Optimal Trading with a Trailing Stop

0.78 MB·

by  Tim Leung

Optimal trading with a trailing stop
Tim Leung∗ Hongzhong Zhang†
January 17, 2017
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Abstract
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Trailing stop is a popular stop-loss trading strategy by which the investor will sell theasset once its
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a price experiences a pre-specified percentage drawdown. In this paper, we study the problem of timing
J buy and then sell an asset subject to a trailing stop. Under a general linear diffusion framework, we
4 studyanoptimaldoublestoppingproblemwitharandompath-dependentmaturity. Specifically,wefirst
1 derivetheoptimalliquidationstrategypriortoagiventrailingstop,andprovetheoptimalityofusinga
sell limit order in conjunction with the trailing stop. Our analytic results for the liquidation problem is

] then used to solve for the optimal strategy to acquire the asset and simultaneously initiate the trailing
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stop. The method of solution also lends itself to an efficient numerical method for computing the the
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optimal acquisition and liquidation regions. For illustration, we implement an example and conduct a
. sensitivity analysis undertheexponential Ornstein-Uhlenbeckmodel.
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- Keywords: trailing stop, stop loss, optimal stopping, drawdown, stochastic floor
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JEL Classification: C41, C61, G11
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Mathematics Subject Classification (2010): 60G40, 62L15, 91G20, 91G80

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∗Department of Applied Mathematics, Computational Finance and Risk Management Program, University of
Washington, Seattle WA 98195. E-mail: [email protected].
†IEORDepartment,ColumbiaUniversity. NewYork,NY10027. E-mail: [email protected]. Correspondingauthor.
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1 Introduction
Trailing stops are a popular trade order widely used by proprietary traders and retail investors to provide
downside protection for an existing position. In contrast to a fixed stop-loss exit, a trailing stop is charac-
terized by a stochastic floor that moves in parallel to the running maximum of the asset price. A trailing
stopistriggeredwhentheprevailingpriceofanassetfallsbelowthe stochasticfloor. Inessence,itallowsan
investorto specify alimit onthe maximumpossiblelosswhile notlimiting the maximumpossible gain. The
downside protectionis alsodynamic as the stochastic flooris raisedwheneverthe assetprice movesupward.
In addition to setting a trailing stop order, the investor can also use a limit order to sell at certain
price target. Indeed, if the price is sufficiently high, the investor may prefer to sell immediately as opposed
to waiting to set off the trailing stop. The investor’s position will be liquidated by either order. In this
paper, we investigate the optimal timing to liquidate a position subject to a trailing stop. Mathematically,
we recognize the trailing stop as a timing constraint in the sense that it installs a path-dependent random
maturityintotheliquidationproblem,rendingtheproblemsignificantlymoredifficulttosolve. Furthermore,
the investor can also decide when to establish the position. This leads us to analyze the optimal timing to
enter the market. In sum, we study an optimal double stopping problem subject to a trailing stop. By
using excursiontheoryof linear diffusion, we derive the value functions using the smallestconcavemajorant
characterization,anddiscusstheeffectoftrailingstoppingontheoptimaltradingstrategiesanalyticallyand
numerically. Amongourresults,wereducetheproblemoffindingtheoptimaltimingstrategiestosolvingan
ODE problem, which forms the basis of our numerical scheme in determining the optimal asset acquisition
and liquidation regions.
In general, a trailing stop can be defined as the first time when the asset price X drops below f(X),
where X is the running maximum process of X, and f is an increasing function such that f(x) < x for all
x in the support of X. In applied probability literature, such a stopping time is related to the drawdown
process and its first passage time. We refer to Lehoczky (1977), Zhang (2015), and Zhang and Hadjiliadis
(2012),forapartiallistofstudiesondrawdownsunderlineardiffusions. Moreover,theoptimalityoftrailing
stopsinexercisingRussianoptionsanddetectingabruptchangescanbefoundinShepp and Shiryaev(1993)
and Zhang et al. (2015), respectively.
