OPTIMAL STOPPING PROBLEMS FOR ASSET MANAGEMENT
SAVAS DAYANIK AND MASAHIKO EGAMI
Abstract.An asset manager invests the savings of some investors in a portfolio of defaultable
bonds.The manager pays the investors coupons at a constant rate and receives management fee
proportional to the value of portfolio.She also has the right to walk out of the contract at any time
with the net terminal value of the portfolio after the payment of investors'initial funds,but is not
responsible for any decit.To control the principal losses,investors may buy from the manager a
limited protection which terminates the agreement as soon as the value of the portfolio drops below
a predetermined threshold.We assume that the value of the portfolio is a jump-diusion process
and nd optimal termination rule of the manager with and without a protection.We also derive
the indierence price of a limited protection.We illustrate the solution method on a numerical
example.The motivation comes from the collateralized debt obligations.
1.Introduction
We study two optimal stopping problems of an institutional asset manager hired by ordinary
investors who do not have access to certain asset classes.The investors entrust their initial funds in
the amount of L to the asset manager.As long as the contract is alive,the investors receive coupon
payments from the asset manager on their initial funds at a xed rate (higher than the risk-free
interest rate).In return,the asset manager collects dividend or management fee (at a xed rate
on the market value of the portfolio).At any time,the asset manager has the right to terminate
the contract and to walk away with the net terminal value of the portfolio after the payment of
the investors'initial funds.However,she is not nancially responsible for any amount of shortfall.
The asset manager's rst problem is to nd a nonanticipative stopping rule which maximizes her
expected discounted total income.
Under the original contract,investors face the risk of losing all or some part of their initial funds.
Suppose that the asset manager oers the investors a limited protection against this risk,in the
form that the new contract will terminate as soon as the market value of the portfolio goes below
a predetermined threshold.The asset manager's second problem is to nd the fair price for the
limited protection and the best time to terminate the contract under this additional clause.
We assume that the market value X of the asset manager's portfolio follows a geometric Brownian
motion subject to downward jumps which occur according to an independent Poisson process.
As explained in detail in the next section,both the problems and the setting are motivated by
those faced by the managers responsible for the portfolios of defaultable bonds,for example,as in
collateralized debt obligations (CDOs).For a detailed description and the valuation of CDOs,we
Date:January 11,2012.
1
2 SAVAS DAYANIK AND MASAHIKO EGAMI
refer the reader to Due and G^arleanu [16],Goodman and Fabozzi [23],Egami and Esteghamat
[18] and Hull and White [19].Brie y,a CDO is a derivative security on a portfolio of bonds,
loans,or other credit risky claims.Cash ows from a collateral portfolio are divided into various
quality/yield tranches which are then sold to investors.In our setting,for example,the times of the
(downward) jumps in the portfolio value process can be thought as the default times of individual
bonds in the portfolio.
The dierence between the real-world CDOs and our setting is that a CDO has a pre-determined
maturity while we assume an innite time horizon.However,a typical CDO contract has a term
of 10-15 years (much longer than,for example,nite-maturity American-type stock options) and
is often extendable with the investors'consent.Hence our perpetuality assumption is a reasonable
approximation of the reality.We believe that our analysis is also applicable in certain other nancial
and real-options settings with no xed maturity,e.g.,open-end mutual funds,outsourcing the
maintenance of computing,printing or internet facilities in a company or in a university.
To nd the solutions of the asset manager's aforementioned problems,we rst model them as
optimal stopping problems for a suitable jump-diusion process under a risk-neutral probability
measure.By separating the jumps from the diusion part by means of a suitable dynamic pro-
gramming operator,similarly to the approach used by Dayanik,Poor,and Sezer [12] and Dayanik
and Sezer [13] for the solutions of sequential statistics problems,we solve the optimal stopping
problems by means of successive approximations,which not only lead to accurate and ecient nu-
merical algorithms but also allow us to establish concretely the form of optimal stopping strategies.
The idea of stripping the jumps from the diusion part of a jump-diusion process was ispired
by the seminal work of Davis [8,9] on piecewise-deterministic Markov processes and the personal
discussions of one of the authors with E.Cnlar (see also his talk on the web,Cnlar [4]).
Without any protection,the optimal rule of the asset manager turns out to terminate the contract
if the market value of the portfolio X becomes too small or too large;i.e.,as soon as X exits an
interval (a;b) for some suitable constants 0 < a < b < 1.
In the presence of limited protection (provided to the investors by the asset manager for a fee)
at some level`2 (a;L],it is optimal for the asset manager to terminate the contract as soon
as the value X of the portfolio exits an interval (`;m) for some suitable m 2 [`;b).Namely,if
the protection is binding,i.e.,`2 (a;b),then the asset manager's optimal continuation region
shrinks.In other words,investors can have limited protection only if they are also willing to give
up in part from the upside-potential of their managed portfolio.\Total protection"(i.e,the case
`= L) wipes out the upside-potential completely since the optimal strategy of the asset manager
becomes\stop immediately"in this extreme case (i.e.,`= m = L).Incidentally,a contract with
a protection at some level is less valuable than an identical contract without a protection.The
dierence between these two values gives the fair price of the investors'protection.The investors
must pay this dierence to the asset manager in order to compensate for the asset manager's lost
potential revenues due to\suboptimal"termination of the contract in the presence of the protection.
OPTIMAL STOPPING PROBLEMS FOR ASSET MANAGEMENT 3
In other words,the asset manager will be willing to provide the protection only if the dierence
between the expected total revenues with and without the protection is cleared by the investors.
Our model also sheds some light on the default timing problem of a single rm.Note that
the lower boundary a of the optimal continuation region in the rst problem's solution may be
interpreted as the\optimal default time"of a CDO.Instead of the value of a portfolio,if X
represents the market value of a rm subject to unexpected\bad news"(downward jumps),then
the asset manager's rst problemand its solution translate into the default and sale timing problem
of the rm and its solution.An action (default or sale) is optimal if the value X of the rm leaves
the optimal continuation region (a;b).It is optimal to default if X reaches (0;a],and optimal to
sell the rm if X reaches [b;1).Our solution extends the work of Due [15,Chapter 11] who
calculates (based on the paper by Leland [22]) the optimal default time for a single rm whose
asset value is modeled by a geometric Brownian motion.
Let us also mention that optimal stopping problems (especially,pricing American-type options)
for Levy processes have been extensively studied;see,for example,Chan [5],Pham [26],Mordecki
[25,24],Boyarchenko and Levendroski
i [3],Kou and Wang [21] and Asmussen et al.[1].
The problems are formulated in Section 2.The solutions of rst and second problems are studied
in Sections 3 and 4,respectively.The solutions methods of Problems 1 and 2 are illustrated on a
numerical example in Section 5.
2.The problem description
Let (
;F;P) be a probability space hosting a Brownian motion B = fB
t
;t 0g and an inde-
pendent Poisson process N = fN
t
;t 0g with the constant arrival rate ,both adapted to some
ltration F = fF
t
g
t0
satisfying the usual conditions.
An asset manager borrows L dollars from some investors and invests in some risky asset X =
fX
t
;t 0g.The process X has the dynamics
dX
t
X
t
= ( )dt +dB
t
y
0
[dN
t
dt];t 0(2.1)
for some constants 2 R, > 0, > 0 and y
0
2 (0;1).We denote by the dividend rate or the
management fee received by the asset manager.Note that the absolute value of relative jump sizes
equals y
0
,and the jumps are downwards.Therefore,the asset price
X
t
= X
0
exp
1
2
2
+y
0
t +B
t
(1 y
0
)
N
t
;t 0
is a geometric Brownian motion subject to downward jumps with constant relative jump sizes.
