to optimality criteria.
Our main point is that the presence of two di¤erent noise terms dW
and dM makes the analysis much more complicated compared to a
scenario with only a Brownian driver.
Thorsten Rheinlander (London School of Economics)
Martingale invariance
Jena, June 2010
15 / 27
Our goal is now to calculate several optimal martingale measures, i.e.
e is a local
probability measures Q such that the price process S
Q-martingale. This
is equivalent to the statement that the stochastic
R
e
e is a local Q-martingale.
logarithm S = d S /S
Assuming the existence of an equivalent martingale measure and a
structure condition, we can write the semi-martingale decomposition
of S uniquely in the form
S =N+
Z
λ d hN i
for a local martingale N and a predictable process λ; here hN i
denotes the predictable compensator of the quadratic variation
process [N ]. In our concrete market model, it is readily computed that
a
dN = b dW + c dM, λ = 2
.
b + µc 2
Moreover, we have
Z
λ dN
=
λ ∆N
=
Thorsten Rheinlander (London School of Economics)
Z
a
(b dW + c dM ) ,
+ µc 2
λc ∆M.
b2
Martingale invariance
Jena, June 2010
16 / 27
Minimal martingale measure
The minimal martingale measure (see Schweizer (1995)), commonly
b is characterized by the property that P-martingales
referred to as P,
b
strongly orthogonal to N under P remain martingales under P.
Its density is given by the Doléans-Dade stochastic exponential
E
Z
= exp
λ dN
T
Z T
0
∏
λt bt dWt
(1
1
2
λt ct ∆Mt ) .
Z T
0
λ2t bt2 dt
0 <t T
As there is one single jump of M with jump size one, the density of
the minimal martingale measure gets negative with non-zero
probability in case a jump occurs at τ T and λτ cτ > 1.
b is in general a signed measure, and we conclude that the
Therefore, P
minimal martingale measure is not a good choice in our situation.
Thorsten Rheinlander (London School of Economics)
Martingale invariance
Jena, June 2010
17 / 27
Linear Esscher measure
R
For every strategy ϑ such that ϑ dS is exponentially special, we
S
de…ne the modi…ed
R Laplace cumulant process K (ϑ) as exponential
compensator of ϑ dS. In particular,
Z ϑ = exp
Z
ϑ dS
K S (ϑ )
is a local martingale with Z0ϑ = 1.
In case Z ϑ is a martingale on [0, T ], it is the density process of a
probability measure P ϑ . If we can …nd ϑ such that S is a local
P ϑ -martingale, P ϑ is called the linear Esscher measure.
Thorsten Rheinlander (London School of Economics)
Martingale invariance
Jena, June 2010
18 / 27
By Kallsen & Shiryaev (2001), we have to solve the equation
DK X (ϑ ) = 0.
This translates into the equation
µ
a = b 2 ϑ +cµe ϑc
By the boundedness of the co-e¢ cients, a bounded solution always
exists and gives a martingale measure by the above exponential tilting
due to a veri…cation result by Protter & Shimbo (2008).
The linear Esscher measure is stable wrt. stopping, and coincides with
the minimal Hellinger measure as studied by Choulli & Stricker
(2005).
Thorsten Rheinlander (London School of Economics)
Martingale invariance
Jena, June 2010
19 / 27
Minimal entropy martingale measure
Entropy measure Q E : Its density can always be written in the form
dQ E
= exp c +
dP
Z T
0
φt dSt
R
for some predictable process φ such that φ dS is a Q E -martingale.
In general, a martingale measure Q for S has a density of the form
dQ
=E
dP
Z
λ dN + L
,
T
where L, L0 = 0, is a local martingale strongly orthogonal to N.
According to martingale representation we can write L as
dL = bL dW + cL dM
for some predictable processes bL , cL . The orthogonality relation
hN, Li = 0 yields
bbL + ccL µ2 = 0.
Thorsten Rheinlander (London School of Economics)
Martingale invariance
Jena, June 2010
20 / 27
Equating the two di¤erent representations, we obtain the optimal
martingale measure equation
c
=
Z
T
+
Z
1 2 2
λ b + µλc
2
φa
T
( φb
+ log (1
µcL
φµc
dt
λb + bL ) dW
( λ τ cτ
Thorsten Rheinlander (London School of Economics)
1 2
b
2 L
cL ( τ )
Martingale invariance
φτ cτ ))
Iτ
T.
Jena, June 2010
21 / 27
Assuming for now formally that there exists a smooth function u
(where t 2 [0, T ] and h 2 f0, 1g) such that
log (1
( λ τ cτ
cL ( τ )
φτ cτ )) = u (t, Ht )
u (T , h) = 0,
u (t, h)
u (t, Ht ) ,
h 2 f0, 1g ,
we write
log (1
(λτ cτ cL (τ ) φτ cτ ))
= u (τ, 1) u (τ, 0)
=
fu (T , 1) u (τ, 1) + u (τ, 0) u (0, 0)g
u (0, 0)
Z T
∂
u (t, Ht ) dt u (0, 0) .
=
0 ∂t
Thorsten Rheinlander (London School of Economics)
Martingale invariance
Jena, June 2010
22 / 27
We work with the hypothesis that the integrals over the dt-terms as
well as the dW -terms vanish at maturity T . This yields the two
equations
1
φa + λ2 b 2
2
1
∂u
µλc + bL2 + µcL + φµc +
= 0,
2
∂t
φb + λb
bL = 0.
Inserting the orthogonality relation, we solve for φ as
Thorsten Rheinlander (London School of Economics)
φ=
λ+
ccL µ2
.
b2
Martingale invariance
Jena, June 2010
23 / 27
Moreover, we get, with the notation ∆t u := u (t, Ht )
that
cL = exp (∆u + φc ) + λc 1
c 2 µ2
= exp ∆u λc + 2 cL
b
+ λc
u (t, Ht ),
1.
Introducing
g (t, ut ) :=
1
φa + λ2 b 2
2
1
µλc + bL2 + µcL + φµc
2
,
t
we derive the system of two ordinary di¤erential equations,
∂u
+ g (t, ut ) = 0,
∂t
u (T , 0) = u (T , 1) = 0.
Thorsten Rheinlander (London School of Economics)
Martingale invariance
Jena, June 2010
24 / 27
The density process Z E associated with Q E de…ned by
ZtE
=
dQ E
dP
= E
Gt
Z
0
λu bu
buL dWu +
Z
0+
λu cuS
cuL dMu
t
is the density process of the minimal entropy martingale measure.
Veri…cation proceeds along the lines of Rheinländer & Steiger (2006).
Thorsten Rheinlander (London School of Economics)
Martingale invariance
Jena, June 2010
25 / 27
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eamericonarticle
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eamericonarticle
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eamericonarticle
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eamericonarticle
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eamericonarticle
Thorsten Rheinlander (London School of Economics)
Martingale invariance
Jena, June 2010
26 / 27
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eamericonarticle
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eamericonarticle
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eamericonarticle
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eamericonarticle
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eamericonarticle
Thorsten Rheinlander (London School of Economics)
Martingale invariance
Jena, June 2010
27 / 27