Despite being commonly used by practitioners, trailing stops have been scarcely studied in the mathe-
maticalfinanceliterature. WetracebacktoGlynn and Iglehart(1995),whostudiedtheexpecteddiscounted
rewardatatrailingstopunderadiscrete-timerandomwalkorageometricBrownianmotion(GBM)model,
andfoundthatitwouldbeoptimaltoneverusethetrailingstopifthestockfollowedaGBMwithapositive
drift. In contrast, our study is conducted in a more general linear diffusion framework, and provides con-
crete illustrative example on how to the use of a trailing stop will affect the optimal timing to sell an asset
under the exponential Ornstein-Uhlenbeck model. In a random walk model, Warburton and Zhang (2006)
performed a probabilistic analysis of a variant of trailing stop. Yin et al. (2010) implemented a stochastic
approximation scheme to determine the optimal percentage trailing stop level that maximizes the expected
discounted simple return from liquidation. The recent study by Imkeller and Rogers (2014) compared the
performanceofanumberoftradingruleswithfixedandtrailingstopsunderanarithmeticBrownianmotion
model. Comparedto these works,we tackle the trading problem from an optimal stopping perspective, and
rigorouslyderivethe optimaltradingstrategy. Mathematically,weintroduceanewoptimaldouble stopping
problem subject to a stopping time constraint induced by the trailing stop. Our method of solution applies
to a generallinear diffusion framework,and our analyticalresults are amenable to computation of the value
function and optimal timing strategies (see Section 5).
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The incorporation of a trailing stop can be viewed as introducing a random maturity or stopping time
constraint to the optimal stopping problem, in the sense that any admissible stopping time must come
before triggering the trailing stop. Related studies by the authors include optimal stopping problems with
maturities determined by an occupation time (Rodosthenous and Zhang (2016, 2017)) or by a default time
(Leung and Yamazaki(2013)),andoptimalmeanreversiontradingwithafixedstop-lossexit(Leung and Li
(2015)). In particular, part of our study (Section 3) generalizes the analytical framework of Leung and Li
(2015)togenerallineardiffusions,andtheresultsfromoptimalstoppingsubjecttoafixedstop-lossexitwill
prove to be directly useful for solving the analogous problem with a trailing stop.
The remaining of the paper is structured as follows. Section 2 presents stochastic framework for our
tradingproblem. InSection3,westudy anoptimaltradingproblemwithafixedstop-loss. Then,inSection
4, we study the optimal stopping problems for trading with a trailing stop. To illustrate our analytical
results,we considertrading under the exponentialOrnstein-Uhlenbeck model, andnumerically compute the
optimalacquisitionandliquidationregionsinSection5. Wealsoprovideasensitivityanalysisontheoptimal
trading strategies with respect to model parameters.
2 Model Formulation
Letus considera riskyassetvalueprocessX = X modeledby alineardiffusiononI (l,r) R with
t t≥0
{ } ≡ ⊂
the infinitesimal generator:
1 ∂2 ∂
= σ2(x) +µ(x) , x I, (1)
L 2 ∂x2 ∂x ∀ ∈
where (µ(),σ()) is a pair of real-valued functions on I such that
· ·
1+ µ()
| · | L1 (I) and σ(x)>0, x I.
σ2() ∈ Loc ∀ ∈
·
For any x¯ I, the running maximum of X is denoted by
∈
X :=x¯ sup X , t 0.
t s
∨ ≥
s∈[0,t]
We denote the unique probabilitylaw of X by P given X =x,X =x¯ for any x,x¯ I with x x¯.
· x,x¯ 0 0
{ } ∈ ≤
The expectation associated with P is denoted by E . In calculations and results where the initial value
x,x¯ x,x¯
X = x¯ is irrelevant, we simply write P and E to denote the probability law of X and the associated
0 x x ·
expectation given X = x . Throughout, we assume that the upper boundary r is natural and the lower
0
{ }
boundary l is either natural or absorbing.
We consider an investor who holds long one unit of the risky asset X. Our objective is to investigate
the optimal trading strategy with a trailing stop. To this end, we consider the problem of optimal early
liquidationofthisriskyasset,givenapre-specifiedtrailingstopmandatoryliquidationorder. Specifically,we
willmodelliquidationtime by astopping time τ ofthe underlying processX, andthe rewardto be realized
·
upon liquidation by h(X ), where h() is a real-valued function on I, such that x I :h(x)>0 = . Fix
τ
· { ∈ }6 ∅
a function f() on I, such that
·
f() is continuous, strictly increasing on I,
· (2)
for all x I,f(x) I,f(x)<x.