An interesting example of our setting is a portfolio of defaultable bonds as in the collateralized
debt obligations.Let X
t
be the value of a portfolio of k defaultable bonds.After every default,
the portfolio loses y
0
percent of its market value.The default times of each bond i constitutes a
Poisson process with the intensity rate
i
independent of others.Therefore,defaults occur at the
rate ,
P
k
i=1
i
at the level of the portfolio.The loss ratio upon a default is the same constant
y
0
across the bonds.The defaulted bond is immediately sold at the market,and a bond with a
4 SAVAS DAYANIK AND MASAHIKO EGAMI
similar default rate is bought using the sales proceeds.Under this assumption,defaults occur at
the xed rate because the number of bonds in the portfolio is xed at k.Egami and Esteghamat
[18] showed that the dynamics in (2.1) are a good approximation of the dynamics of the aggregate
value of individual defaultable bonds when priced in the\intensity-based"modeling framework
(see,e.g.,Due and Singleton [17]).The jump size y
0
on the portfolio level has to be calibrated.
Suppose that the asset manager pays the investors a coupon of c percent on the face value of the
initial borrowing L on a continuously compounded basis.We assume c < .The asset manager
has the right to terminate the contract at any time 2 R
+
and receive (X
L)
+
.Dividend and
coupon payments to the parties cease upon the termination of the contract.Let 0 < r < c be the
risk-free interest rate,and S be the collection of all F-stopping times.The asset manager's rst
problem is to calculate her maximum expected discounted total income
U(x),sup
2S
E
x
e
r
(X
L)
+
+
Z
0
e
rt
(X
t
cL)dt
;x 2 R
+
;(2.2)
where E
is taken under the equivalent martingale measure P
for a specied market price of the
jump risk,and to nd some
2 S that attains the supremum (if it exists) under the condition
0 < r < c < :
In the case of real CDOs,the dividend payment is often subordinated to the coupon payment.
But since we allow the possibility that the asset manager's net running cash ow X
t
cL becomes
negative,our formulation has more stringent requirement on the asset manager than a simple
subordination.
In the asset manager's second problem,the investors'assets have limited protection.In the
presence of the limited protection at level`> 0,the contract terminates at time e
(`;1)
,infft 0:
X
t
62 (`;1)g automatically.The asset manager wants to maximize her expected total discounted
earnings as in (2.2),but now the supremum has to be taken over all F-adapted stopping times
2 S which are less than or equal to e
(`;1)
almost surely.
3.The solution of the asset manager's first problem
In the no-arbitrage pricing framework,the value of a contract contingent on the asset X is
the expectation of the total discounted payo of the contract under some equivalent martingale
measure.Since the dynamics of X in (2.1) contain jumps,there are more than one equivalent mar-
tingale measure;see,e.g.Schoutens [27] and Nunno [14].The restriction to F
t
of every equivalent
martingale measure P
in a large class admits a Radon-Nikodym derivative in the form of
dP
dP
F
t
,
t
and
d
t
t
= dB
t
+( 1)[dN
t
dt];t 0 (
0
= 1);(3.1)
which has the solution
t
= exp
B
t
1
2
2
t + N
t
log ( 1)t
,t 0 for some constants
2 R and > 0.The constants and are known as the market price of the diusion risk and
the market price of the jump risk,respectively,and satisfy the drift condition
> 0 and r + y
0
( 1) = 0:(3.2)
OPTIMAL STOPPING PROBLEMS FOR ASSET MANAGEMENT 5
Then the discounted value process fe
(r)t
X
t
:t 0g before the dividends are paid is a (P
;F)-
martingale;see,e.g.,Pham [26],Colwell and Elliott [6],Cont and Tankov [7].Girsanov theorem
implies that B
t
,B
t
t,t 0 is a standard Brownian motion,and N
t
,t 0 is a homogeneous
Poisson process with intensity independent of B
under the new measure P
.Then
dX
t
X
t
= (r )dt +dB
t
y
0
[dN
t
dt];t 0;(3.3)
where + y
0
( 1) = r follows from the drift condition in (3.2).It^o's rule implies
X
t
= X
0
exp
r
1
2
2
+ y
0
t +B
t
(1 y
0
)
N
t
;t 0:(3.4)
The innitesimal generator of the process X under the probability measure P
coincides with the
second order dierential-dierence operator
(A
f)(x),(r + y
0
) xf
0
(x) +
1
2
2
x
2
f
00
(x) + [f(x(1 y
0
)) f(x)](3.5)
on the collection of twice-continuously dierentiable functions f().
Because fe
(r)t
X
t
;t 0g is a martingale under P
,we have E
x
[
R
1
0
X
t
e
rt
dt] =
R
1
0
xe
t
dt =
x,and for every stopping time 2 S,the strong Markov property implies that E
x
[
R
0
X
t
e
rt
dt] =
E
x
[
R
1
0
X
t
e
rt
dt]E
x
[
R
1
X
t
e
rt
dt] = xE
x
[e
r
E
X
(
R
1
0
X
s
e
rs
ds)] = xE
x
[e
r
X
],x 2 R
+
.
Because E
x
[
R
0
cLe
rt
dt] =
cL
r
E
x
[
cL
r
e
r
] for every 2 S and x 2 R
+
,(2.2) can be rewritten as
U(x) = V (x) +x
cL
r
;x 2 R
+
;where V (x),sup
2S
E
x
e
r
(X
L)
+
X
+
cL
r
(3.6)
is a discounted optimal stopping problem with the terminal reward function
h(x),(x L)
+
x +
cL
r
;x 2 R
+
:(3.7)
We x the market price of jump risk,and the market price is determined by the drift condition
in (3.2).In the remainder,we shall describe the solution of the optimal stopping problem (3.6).
Let T
1
;T
2
;:::be the arrival times of process N.Observe that X
T
n+1
= (1 y
0
)X
T
n+1
and
X
T
n
+t
X
T
n
= exp
r + y
0
2
2
t +(B
T
n
+t
B
T
n
)
;0 t < T
n+1
T
n
;n 1:
Le us dene for every n 0 the standard Brownian motion B
;n
t
:= B
T
n
+t
B
T
n
,t 0 and Poisson
process T
(n)
k
:= T
n+k
T
n
,k 0,respectively,under P
and the one-dimensional diusion process
Y
y;n
t
,y exp
r + y
0
2
2
t +B
;n
t
;t 0;(3.8)
which has dynamics
Y
y;n
0
= y and dY
y;n
t
= Y
y;n
t
[(r + y
0
)dt +dB
;n
t
];t 0(3.9)
and innitesimal generator (under P
x
)
(A
0
f)(y) =
2
y
2
2
f
00
(y) +(r + y
0
)yf
0
(y)(3.10)
6 SAVAS DAYANIK AND MASAHIKO EGAMI
acting on twice-continuously dierentiable functions f:R
+
7!R.Then X coincides with Y
X
T
n
;n
on [T
n
;T
n+1
) and jumps to (1 y
0
)Y
X
T
n
;n
T
n+1
T
n
at time T
n+1
for every n 0;namely,
X
T
n
+t
=
8
<
:
Y
X
T
n
;n
t
;0 t < T
n+1
T
n
;
(1 y
0
)Y
X
T
n
;n
T
n+1
T
n
;t = T
n+1
T
n
:
For n = 0,we shall write Y
y;0
Y
y
= y exp
(r y
0
2
=2)t +B
t
and Y
X
0
;0
Y
X
0
.
3.1.A dynamic programming operator.Let S
B
denote the collection of all stopping times of
the diusion process Y
X
0
,or equivalently,Brownian motion B.Let us take any arbitrary but xed
stopping time 2 S
B
and consider the following stopping strategy toward the solution of (3.6):
(i) on f < T
1
g stop at time ,
(ii) on f T
1
g,update X at time T
1
to X
T
1
= (1y
0
)Y
X
0
T
1
and continue optimally thereafter.
The value of this new strategy is E
x
[e
r
h(X
)1
f<T
1
g
+e
rT
1
V (X
T
1
)1
fT
1
g
] and equals
E
x
h
e
r
h(Y
X
0
)1
f<T
1
g
+e
rT
1
V ((1 y
0
)Y
X
0
T
1
)1
fT
1
g
i
= E
x
e
(r+ )
h(Y
X
0
) +
Z
0
e
(r+ )t
V ((1 y
0
)Y
X
0
t
)dt
:
If for every bounded function w:R
+
7!R
+
we introduce the operator
(Jw)(x),sup
2S
B
E
x
e
(r+ )
h(Y
X
0
) +
Z
0
e
(r+ )t
w((1 y
0
)Y
X
0
t
)dt
;x 0;(3.11)
then we expect the value function V () of (3.6) to be the unique xed point of operator J;namely,
V () = (JV )(),and that V () is the pointwise limit of the successive approximations
v
0
(x),h(x) = (x L)
+
x +
cL
r
;x 0;
v
n
(x),(Jv
n1
)(x);x 0;n 1:
Lemma 1.Let w
1
;w
2
:R
+
7!R be bounded.If w
1
() w
2
(),then (Jw
1
)() (Jw
2
)().If w() is
nonincreasing convex function such that h() w() cL=r,then (Jw)() has the same properties.