∈ ∈
3
3.5
3.0
2.5
2.0
1.5
1.0
0.5
2 4 6 8 10
Figure 1: Sample paths of the asset price (solid black), its running maximum (gray dashed), and the 30%-
drawdown floor representing the trailing stop (red dashed).
Then, we define the stochastic floor by f(X), where X is the running maximum of X. The trailing stop,
denoted by ρ , is defined as the first time the asset value X reaches the stochastic floor f(X) from above.
f
That is,1
ρ :=inf t>0:X <f(X ) . (3)
f t t
{ }
Therefore, the investor faces the following optimal stopping problem:
v (x,x¯):= sup E (e−qτh(X )1 ). (4)
f x,x¯ τ {τ<∞}
τ∈TT
f
T
where q >0 is a subjective discounting rate, and is the set of all stopping times of X that stop no later
Tf
than the trailing stop ρ . Notice that ρ puts a mandatory selling order of the risky asset, pre-specified by
f f
the investor.
To quantify the gain in terms of expected discounted reward from liquidating earlier than the trailing
stop time ρ , we define the early liquidation premium by the difference
f
p (x,x¯):=v (x,x¯) g (x,x¯), (5)
f f f
−
where the second term represents the expected discounted reward from waiting to sell at the trailing stop,
that is,
gf(x,x¯):=Ex,x¯(e−qρfh(Xρf)1{ρf<∞}). (6)
As a convention, we define = if both terms on the right-hand side of (5) are infinity. Clearly,
∞−∞ ∞
we have p (x,x¯) 0 for all x,x¯ I with x x¯. For our study, the early liquidation premium turns
f
≥ ∈ ≤
out to be amenable to analysis and give intuitive interpretations. The related concepts of early/delayed
exercise/purchase premium have been analyzed in pricing American options (see Carr et al. (1992)) and
derivatives trading (Leung and Ludkovski (2011)), among other applications.
Remark 2.1. We give two standard choices of the floor function f() here with I =R. For example, setting
·
f(x) = x a for some a > 0 gives the absolute drawdown floor, and ρ is the first time X falls from its
f
−
maximum X by a units. Another specification, f(x) = (1 α)x for some α (0,1), gives the percentage
− ∈
1Asusual,wesetinf∅=∞.
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drawdown, and ρ is the first time X falls from its maximum X by (100 α)%, as depicted in Figure 1 with
f
×
α=0.3.
Remark 2.2. If floor functions f (),f () both satisfy (2), and f (x) f (x) for all x I, then for every
1 2 1 2
· · ≤ ∈
fixed x I, we have the inequalities:
∈
h(x) v (x,x¯) v (x,x¯) supE (e−rτh(X )1 ),
≤ f1 ≤ f2 ≤ x τ {τ<∞}
τ∈T
where is the set of all stopping times of X.
T
Given the optimal value v (x,x¯), another related problem is
f
v(1)(x)= supE (e−qˆτ(v (X ,X ) h(X ) c)1 ), (7)
f x f τ τ − τ − {τ<∞}
τ∈T
where qˆ (0,q] is the discounting rate, c [0,sup (v (x,x) h(x))) is the transaction fee,2 is the
∈ ∈ x∈I f − T
set of all stopping times w.r.t. the filtration generated by X. The problem arises, for example, in optimal
acquisition of the asset X when h(x)=x. In general, if we assume that h(X) is the price the investor need
to pay to acquire one unit of the risky asset, then (7) represents the problem of finding the optimal time to
purchase this risk asset. Note that the investor will select the optimal time to sell but subject to a trailing
stop exit. For this reason, we will call the problem in (7) the optimal acquisition problem with a trailing
stop, even for a general reward h().
·
Remark 2.3. Note that in (7), we apply the value function v (x,x¯) only with x=x¯. From a practical point
f
of view, this is the most relevant case since a trailing stop should be placed based on the price at which the
asset was purchased, rather than an arbitrary reference price.
In summary, the solutions to (7) and (4) yield the optimal trading strategy that involves buying a risky
asset and selling it later while being protected by a trailing stop.
2.1 Preliminaries of linear diffusions
It is well known that, for any q >0, the Sturm-Liouville equation ( q)u(x)=0 has a positive increasing
L−
solution φ+() and a positive decreasing solution φ−(). In fact, for an arbitrary fixed κ R , the solutions
q · q · ∈ +
can be expressed as
Ex(e−qτX+(κ)), if x κ 1 , if x κ
φ+q (x)=Eκ(e−1qτX+(x)), if x≤>κ, φ−q (x)=EExκ((ee−−qqττX−X−((κx)))), if x≤>κ, ∀x∈I, (8)
where τ+(y) and τ−(y) are the first passage times of X tolevel y I from below and above, respectively,
X X ∈
τ±(y):=inf t>0:X ≷y , y I. (9)
X { t } ∀ ∈
The functions φ±() are also closely related to two-sided exit problems of X. Specifically, we have:
q ·
2Ifcistoolargethenitisneveroptimaltoexercisesotheproblemhas0value.
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Lemma 2.1. [Lehoczky (1977)] Suppose that l<y x z <r, then for q >0, we have
≤ ≤
Ex(e−qτX−(y)1{τX−(y)<τX+(z)})= φφ−q−q((xy))ψψqq((zz))−ψψqq((xy)),
−
Ex(e−qτX+(z)1{τX−(y)>τX+(z)})= φφ−q−q((xz))ψψqq((xz))−ψψqq((yy)),
−
where ψ :I R is a strictly increasing function defined as
q +
7→
φ+(x)
q
ψ (x):= , x I. (10)
q φ−(x) ∀ ∈
q
Remark 2.4. By the boundary behavior of X, we have ψ (l+)=0 and ψ (r )= .
q q
− ∞
2.2 Standing assumption
We now discuss the following standing assumption on the reward function h().
·
Assumption 2.1. Define the function
h(x)
H(z):= , where z =ψ (x) R . (11)
φ−(x) q ∈ +
q
We assume that:
(i) There is z R such that H() is convex over (0,z ) and concave over [z , );
0 + 0 0
∈ · ∞
(ii) We have
H(z)
H(0):=limH(z)=0, and sup >H′ ( ), (12)
z↓0 z>z0 z + ∞
where H′ (z) is the right derivative of H() at z.
+ ·
Remark 2.5. Recall that x I : H(ψ (x)) > 0 x I : h(x) > 0 = , so the convexity of H() in
q
{ ∈ } ≡ { ∈ } 6 ∅ ·
Assumption 2.1 implies that there exists a z [0, ) such that H(z)>0 for all z (z , ) and H(z)<0
1 1
∈ ∞ ∈ ∞
for all z (0,z ). Moreover, we must have z <z (otherwise H(z)<0 on R ).
1 1 0 +
∈
Weremarkthat,whentherewardfunctionh()istwicedifferentiable,Assumption2.1canbeconveniently
·
verified by the super/sub-harmonic property of h().
·
Lemma 2.2. Let the reward function h() be twice differentiable and satisfy (12). Then, Assumption 2.1
·
holds if there exists a constant x I such that ( q)h(x) 0 if and only if x x .
0 0
∈ L− ≥ ≤
Proof. The claim follows from Section 6 of Dayanik and Karatzas (2003). Indeed, we have
2
H′′(z)= (( q)h(x)), for z =ψ (x).
σ2(x)φ−(x)(ψ′(x))2 L− q
q q
So H′′(z) 0 if and only if z ψ (x ).
q 0
≥ ≤
6
3 Optimal trading with a fixed stop-loss
To gain some intuition for our solution method for the problems in (4) and (7) with a trailing stop, we first
consider the optimal stopping problems when the investor uses a fixed stop-loss exit instead of a trailing
stop. Precisely, arbitrarily fix a y I, we consider the following class of problems indexed by y:
∈
V (x):= sup E (e−qτh(X )1 ), (13)
y x τ {τ<∞}
τ∈TS
y
V(1) :=supE (e−qτ(V (X ) h(X ) c)1 ), (14)
y x y τ − τ − {τ<∞}
τ∈T
where S is the set of all stopping times of X that stops no later than the first passage time to level y, i.e.