The proof easily follows from the linearity of y 7!Y
y
t
for every xed t 0 and the denition of
the operator J.The next proposition guarantees the existence of unique xed point of J.
Proposition 2.For every bounded w
1
;w
2
:R
+
7!R,we have kJw
1
Jw
2
k
r+
kw
1
w
2
k,
where kwk = sup
x2R
+
jw(x)j;namely,J acts as a contraction mapping on the bounded functions.
Proof.Because w
1
();w
2
() are bounded,(Jw
1
)() and (Jw
2
)() are nite,and for every"and x > 0,
there are"-optimal stopping times
1
(";x) and
2
(";x),which may depend on"and x,such that
(Jw
i
)(x) " E
x
h
e
(r+ )
i
(";x)
h(Y
X
0
i
(";x)
) +
Z
i
(";x)
0
e
(r+ )t
w
i
((1 y
0
)Y
X
0
t
)dt
i
;i = 1;2:
Therefore,(Jw
1
)(x) (Jw
2
)(x) "+ kw
1
w
2
k
R
1
0
e
(r+ )t
dt ="+ kw
1
w
2
k
r+
:Inter-
changing the roles of w
1
() and w
2
() gives j(Jw
1
)(x) (Jw
2
)(x)j "+kw
1
w
2
k
r+
for every
x > 0 and"> 0.Taking the supremum of both sides over x 0 completes the proof.
OPTIMAL STOPPING PROBLEMS FOR ASSET MANAGEMENT 7
Lemma 3.The sequence (v
n
)
n0
of successive approximations is nondecreasing.Therefore,the
pointwise limit v
1
(x),lim
n!1
v
n
(x),x 0 exists.Every v
n
(),n 0 and v
1
() are nonincreas-
ing,convex,and bounded between h() and cL=r.
Lemma 3 follows from repeated applications of Lemma 1.Proposition 4 below shows that the
unique xed point of J is the uniform limit of successive approximations.
Proposition 4.The limit v
1
() = lim
n!1
v
n
() = sup
n0
v
n
() is the unique bounded xed point
of operator J.Moreover,0 v
1
(x) v
n
(x)
cL
r
(
r+
)
n
for every x 0.
Proof.Since v
n
() % v
1
() as n!1,and every v
n
() is bounded from below by
cr
r
L,and
E
R
0
e
(r+ )t cr
r
Ldt
< 1for every 2 S
B
,the monotone convergence theorem implies that
v
1
(x) = sup
n0
v
n
(x) = sup
2S
B
lim
n!1
E
x
h
e
(r+ )
h(Y
X
0
) +
Z
0
e
(r+ )t
v
n
((1 y
0
)Y
X
0
t
)dt
i
= sup
2S
B
E
x
h
e
(r+ )
h(Y
X
0
) +
Z
0
e
(r+ )t
v
1
((1 y
0
)Y
X
0
t
)dt
i
= (Jv
1
)(x):
Thus,v
1
() is the bounded xed point of contraction mapping J.Lemma 3 implies 0 v
1
()v
n
(),
and kv
1
v
n
k = kJv
1
Jv
n1
k
r+
kv
1
v
n1
k ::: (
r+
)
n
cL
r
for every n 1.
3.2.The solution of the optimal stopping problemin (3.11).We shall next solve the optimal
stopping problemJw in (3.11) for every xed w:R
+
7!R which satises the following assumption:
Assumption 5.Let w:R
+
7!R be nonincreasing,convex,bounded between h() and cL=r,and
w(+1) =
cr
r
L and w(0+) =
c
r
L.
We shall calculate the value function (Jw)() and explicitly identify an optimal stopping rule.
Because w() is bounded,we have
E
x
Z
1
0
e
(r+ )t
jw((1 y
0
)Y
X
0
t
)jdt
kwk
Z
1
0
e
(r+ )t
dt =
kwk
r +
< 1;x 0;
and for every stopping time 2 S
B
,the strong Markov property of Y
X
0
at time implies that
(Hw)(x),E
x
Z
1
0
e
(r+ )t
w((1 y
0
)Y
X
0
t
)dt
(3.12)
= E
x
Z
0
e
(r+ )t
w((1 y
0
)Y
X
0
t
)dt
+E
x
h
e
(r+ )
(Hw)(Y
X
0
)
i
:
Therefore,E
x
[
R
0
e
(r+ )t
w((1 y
0
)Y
X
0
t
)dt] = (Hw)(x) E
x
[e
(r+ )
(Hw)(Y
X
0
)],and we can
write the expected payo E
x
[e
(r+ )
h(Y
X
0
) +
R
0
e
(r+ )t
w((1 y
0
)Y
X
0
t
)dt] in (3.11) as
(Hw)(x) +E
x
e
(r+ )
fh (Hw)g (Y
X
0
)
for every 2 S
B
and x > 0.If we dene
(Gw)(x),sup
2S
B
E
x
h
e
(r+ )
fh (Hw)g (Y
X
0
)
i
;x > 0;(3.13)
then the value function in (3.11) can be calculated by
(Jw)(x) = (Hw)(x) +(Gw)(x);x > 0:(3.14)
8 SAVAS DAYANIK AND MASAHIKO EGAMI
We take R
+
= [0;1) everywhere.The state 0 is a natural boundary point for the geometric
Brownian motion Y:it starts from 0,then it stays there forever and cannot get into the interior
of the state space with probability one.If it starts in the interior of its state space (namely,Y
0
2
(0;1)),then it can never reach 0.For all practical purposes,we can neglect state 0 and the values
at 0 of any functions related to Y.For completeness,we can for example dene G(0) = G(0+),
(Hw)(0) = (Hw)(0+),and (Jw)(0) = (Jw)(0+).
Let us rst calculate (Hw)().Let () and'() be,respectively,the increasing and decreasing
solutions of the second order ordinary dierential equation (A
0
f)(y) (r + )f(y) = 0,y > 0
with boundary conditions,respectively, (0+) = 0 and'(+1) = 0,where A
0
is the innitesimal
generator in (3.10) of diusion process Y
X
0
Y
X
0
;0
.One can easily check that
(y) = y
1
and'(y) = y
0
for every y > 0,(3.15)
with the Wronskian
W(y) =
0
(y)'(y) (y)'
0
(y) = (
0
+
1
)y
0
+
1
1
;y > 0;(3.16)
where
0
<
1
are the roots of the characteristic function g() =
2
2
( 1) +(r + y
0
)
(r + ) of the above ordinary dierential equation.Because both g(0) < 0 and g(1) < 0,we have
0
< 0 < 1 <
1
:
Let us denote the hitting and exit times of diusion process Y
X
0
,respectively,by
a
,infft 0;Y
X
0
t
= ag;a >0;
ab
,infft 0;Y
X
0
t
62 (a;b)g;0 < a < b < 1;
and dene operator
(H
ab
w)(x),E
x
Z
ab
0
e
(r+ )t
w((1 y
0
)Y
X
0
t
)dt +1
f
ab
<1g
e
(r+ )
ab
h(Y
X
0
ab
)
and
a
(y), (y)
(a)
'(a)
'(y) and'
b
(y),'(y)
'(b)
(b)
(y) for every y > 0,
which are,respectively,the increasing and decreasing solutions of (A
0
f)(y) (r + )f(y) = 0,
a < y < b with boundary conditions,respectively,f(a) = 0 and f(b) = 0.In terms of W() in
(3.16),the Wronskian of
a
() and'
a
() becomes
W
ab
(y) =
0
a
(y)'
b
(y)
a
(y)'
0
b
(y) =
1
(a)
'(a)
'(b)
'(b)
W(y);y > 0:(3.17)
Taylor and Karlin [20,Chapter 15],Borodin and Salminen [2] prove Lemma 6 below.