Ty
τ−(y)=inf t>0:X <y , (15)
X { t }
is the set of all stopping times of X, and c [0,sup (V (x) h(x))) is a transaction fee for asset
T ∈ x∈I y −
acquisition. The problem in (13) puts a mandatory liquidation constraint upon hitting the fixed stop-loss
level y from above.
The special cases of the problems in (13) and (14) with the reward function h(x) =x c driven by the
−
OU and CIR processes have been studied in Cartea et al. (2015); Leung and Li (2015); Leung et al. (2014,
2015). Inthis section,we presentthe analysisofproblems(13)and (14) drivenby a generallinear diffusion.
3.1 Optimal liquidation subject to a stop-loss exit
We now study the optimal liquidation problem (13) where X follows a general linear diffusion (see (1)). To
facilitate our analysis, we also consider the extended case of (13) for y =l, in which case we have
V (x)= supE (e−qτh(X )1 ). (16)
l x τ {τ<∞}
τ∈T
Remark 3.1. For any x I, the mapping y V (x) is obviously non-increasing over [0, ).
y
∈ 7→ ∞
Remark 3.2. The connection between (4) and (13) can be seen as follows. For any x,x¯ I such that
∈
x (f(x¯),x¯], by the P -a.s. inequality that ρ τ−(f(x¯)), we know that T S . Hence, v (x,x¯)
∈ x,x¯ f ≤ X Tf ⊂ Tf(x¯) f ≤
V (x). As a consequence, if we define the optimal liquidation regions
f(x¯)
T,L(x¯):= x (l,x¯]:v (x,x¯)=h(x) , x¯ I, (17)
Sf { ∈ f } ∀ ∈
S,L := x I :V (x)=h(x) , y I, (18)
Sy { ∈ y } ∀ ∈
then we have
S,L (l,x¯] T,L(x¯), x¯>0.
Sf(x¯)∪ ⊂Sf ∀
(cid:16) (cid:17)
Additionally, if x¯ S,L then we have S,L (l,x¯] = T,L(x¯), since in this case it is optimal to liquidate
∈Sf(x¯) Sf(x¯)∪ Sf
before X reaching a new maximum. (cid:16) (cid:17)
Theorem 3.1. Under Assumption 2.1, for any fixed y (l,ψ−1(z )), there is a finite threshold b(y) such
∈ q 0
7
that3
Vy(x)=Ex(e−q(τX+(b(y))∧τX−(y))h(XτX+(b(y))∧τX−(y))), ∀x∈I. (19)
Moreover, the mapping y : b(y) is non-increasing over (l,ψ−1(z )), with limits b(ψ−1(z ) ) = ψ−1(z ),
7→ q 0 q 0 − q 0
and b(l+)<r.
Proof. AccordingtoDayanik and Karatzas(2003),V (ψ−1(z))/φ−(ψ−1(z))isthesmallestconcavemajorant
y q q q
ofH(z)forallz >ψ (y),whichwedenotebyHˆ(z). BytheconvexityofH(),weknowthisconcavemajorant
q
·
is given by
z(y) z z ψ (y)
q
H(ψ (y)) − +H(z(y)) − , z (ψ (y),z(y)],
q q
Hˆy(z)= z(y)−ψq(y) z(y)−ψq(y) ∀ ∈ (20)
H(z), z (0,ψq(y)] (z(y), ),
∀ ∈ ∪ ∞

where z(y) is defined as
H(z) H(ψ (y))
q
z(y):=infargmax − . (21)
z>z0 z ψq(y)
−
Now define the barrier b(y):= ψ−1(z(y)), we have S,L =(l,y] [b(y),r). From Remark 3.1 we know that,
q Sy ∪
for l y <y <ψ−1(z ), the equalities hold:
≤ 1 2 q 0
((l,y ] [b(y ),r)) S,L S,L ((l,y ] [b(y ),r)).
1 ∪ 1 ≡Sy1 ⊂Sy2 ≡ 2 ∪ 2
Thus necessarily, b(y ) b(y ). To show the boundedness of b(y), we consider the special case that y = l.
2 1
≤
In this case, define the function F(z):=H(z)/z, which is continuous by the convexity of H(). We need to
·
show that F() attains its supremum over (z , ) at a finite point. Suppose this is not true, which means
0
· ∞
that F(z) must be maximized as z . However, by Assumption 2.1, we find a sufficiently large z > z
1 0
→ ∞
such that
sup F(z)>H′ (z ).
+ 1
z>z0
This implies that (using concavity of H() at z )
1
·
H(z )+H′ (z )∆
sup F(z)=limsupF(z) limsup 1 + 1 =H′ (z ),
z>z0 z→∞ ≤ ∆→∞ z1+∆ + 1
which is a contradiction to our choice of z . Hence we must have z(l)< so b(l)<r, and b(y) b(l+)
1
∞ ≤ ≤
b(l)<r by the monotonicity.
As y ψ−1(z ), b(y) converges to some limit in [ψ−1(z ),r). Suppose b(ψ−1(z ) ) b >ψ−1(z ), then
↑ q 0 q 0 q 0 − ≡ q 0
the concavity of H() over (z , ) implies that
0
· ∞
H(b) H(z )
H′ (b) − 0 .
+ ≥ ψ (b) z
q 0
−
However, by the definition of z(y), this inequality is in fact an equality. This implies that H() is in fact a
·
straightlineover[z ,ψ−1(b)],butthen(bythedefinitionofz(y),again)wemusthaveb(ψ−1(z ) )=ψ−1(z )
0 q q 0 − q 0
instead.
3Noticethatintheexpectation (19)theindicator1{τ+(b(y))∧τ−(y)<∞},asitisequal to1almostsurely.
X X
8
Remark 3.3. The mapping y b(y) is not necessarily continuous on (0,ψ−1(z )). For example, if H() is
7→ q 0 ·
piecewise linear and concave on (z , ), and we denote set of all kinks’ x-coordinates by . Then b(y) may
0
∞ K
be a piecewise constant function that maps onto . In this case, the function b(y) is right continuous with
K
a left limit at every y (l,ψ−1(z )].
∈ q 0
The proof of Theorem 3.1 gives an effective way to calculate the optimal threshold b(y) when H() (or
·
equivalently h()) is continuously differentiable.
·
Corollary3.1. InadditiontoAssumption2.1,assumethath()iscontinuouslydifferentiableover(ψ−1(z ),r),
· q 0
then for any y [l,ψ−1(z )), the optimal threshold b(y) is the smallest solution over (ψ−1(z ),r) to
∈ q 0 q 0
1 h′(b) h(b)φ−,′(b) 1 h(b) h(y)
q
= .
ψq′(b)(cid:18)φ−q (b) − (φ−q (b))2 (cid:19) ψq(b)−ψq(y)(cid:18)φ−q (b) − φ−q (y)(cid:19)
Remark 3.4. A non-continuously differentiable reward function h() may be chosen because one can still
·
obtain a similar characterization of the optimal barrier z(y) = ψ−1(b(y)) using the left/right derivative of
q
H(). Indeed, one way to construct the smallest concave majorant of H() over [ψ (y), ) is to tune the
q
· · ∞
z-value in (z , ) so that the line segment connecting points (ψ (y),H(ψ (y))) and (z,H(z)) does not go
0 q q
∞
below the graph of H(). But this line segments go below the graph of H() if and only if
· ·
H(z) H(ψ (y))
H′ (z)< − q .
− z ψ (y)
q
−
On the other hand, as a concave majorant, we automatically have that
H(z(y)) H(ψ (y))
H′ (z(y)) − q .
+ ≤ z(y) ψ (y)
q
−
In summary, the optimal barrier z(y) is determined by
H(z) H(ψ (y))
z(y)=inf z >z :H′ (z) − q H′ (z) . (22)
0 + ≤ z ψ (y) ≤ −
q
−
(cid:8) (cid:9)
Corollary 3.2. If y ψ−1(z ), then the stopping region S,L =I, i.e. there is no continuation region.