Lemma 6.For every x > 0,we have
(Hw)(x),E
x
Z
1
0
e
(r+ )t
w((1 y
0
)Y
X
0
t
)dt
= lim
a#0;b"1
(H
ab
w)(x)
='(x)
Z
x
0
2 ()w((1 y
0
))
p
2
()W()
d + (x)
Z
1
x
2'()w((1 y
0
))
p
2
()W()
d;
OPTIMAL STOPPING PROBLEMS FOR ASSET MANAGEMENT 9
which is twice-continuously dierentiable on R
+
and satises the ordinary dierential equation
(A
0
f)(x) (r + )f(x) +w((1 y
0
)x) = 0.
Using the potential theoretic direct methods of Dayanik and Karatzas [11] and Dayanik [10],we
shall nowsolve the optimal stopping problem(Gw)() (3.13) with payo function (h (Hw))(x) =
(x L)
+
x +
cL
r
h
'(x)
Z
x
0
2 ()w((1 y
0
))
p
2
()W()
d + (x)
Z
1
x
2'()w((1 y
0
))
p
2
()W()
d
i
= (xL)
+
x+
cL
r
2
2
(
1
0
)
h
x
0
Z
x
0
1
0
w((1y
0
))d+x
1
Z
1
x
1
1
w((1y
0
))d
i
;
where (x) = x
1
,'(x) = x
0
,p
2
() =
2
2
,W() =
0
()'() ()'
0
() = (
1
0
)
0
+
1
1
.
We observe that 0 (Hw)(x) = E
x
[
R
1
0
e
(r+ )t
w((1 y
0
)Y
X
0
t
)dt]
cL
r
R
1
0
e
(r+ )t
dt =
cL
r(r+ )
< 1.Hence,(h (Hw))() is bounded,and because (+1) ='(0+) = +1,we have
limsup
x#0
(h (Hw))
+
(x)
'(x)
= 0 and limsup
x"1
(h (Hw))
+
(x)
(x)
= 0:
By Propositions 5.10 and 5.13 of Dayanik and Karatzas [11],value function (Gw)() is nite;the
set
[w],fx > 0;(Gw)(x) = (h (Hw))(x)g = fx > 0;(Jw)(x) = h(x)g(3.18)
is the optimal stopping region,and
[w],infft 0;Y
X
0
t
2 [w]g(3.19)
is an optimal stopping time for (3.13)|and for (3.11) because of (3.14).According to Proposition
5.12 of Dayanik and Karatzas [11],we have
(Gw)(x) ='(x)(Mw)(F(x));x 0;and [w] = F
1
(f > 0;(Mw)() = (Lw)()g);
where F(x), (x)='(x) and (Mw)() is the smallest nonnegative concave majorant on R
+
of
(Lw)(),
8
>
<
>
:
h (Hw)
'
F
1
(); > 0;
0; = 0:
(3.20)
To describe explicitly the formof the smallest nonnegative concave majorant (Mw)() of (Lw)(),
we shall rstly identify a few useful properties of function (Lw)().Because Y
X
0
X
0
Y
1
by (3.8)
and w() is bounded,the bounded convergence theorem implies that
lim
x"1
(Hw)(x) = E
1
Z
1
0
e
(r+ )t
lim
x"1
w((1 y
0
)xY
1
t
)dt
=
w(+1)
r +
cL
r
;
and lim
x"1
(h (Hw))(x) = lim
x"1
((x L)
+
x +
cL
r
(Hw)(x))
cr
r+
L > 0.Therefore,
(Lw)(+1) = lim
x"1
(h (Hw))(x)
'(x)
= +1:(3.21)
Note also that
(Lw)
0
() =
d
d
h (Hw)
'
F
1
()
=
1
F
0
h (Hw)
'
0
F
1
():
10 SAVAS DAYANIK AND MASAHIKO EGAMI
Because F() is strictly increasing,we have F
0
> 0.Because w() is nonincreasing,the mapping
x 7!E
x
[
R
1
0
e
(r+ )t
w((1 y
0
)Y
X
0
t
)dt] = E
x
[
R
1
0
e
(r+ )t
w((1 y
0
)X
0
Y
1
t
)dt] is decreasing.Then
for x > L,because h() cL=r is constant,the mapping x 7!(
h (Hw)
'
)(x) is increasing.
For every 0 < x < L,we can calculate explicitly that [
1
F
0
(
h (Hw)
'
)
0
](x) =
x
1
1
0
(
0
)
cL
r
(1
0
)x
(
0
)
1
r +
x
1
Z
1
x
1
1
w((1 y
0
))d
;
and because lim
x#0
x
1
R
1
x
1
1
w((1 y
0
))d =
w(0+)
1
and
1
> 1,we have
lim
x#0
1
F
0
h (Hw)
'
0
(x) = +1:
Let us also study the sign of the second derivative (Lw)
00
().For every x 6= L,Dayanik and
Karatzas [11,page 192] show that
(Lw)
00
(F(x)) =
2'(x)
p
2
(x)W(x)F
0
(x)
(A
0
(r + ))(h (Hw))(x)(3.22)
and'();p
2
();W();F
0
() are positive.Therefore,
sgn[(Lw)
00
(F(x))] = sgn[(A
0
(r + ))(h (Hw))(x)]:
Recall from Lemma 6 that (A
0
(r + ))(Hw)(x) = w((1 y
0
)x) and because h(x) = (x +
cL
r
)1
fxLg
+
(cr)L
r
1
fx>Lg
,we have (A
0
(r + ))(h (Hw))(x) =
h
(1 y
0
)x (r + )
cL
r
+ w((1 y
0
)x)
i
1
fxLg
+
h
w((1 y
0
)x) (r + )
(c r)L
r
i
1
fx>Lg
:
Note that lim
x#0
(A
0
(r + ))(h (Hw))(x) = cL < 0 and lim
x"1
(A
0
(r + ))(h
(Hw))(x) = (c r)L < 0.Note also that x 7!(A
0
(r + ))(h (Hw))(x) is convex and
continuous on x 2 (0;L) and x 2 (L;1).Therefore,(A
0
(r + ))(h (Hw))(x) is strictly
negative in some open neighborhoods of 0 and +1,and in the complement of their unions,whose
closure contains L if it is not empty,it is nonnegative.Therefore,(3.22) implies that (Lw)() is
strictly concave in some neighborhood of = 0 and = 1,and in the complement of their unions,
whose closure contains F(L) if it is not empty,this function is convex.Earlier we also showed that
7!(Lw)() is increasing at every > F(L) and (Lw)(+1) = (Lw)
0
(0+) = +1.Moreover,
(Lw)
0
(F(L)) (Lw)
0
(F(L)+) =
L
1
1
1
0
< 0;
namely,(Lw)
0
(F(L)) < (Lw)
0
(F(L)+).Two possible forms of 7!(Lw)() and their smallest
nonnegative concave majorants 7!(Mw)() are depicted by two pictures of Figure 1.
The properties of the mapping 7!(Lw)() imply that there are unique numbers 0 <
1
[w] <
F(L) <
2
[w] < 1 such that
(Lw)
0
(
1
[w]) =
(Lw)(
2
[w]) (Lw)(
1
[w])
2
[w]
1
[w]
= (Lw)
0
(
2
[w]);
OPTIMAL STOPPING PROBLEMS FOR ASSET MANAGEMENT 11
concave
(Lw)()
(Mw)()
1
[w]
2
[w]F(L) 0
concave concave
(Lw)()
(Mw)()
1
[w]
2
[w]F(L) 0
concave
convex
Figure 1.Two possible forms of (Lw)() and their smallest nonnegative concave majorants (Mw)().
and the smallest nonnegative concave majorant (Mw)() of (Lw)() on (0;
1
[w]] [ [
2
[w];1) coin-
cides with (Lw)(),and on (
1
[w];
2
[w]) with the straight-line that majorizes (Lw)() everywhere
on R
+
and is tangent to (Lw)() exactly at =
1
[w] and
2
[w];see Figure 1.More precisely,
(Mw)() =
8
>
<
>
:
(Lw)(); 2 (0;
1
[w]] [[
2
[w];1);
2
[w]
2
[w]
1
[w]
(Lw)(
1
[w]) +
1
[w]
2
[w]
1
[w]
(Lw)(
2
[w]); 2 (
1
[w];
2
[w]):
Let us dene x
1
[w],F
1
(
1
[w]) and x
2
[w],F
1
(
2
[w]).Then by Proposition 5.12 of Dayanik
and Karatzas [11],the value function of the optimal stopping problem in (3.13) equals
(3.23) (Gw)(x) ='(x)(Mw)(F(x))
=
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
(h (Hw))(x);x2 (0;x
1
[w]] [[x
2
[w];1);
(x
2
[w])
1
0
x
1
0
(x
2
[w])
1
0
(x
1
[w])
1
0
(h (Hw))(x
1
[w])
+
x
1
0
(x
1
[w])
1
0
(x
2
[w])
1
0
(x
1
[w])
1
0
(h (Hw))(x
2
[w]);
x 2 (x
1
[w];x
2
[w]):
The optimal stopping region in (3.18) becomes [w] = fx > 0;(Gw)(x) = (h )(Hw)(x)g =
(0;x
1
[w]] [[x
2
[w];1),and the optimal stopping time in (3.19) becomes
[w] = infft 0;Y
X
0
t
2 (0;x
1
[w]] [[x
2
[w];1)g:
Proposition 7.The value function x 7!(Gw)() of (3.13) is continuously dierentiable on R
+
and twice-continuously dierentiable on R
+
n fx
1
[w];x
2
[w]g.Moreover,(Gw)() satises
(i) (A
0
(r + ))(Gw)(x) = 0;x 2 (x
1
[w];x
2
[w]);
(ii) (Gw)(x) > h(x) (Hw)(x);x 2 (x
1
[w];x
2
[w]);
(iii) (A
0
(r + ))(Gw)(x) < 0;x 2 (0;x
1
[w]) [(x
2
[w];1);
(iv) (Gw)(x) = h(x) (Hw)(x);x 2 (0;x
1
[w]] [[x
2
[w];1):
12 SAVAS DAYANIK AND MASAHIKO EGAMI
The dierentiability of (Gw)() is clear from (3.23).The variational inequalities can be veried
directly.For (iii) note that,if x 2 (0;x
1
[w]) [ (x
2
[w];1),then sgnf(A
0
(r + ))(Gw)(x)g =
sgnf(A
0
(r + ))(h (Hw))(x)g = sgnf(Lw)
00
(F(x))g < 0.
Because (Hw)() is twice-continuously dierentiable and (A
0
(r+ )(Hw))(x) = w((1y
0
)x)
for every x > 0 by Proposition 6,Proposition 7 and (3.14) lead directly to the next proposition.
Proposition 8.The value function x 7!(Jw)() of (3.11) is continuously dierentiable on R
+
and
twice-continuously dierentiable on R
+
n fx
1
[w];x
2
[w]g.Moreover,(Jw)() satises
(i) (A
0
(r + ))(Jw)(x) + w((1 y
0
)x) = 0;x 2 (x
1
[w];x
2
[w]);
(ii) (Jw)(x) > h(x);x 2 (x
1
[w];x
2
[w]);
(iii) (A
0
(r + ))(Jw)(x) + w((1 y
0
)x) < 0;x 2 (0;x
1
[w]) [(x
2
[w];1);
(iv) (Jw)(x) = h(x);x 2 (0;x
1
[w]] [[x
2
[w];1):
By Lemma 3,every v
n
(),n 0 and v
1
() are nonincreasing,convex,and bounded between
h() and cL=r.Moreover,by using induction on n,we can easily show that v
n
(0+) = cL=r and
v
n
(+1) = (c r)L=r for every n 2 f0;1;:::;1g.Therefore,Proposition 8,applied to w = v
1
,
and Proposition 4 directly lead to the next theorem.
Theorem 9.The function x 7!v
1
(x) = (Jv
1
)(x) is continuously dierentiable on R
+
and twice-
continuously dierentiable on R
+
n fx
1
[v
1
];x
2
[v
1
]g and satises the variational inequalities
(i) (A
0
(r + ))v
1
(x) + v
1
((1 y
0
)x) = 0;x 2 (x
1
[v
1
];x
2
[v
1
]);
(ii) v
1
(x) > h(x);x 2 (x
1
[v
1
];x
2
[v
1
]);
(iii) (A
0
(r + ))v
1
(x) + v
1
((1 y
0
)x) < 0;x 2 (0;x
1
[v
1
]) [(x
2
[v
1
];1);
(iv) v
1
(x) = h(x);x 2 (0;x
1
[v
1
]] [[x
2
[v
1
];1);
which can be expressed in terms of the generator A
in (3.5) of the jump-diusion process X as
(i)
0
(A
r)v
1
(x) = 0;x 2 (x
1
[v
1
];x
2
[v
1
]);
(ii)
0
v
1
(x) > h(x);x 2 (x
1
[v
1
];x
2
[v
1
]);
(iii)
0
(A
r)v
1
(x) < 0;x 2 (0;x
1
[v
1
]) [(x
2
[v
1
];1);
(iv)
0
v
1
(x) = h(x);x 2 (0;x
1
[v
1
]] [[x
2
[v
1
];1):
The next theorem identies the value function and an optimal stopping time for the optimal
stopping problem in (3.6).For every w:R
+
7!R satisfying Assumption 5 let us denote by e[w]
the stopping time of jump-diusion process X dened by
e[w],infft 0;X
t
2 (0;x
1
[w]] [[x
2
[w];1)g:
Theorem 10.For every x 2 R
+
,we have V (x) = v
1
(x) = E
x
e
re[v
1
]
h(X
e[v
1
]
)
,and e[v
1
] is
an optimal stopping time for (3.6).
OPTIMAL STOPPING PROBLEMS FOR ASSET MANAGEMENT 13
Proof.Let e
ab
= infft 0;X
t
2 (0;a] [[b;1)g for every 0 < a < b < 1.By It^o's rule,we have
e
r(t^^e
ab
)
v
1
(X
t^^e
ab
) = v
1
(X
0
) +
Z
t^^e
ab
0
e
rs
(A
r)v
1
(X
s
)ds
+
Z
t^^e
ab
0
e
rs
v
1
(X
s
)X
s
dB
s
+
Z
t^^e
ab
0
e
rs
[v
1
((1 y
1
)X
s
) v
1
(X
s
)](dN
s
ds)
for every t 0, 2 S,and 0 < a < b < 1.Because v
1
() and v
0
1
() are continuous and bounded
on every compact subinterval of (0;1),both stochastic integrals are square-integrable martingales,
and taking expectations of both sides gives
E
x
[e
r(t^^e
ab
)
v
1
(X
t^^e
ab
)] = v
1
(x) +E
x
h
Z
t^^e
ab
0
e
rs
(A
r)v
1
(X
s
)ds
i
:(3.24)
Because (A
r)v
1
() 0 and v
1
() h() by the variational inequalities of Theorem 9,we
have E
[e
r(t^^e
ab
)
v
1
(X
t^^e
ab
)] v
1
(x) for every t 0, 2 S,and 0 < a < b < 1.Because
lim
a#0;b"1
e
ab
= 1 a.s.and h() is continuous and bounded,we can take limits of both sides as
t"1,a#0,b"1and use the bounded convergence theorem to get E
[e
r
v
1
(X
)] v
1
(x) for
every 2 S.Taking supremum over all 2 S gives V (x) = sup
2S
E
[e
r
v
1
(X
)] v
1
(x).
In order to show the reverse inequality,we replace in (3.24) and e
ab
with e[v
1
].Because
(A
r)v
1
(x) = 0 for every x 2 (x
1
[v
1
];x
2
[v
1
]) by Theorem9 (i)
0
,E
x
[e
r(t^e[v
1
])
v
1
(X
t^e[v
1
]
)] =
v
1
(x)+E
x
[
R
t^e[v
1
]
0
e
rs
(A
r)v
1
(X
s
)ds] = v
1
(x) for every t 0.Since v
1
is bounded and con-
tinuous,taking limits as t"1and the bounded convergence give v
1
(x) = E
x
[e
re[v
1
]
v
1
(X
e[v
1
]
)] =
E
x
[e
re[v
1
]
h(X
e[v
1
]
)] V (x) by Theorem 9 (iv)
0
,which completes the proof.