≥ q 0 Sy
Next, we establish the monotonicity of the value function V () over the continuation region for y
y
· ∈
(l,ψ−1(z )).
q 0
Proposition 3.1. For any y (l,ψ−1(z )), the value function V () is strictly increasing over [y,b(y)].
∈ q 0 y ·
Proof. We begin by studying the limiting case of y =l. In the case y =l, the smallest concaveHˆ () in (20)
l
·
must be strictly increasing over [0,z(l)]. Otherwise, we will have H(z) Hˆ (z) 0 for all z R , which is
0 +
≤ ≤ ∈
contradiction to Remark 2.5. Therefore, we may write
Hˆ (z)=β z, z [0,z(l)],
0 1
∀ ∈
for some β >0. But then it follows from Dayanik and Karatzas (2003) that
1
V (x)=φ−(x)Hˆ (ψ (x))=β φ+(x), x (l,b(l)).
0 q 0 q 1 q ∀ ∈
9
So the claim holds for the limiting case.
Now for any y (l,ψ−1(z )), by (22) and the concavity of H() over [z , ), we know that
∈ q 0 · 0 ∞
H(z(y)) H(ψ (y))
(Hˆ )′ (ψ (y))= − q is increasing in y.
y + q z(y) ψ (y)
q
−
Hence Hˆ′(z) > Hˆ′(z) for all z (ψ (y),z(y)) (0,z(l)). Then by Section 6 of Dayanik and Karatzas
y 0 ∈ q ⊂
(2003), this means that for all x (y,b(y)) (l,b(l)),
∈ ⊂
1 V′(x) φ−,′(x) 1 V′(x) φ−,′(x)
y V (x) q =Hˆ′(x)>Hˆ′(x)= 0 V (x) q .
ψq′(x)(cid:18)φq(x) − y (φ−q (x))2(cid:19) y 0 −ψq′(x)(cid:18)φq(x) − 0 (φ−q (x))2(cid:19)
Simplifying the above inequalities, we obtain
φ−,′(x)
V′(x) V′(x)> q (V (x) V (x)).
y − 0 − φ−(x) 0 − y
q
Since φ−() is positive and strictly decreasing, by Remark 3.1 we find that the right-hand side of the above
q ·
inequality is nonnegative. As a consequence, V′(x)>V′(x)>0, as claimed.
y 0
If we define the early liquidation premium subject to the (fixed) stop-loss exit τ−(y) (see (15)) by the
X
difference
Py(x):=Vy(x)−Ex(e−qτX−(y)h(XτX−(y))1{τX−(y)<∞}),
then we have the following characterizations.
Corollary 3.3. We have
1. If l<y <ψ−1(z ) and y <x<b(y), then
q 0
ψ (x) ψ (y)
P (x)=φ−(x)(H(ψ (b(y))) H(ψ (y))) q − q .
y q q − q ψ (b(y)) ψ (y)
q q
−
2. If y <ψ−1(z ) and x b(y) or y ψ−1(z ), then
q 0 ≥ ≥ q 0
φ−(x)
q
P (x)=h(x) h(y) .
y − φ−(y)
q
Proof. Suppose that 0 < y < ψ−1(z ) and y < x < b(y). Then, the strong Markov property of X implies
q 0
that
Py(x)=Ex([h(b(y))e−qτX+(b(y))−h(y)e−qτX−(y)]1{τX+(b(y))<τX−(y)})
=Ex(e−qτX+(b(y))[h(b(y))−h(y)Eb(y)(e−qτX−(y))]1{τX+(b(y))<τX−(y)})
φ−(b(y)) φ−(x) ψ (x) ψ (y)
q q q q
= h(b(y)) h(y) − (23)
(cid:18) − φ−q (y) (cid:19)φ−q (b(y))ψq(b(y))−ψq(y)
ψ (x) ψ (y)
=φ−(x)(H(ψ (b(y))) H(ψ (y))) q − q ,
q q − q ψ (b(y)) ψ (y)
q q
−
where we have used (8) and Lemma 2.1 in (23).
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