Proposition 11.The optimal stopping regions [v
n
] = fx > 0;(Jv
n
)(x) h(x)g = (0;x
1
[v
n
]] [
[x
2
[v
n
];1),n 2 f0;1;:::;1g are decreasing,and 0 < x
1
[v
1
] ::: x
1
[v
1
] x
1
[v
0
] L
x
2
[v
0
] x
2
[v
1
] ::: x
2
[v
1
] < 1.Moreover,x
1
[v
1
] = lim
n!1
x
1
[v
n
] and x
2
[v
1
] = lim
n!1
x
2
[v
n
].
The proof follows from the monotonicity of operator J and that v
n
(x)"v
1
(x) as n!1
uniformly in x > 0.The next proposition and its corollary identify the optimal expected reward
and nearly optimal stopping strategies for the asset manager in the rst problem.
Proposition 12.For all n 0,we have v
1
(x) E
x
[e
re[v
n
]
h(X
e[v
n
]
)] +
cL
r
(
r+
)
n+1
.Hence,for
every"> 0 and n 0 such that
cL
r
(
r+
)
n+1
",the stopping time e[v
n
] is"-optimal for (3.6).
Proof.Recall that ~[v
n
] = infft 0;X
t
2 [v
n
]g = infft 0;X
t
2 (0;x
1
[v
n
]] [ [x
2
[v
n
];1).If we
replace and e
ab
in (3.24) with e[v
n
],then for every t 0 we obtain E
x
[e
r(t^e[v
n
])
v
1
(X
t^e[v
n
]
)] =
v
1
(x) + E
x
[
R
t^e[v
n
]
0
e
rs
(A
r)v
1
(X
s
)ds] = v
1
(x),because,for every 0 < t < e[v
n
] we have
X
t
2 (x
1
[v
n
];x
2
[v
n
]) (x
1
[v
1
];x
2
[v
1
]),at every element x of which (A
r)v
1
(x) equals 0
according to 9 (i)
0
.Because v
1
() is continuous and bounded,taking limits as t"1 and the
bounded convergence theorem give v
1
(x) = E
x
[e
re[v
n
]
v
1
(X
e[v
n
]
)].Because (Jv
n
)() = h() on
[v
n
] 3 X
e[v
n
]
on fe[v
n
] < 1g,Proposition 4 implies
v
1
(x) E
x
h
e
re[v
n
]
v
n+1
(X
e[v
n
]
) +
cL
r
r +
n+1
i
E
x
h
e
re[v
n
]
(Jv
n
)(X
e[v
n
]
)
i
14 SAVAS DAYANIK AND MASAHIKO EGAMI
+
cL
r
r +
n+1
= E
x
h
e
re[v
n
]
h(X
e[v
n
]
)
i
+
cL
r
r +
n+1
:
Corollary 13.The maximum expected reward of the asset manager is given by U(x) = x
cL
r
+
V (x) = x
cL
r
+ v
1
(x) for every x 0.The stopping rule e[v
1
] is optimal,and e[v
n
] is"-
optimal for every"> 0 and n 0 such that
cL
r
(
r+
)
n+1
<":U(x) = E
x
[e
re[v
1
]
(X
e[v
1
]
L)
+
+
R
e[v
1
]
0
e
rt
(X
t
cL)dt] and U(x) " E
x
[e
re[v
n
]
(X
e[v
n
]
L)
+
+
R
e[v
n
]
0
e
rt
(X
t
cL)dt],x > 0.
4.The solution of the asset manager's second problem
In the asset manager's second problem,the investors'assets have limited protection.In the
presence of the limited protection at level`> 0,the contract terminates at time e
`;1
,infft 0:
X
t
=2 (`;1)g automatically.The asset manager wants to maximize her expected total discounted
earnings as in (2.2),but now the supremum has to be taken over all stopping times 2 S which
are less than or equal to e
`;1
almost surely.Namely,we would like to solve the problem
U
`
(x),sup
2S
E
x
h
e
r(e
`;1
^)
(X
e
`;1
^
L)
+
+
Z
e
`;1
^
0
e
rt
(X
t
cL)dt
i
;x 2 R
+
:(4.1)
If`< x
1
[v
1
],then U
`
(x) = U(x) = E
x
[e
r(e[v
1
])
(X
e
`;1
[v
1
]
L)
+
+
R
e[v
1
]
0
e
rt
(X
t
cL)dt] for
every x > 0.On the one hand,because for every 2 S,e[v
1
] ^ also belongs to S,we have
U
`
(x) U(x).On the other hand,because` x
1
[v
1
],we have a.s.e[v
1
] = e
`;1
^ e[v
1
] 2 S
and U
`
(x) E
x
[e
re[v
1
]
(X
e[v
1
]
L)
+
+
R
e[v
1
]
0
e
rt
(X
t
cL)dt] = U(x) for every x.Therefore,
U
`
(x) = U(x) for every x > 0 if` x
1
[v
1
].
Assumption 14.In the remainder,suppose that the protection level`is such that x
1
[v
1
] <` L.
The strong Markov property of X can be used to similarly show that
U
`
(x) = x
cL
r
+V
`
(x);x 0;where V
`
(x),sup
2S
E
x
h
e
r(e
`;1
^)
h(X
e
`;1
^
)
i
;x > 0(4.2)
is the discounted optimal stopping problem for the stopped jump-diusion process X
e
`;1
^t
,t 0
with the same terminal payo function h() as in (3.7).Let us dene the stopping time
`;1
,
infft 0;Y
X
0
t
=2 (`;1)g of diusion process Y
X
0
and the operator
(4.3) (J
`
w)(x),sup
2S
B
E
x
h
e
r
h(X
`;1
^
)1
f
`;1
^<T
1
g
+e
rT
1
w(X
T
1
)1
f
`;1
^T
1
g
i
= sup
2S
B
E
x
h
e
(r+ )(
`;1
^)
h(Y
X
0
`;1
^
) +
Z
`;1
^
0
e
(r+ )t
w((1 y
0
)Y
X
0
t
)dt
i
;x 0:
We expect that V
`
() = (JV
`
)();namely,that V
`
() is one of the xed points of operator J
`
.We
can nd one of the xed points of J
`
by taking limit of successive approximations dened by
v
`;0
(x),h(x) and v
`;n
(x),(J
`
v
`;n1
)(x);n 1;x > 0:
Lemmas 1 and 3 and Propositions 2 and 4 hold with obvious changes.Let w:R
+
7!R be a
function as in Assumption 5.Then
(J
`
w)(x) = (Hw)(x) +(G
`
w)(x);x > 0;where(4.4)
OPTIMAL STOPPING PROBLEMS FOR ASSET MANAGEMENT 15
(M
`
w)() (Mw)()
`;1
[w]
1
[w]
`;2
[w]
2
[w]
F(`)
(Mw)()
1
[w]
2
[w]F(L) 0
(Lw)()
`;2
[w]
(M
`
w)()
F(`)
`;1
[w]
F(L) 0
(Lw)()
(Lw)()
u;2
[w]
(M
`
w)()
(M
u
w)()
F(u)
u;1
[w]
F(`)
`;1
[w]
(Mw)()
0
1
[w] F(L)
2
[w]
`;2
[w]
Figure 2.Sketches of (Lw)() and (M
`
w)().On the left:F(`)
1
[w].In the middle:
1
[w] < F(`)
F(L).On the right:the comparison of (M
`
w)() and (M
`
w)() for 0 <`< u < L.
(G
`
w)(x),sup
2S
B
E
x
h
e
(r+ )
`;1
^
fh (Hw)g (Y
X
0
`;1
^
)
i
;x > 0:(4.5)
We obviously have (G
`
w)(x) = h(x) (Hw)(x) for every x 2 (0;`].If the initial state X
0
of
Y
X
0
`;1
^t
,t 0 is in (`;1),then`becomes an absorbing left-boundary for the stopped process
Y
X
0
`;1
^t
,t 0.
Let (M
`
w)() be the smallest concave majorant on [F(`);1) of (Lw)() dened by (3.20) and
equal on (0;F(`)) identically to (Lw)().Then by Proposition 5.5 of Dayanik and Karatzas [11]
(G
`
w)(x) ='(x)(M
`
w)(F(x)),x > 0 and
`
[w] = F
1
(f > 0;(M
`
w)() = (Lw)()g) are value
function and optimal stopping region for (4.5).The analysis of the shape of (Lw)() prior to Figure
1 implies that there are unique numbers 0 <
`;1
[w] < F(L) <
`;2
[w] < 1such that
8
>
<
>
:
(Lw)
0
(
`;1
[w]) =
(Lw)(
`;2
[w]) (Lw)(
`;1
[w])
`;2
[w]
`;1
[w]
= (Lw)
0
(
`;2
[w])
namely,
`;1
[w]
1
[w] and
`;2
[w]
2
[w]
9
>
=
>
;
if F(`)
1
[w],
`;1
[w] =`and
(Lw)(
`;2
[w]) (Lw)(
`;1
[w])
`;2
[w]
`;1
[w]
= (Lw)
0
(
`;2
[w]) if F(`) >
1
[w],
and
(M
`
w)() =
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
(Lw)(); 2 (0;
`;1
[w]] [[
`;2
[w];1);
`;2
[w]
`;2
[w]
`;1
[w]
(Lw)(
`;1
[w])
+
`;1
[w]
`;2
[w]
`;1
[w]
(Lw)(
`;2
[w]);
2 (
`;1
[w];
`;2
[w]):
Let us dene x
`;1
[w] = F
1
(
`;1
[w]) and x
`;2
[w] = F
1
(
`;2
[w]).Then the value function equals
(4.6) (G
`
w)(x) ='(x)(M
`
w)(F(x))
16 SAVAS DAYANIK AND MASAHIKO EGAMI
=
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
(h (Hw))(x);x2 (0;x
`;1
[w]] [[x
`;2
[w];1);
(x
`;2
[w])
1
0
x
1
0
(x
`;2
[w])
1
0
(x
`;1
[w])
1
0
(h (Hw))(x
`;1
[w])
+
x
1
0
(x
`;1
[w])
1
0
(x
`;2
[w])
1
0
(x
`;1
[w])
1
0
(h (Hw))(x
`;2
[w]);
x 2 (x
`;1
[w];x
`;2
[w])
and the optimal stopping region and an optimal stopping time are given by
`
[w] = fx > 0;(G
`
w)(x) = (h (Hw))(x)g = (0;x
`;1
[w]] [[x
`;2
[w];1);(4.7)
`
[w],inffx > 0;Y
X
0
t
2
`
[w]g = inffx > 0;Y
X
0
t
2 (0;x
`;1
[w]] [[x
`;2
[w];1)g(4.8)
for the problem in (4.5).A direct verication together with the chain of equalities sgnf(A
0
(r + ))(G
`
w)(x)g = sgnf(A
0
(r + ))(h (Hw))(x)g = sgnf(Lw)
00
(F(x))g < 0 for every
x 2 (`;x
`;1
[w]) [ (x
`;2
[w];1) from Dayanik and Karatzas [11,page 192] prove the versions of
Propositions 7 and 8 and Theorem9 for the second problemobtained after G,H,J are replaced with
G
`
,H
`
,J
`
and all functions are restricted to [`;1).By the next theorem,optimal stopping time for
asset manager's second problem is of the form e
`
[w],infft 0;X
t
2 (0;x
`;1
[w]] [[x
`;2
[w];1)g.
Theorem 15.For every x 2 R
+
,we have V
`
(x) = v
`;1
(x) = E
x
e
re
`
[v
`;1
]
h(X
e
`
[v
`;1
]
)
,and
e
`
[v
`;1
] is an optimal stopping time for (4.2).
The proof is similar to that of Theorem 10,and Propositions 11 and 12 and Corollary 13 hold
with obvious changes.We expect that the value of the limited protection at level`to increase as`
increases.We also expect that the asset manager quits early as the protection limit`increases to
L.Those expectations are validated by means of the next lemma.
Lemma 16.Let w:R
+
7!R be as in Assumption 5.Suppose that 0 <`< u < L.Then
(i) (M
`
w)() (M
u
w)() on R
+
;(ii) 0 <
`;1
[w] <
u;1
[w] < F(L) <
u;2
[w] <
`;2
[w] < 1;
(iii) (J
`
w)() (J
u
w)() on R
+
;(iv) 0 < x
`;1
[w] < x
u;1
[w] < L < x
u;2
[w] < x
`;2
[w] < 1:
Recall that (M
`
w)() and (M
u
w)() coincide,respectively,on (0;F(`)] and (0;F(u)] with (Lw)()
and on (F(`);1) and (F(u);1) with the smallest nonnegative concave majorants of (Lw)(),
respectively,over (F(`);1) and (F(u);1).Therefore,(i) and (ii) of Lemma 16 immediately
follow;see the picture on the right in Figure 2.Finally,(iii) and (iv) follow from (i) and (ii) by
the relation (4.4):(J
`
w)(x) = (Hw)(x) +(G
`
w)(x) = (Hw)(x) +'(x)(M
`
w)(F(x)) for every
x;x
`;1
[w] = F
1
(
`;1
[w]),x
`;2
[w] = F
1
(
`;2
[w]),and that F() is strictly increasing.
Proposition 17 shows that demanding higher portfolio insurance or limiting more severely the
downward risks or losses also limits the upward potential and reduces the total value of the portfolio.
Proposition 17.For every 0 <`< u < L,(i) v
`;n
(x) v
u;n
(x) for all 0 n 1,(ii) U
`
(x)
U
u
(x) for every x 2 R
+
,and (iii) 0 < x
`;1
[v
`;n
] x
u;1
[v
u;n
] < L < x
u;2
[v
u;n
] x
`;2
[v
`;n
] < 1.
Proof.Note rst that v
`;0
(x) = h(x) = v
u;0
(x) for every x 2 R
+
.Suppose that v
`;n
() v
u;n
() for
some n 0.Then by the monotonicity and Lemma 16 (iii),v
`;n+1
() = (J
`
v
`;n
)() (J
`
v
u;n
)()
OPTIMAL STOPPING PROBLEMS FOR ASSET MANAGEMENT 17
(J
u
v
u;n
)() = v
u;n+1
().Therefore,for every n 0 v
`;n
() v
u;n
(),and v
`;1
() = lim
n!1
v
`;n
()
lim
n!1
v
u;n
() = v
u;1
(),which proves (i).By (4.2),U
`
(x) = x
cL
r
+v
`;1
(x) x
cL
r
+v
u;1
(x) =
U
u
(x) for every x > 0,and (ii) follows.Finally,(4.7) and (i) imply (0;x
`;1
[v
`;1
]] [[x
`;1
[v
`;1
];1) =
fx > 0;v
`;1
(x) h(x)g fx > 0;v
u;1
(x) h(x)g = (0;x
u;1
[v
u;1
]] [ [x
u;1
[v
u;1
];1).Hence,0 <
x
`;1
[v
`;1
] x
u;1
[v
u;1
] < L < x
u;2
[v
u;1
] x
`;2
[v
`;1
] < 1.Similarly,(0;x
`;1
[v
`;n
]] [[x
`;1
[v
`;n
];1) =
fx > 0;v
`;n+1
(x) h(x)g fx > 0;v
u;n+1
(x) h(x)g = (0;x
u;1
[v
u;n
]] [ [x
u;1
[v
u;n
];1),which
implies 0 < x
`;1
[v
`;n
] x
u;1
[v
u;n
] < L < x
u;2
[v
u;n
] x
`;2
[v
`;n
] < 1for every nite n 0.
5.Numerical illustration
For illustration,we take L = 1, = 0:275,r = 0:03,c = 0:05, = 0:08, = 0:01,y
0
= 0:03.
Observe that 0 < r < c < .We obtain
0
= 0:3910 and
1
= 2:7054.We implemented the suc-
cessive approximations of Sections 3 and 4 in R in order to use readily available routines to calculate
the smallest nonnegative concave majorants of functions.We have used gcmlcm function fromthe R
package fdrtool developed by Korbinian Strimmer for that purpose.The approximation functions
approxfun and splinefun were also useful to compactly represent the functions we evaluated on
appropriate grids placed on state space and their F-transformations.By trial-and-error,we nd
out that optimal continuation region lies strictly inside [0;10L].Because F(L) turns out to be
signicantly smaller than the upper bound 10L,for the accuracy of the results it proved useful to
put a grid on the interval [0;F(L)] one hundred times ner than the grid put on [F(L);F(10L)].
In the implementation of the successive approximations of Sections 3 and 4,we decided to stop
the iterations as soon as the maximum absolute dierence between the last two approximations
over the grid placed on [0;10L] is less than 0:01.We obtain a good approximation for the rst
problem after three iterations with the maximum absolute dierence kv
3
v
2
k 0:0011 and
returns v
3
(),(0;x
1
[v
2
]] [ [x
2
[v
2
];1) = (0;0:3874] [ [4:7968;1),and e[v
3
] = infft 0;X
t
=2
(0;0:3874] [ [4:7968;1)g as the approximate value function,approximate stopping region,and
nearly optimal stopping rule for (3.6),respectively.The bound of Corollary 13 also guarantees
that kV () v
3
()k
cL
r
(
r+
)
3
= 0:026.The leftmost picture in Figure 3 suggests that the
algorithm actually converges faster than what this upper bound implies.The second and third
pictures illustrate how the solution of each auxiliary problem is found by constructing the smallest
nonnegative concave majorants M of the transformations with operator L.The insets give closer
look over the small interval [0;F(L)] at the same pictures which are otherwise harder to identify.
The rst three pictures in Figure 3 are consistent with the general form sketched in Figure 1.
The last three pictures in Figure 3 similarly illustrate the solution of the second problem of
the asset manager when the investors hold a limited protection of their assets with lower bound
`= 0:69 on the market value of the asset manager's portfolio.Because x
1
[v
1
] x
1
[v
2
] = 0:3874 <
`< 4:7968 = x
2
[v
2
] x
2
[v
1
],the unconstrained solution of Problem 1 (corresponding to`= 0) is
not any more optimal.Therefore,we calculate the successive approximations of Section 4,which
converge in two iterations because kv
`;2
v
`;1
k 0:0063 < 1=100.Hence,v
`;2
(),(0;x
`;1
[v
`;1
]] [
18 SAVAS DAYANIK AND MASAHIKO EGAMI
0
1
2
3
4
5
0.8
1.0
1.2
1.4
x
x
1
[
v
2
]
x
2
[
v
2
]
v
0
(
x
)
h
(
x
)
v
1
(
x
)
v
2
(
x
)
v
3
(
x
)
0
50
100
150
0.4
0.5
0.6
0.7
0.8
=
F
(
x
)
L
v
0
(
)
L
v
1
(
)
L
v
2
(
)
0.0
0.5
1.0
1.5
0.35
0.40
0.45
0.50
0.55
0.60
0
50
100
150
0.4
0.5
0.6
0.7
0.8
=
F
(
x
)
1
[
v
2
]
2
[
v
2
]
M
v
0
(
)
M
v
1
(
)
M
v
2
(
)
0.0
0.5
1.0
1.5
0.35
0.40
0.45
0.50
0.55
0.60
1
[
v
2
]
0
1
2
3
4
5
0.8
1.0
1.2
1.4
x
x
1
[
v
l
,
1
]
x
2
[
v
l
,
1
]
l
v
l
,
0
(
x
)
h
(
x
)
v
l
,
1
(
x
)
v
l
,
2
(
x
)
0
50
100
150
0.4
0.5
0.6
0.7
0.8
=
F
(
x
)
F
(
l
)
L
v
l
,
0
(
)
L
v
l
,
1
(
)
0.0
0.5
1.0
1.5
0.35
0.40
0.45
0.50
0.55
0.60
F
(
l
)
0
50
100
150
0.4
0.5
0.6
0.7
0.8
=
F
(
x
)
F
(
l
)
M
v
l
,
0
(
)
M
v
l
,
1
(
)
0.0
0.5
1.0
1.5
0.35
0.40
0.45
0.50
0.55
0.60
1
[
v
l
,
1
]
F
(
l
)
Figure 3.Numerical illustrations of the solutions of the auxiliary optimal stopping problems (3.6) on the
left and (4.2) on the right in the rst and second problems (with`= 0:69),respectively.
0
1
2
3
4
5
0.00
0.02
0.04
0.06
0.08
0.10
market value of portfolio, x
The value of protection at level
l
=
0.69
v
3
(
.
)
v
l
,
2
(
.
)
x
1
[
v
2
]
x
2
[
v
2
]
x
1
[
v
l
,
1
]
x
2
[
v
l
,
1
]
L
No-difference price at level
l
=
0.69
v
3
(
L
)
v
l
,
2
(
L
)
=
0.087
0.0
0.2
0.4
0.6
0.8
1.0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
protection level, l
No difference price
x
1
[
v
2
]
l
=
0.69
L
0.087
Figure 4.On the left,the value of the limited protection at level`= 0:69 as the market value of portfolio
changes,and on the right,no-dierence prices of the protections for dierent protection limits.
[x
`;2
[v
`;1
];1) = (0;0:69] [ [3:4724;1),and e
`
[v
`;1
] = infft 0;X
t
62 (0;0:69] [ [3:4724;1)g are
approximate value function and stopping region,and nearly optimal stopping rule for (4.2).
Observe that the stopping region of Problem 2 contains the stopping region of Problem 1:
(0;x
`;1
[v
`;1
]] [ [x
`;2
[v
`;1
];1) = (0;0:69] [ [3:4724;1) (0;x
1
[v
2
]] [ [x
2
[v
2
];1) = (0;0:3874] [
[4:7968;1).Thus,asset manager stops early in the presence of portfolio protection at level
`= 0:69.Because U(x) x
cL
r
+ v
2
(x) and U
`
(x) x
cL
r
+ v
`;1
(x) are approximately
the value functions of Problems 1 and 2,the value of the limited protection at level`when
stock price is x equals U(x) U(`)(x) v
3
(x) v
`;2
(x),which is plotted on the left in Fig-
ure 4.Therefore,the no-dierence price of this protection at the initiation of the contract equals
U(L)U(`)(L) v
3
(L)v
`;2
(L) = 0:087.The plot on the right in Figure 4 shows the no-dierence
prices of the protection at levels`changing between 0 and L = 1.The protection has no value at
the protection levels less than or equal to x
1
[v
1
] x
1
[v
2
],because the optimal policy,even in the
absence of protection clause,instructs the asset manager to quit as soon as the market value of the
portfolio goes below x
1
[v
1
] x
1
[v
2
].
OPTIMAL STOPPING PROBLEMS FOR ASSET MANAGEMENT 19
Let us nish with a nal remark about the role of L.Let us replace U() in (2.2) with U
L
() to
emphasize its dependence on L > 0.Then
U
L
(x) = sup
2S
E
x
e
r
(X
L)
+
+
Z
0
e
rt
(X
t
cL)dt
= sup
2S
LE
x
e
r
(X
=L1)
+
+
Z
0
e
rt
(X
t
=Lc)dt
= sup
2S
LE
x=L
e
r
(X
1)
+
+
Z
0
e
rt
(X
t
c)dt
= LU
1
(x=L) for every x > 0:
Therefore,we can in fact choose L = 1 in (2.2) without loss of generality and solve it for U
1
() and
obtain the solutions for all other L > 0 values by the transformation U
L
(x) = LU
1
(x=L) for every
x > 0.
Acknowledgments
Savas Dayanik's research was partly supported by the T
UB
_
ITAK Research Grants 109M714 and
110M610.The authors thank two anonymous referees and the editors for the suggestions that
improved the presentation of the paper.
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Bilkent University,Departments of Industrial Engineering and Mathematics,Turkey (sdayanik@
bilkent.edu.tr)
Graduate School of Economics,Kyoto University,Japan (egami@econ.kyoto-u.ac.jp)
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