Liquidation in Limit Order Books with Controlled Intensity

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by  Erhan Bayraktar

LIQUIDATION IN LIMIT ORDER BOOKS WITH CONTROLLED INTENSITY
ERHANBAYRAKTARANDMICHAEL LUDKOVSKI
2 Abstract. Weconsideraframeworkforsolvingoptimalliquidationproblemsinlimitorderbooks.
1
Inparticular,orderarrivalsaremodeledasapointprocesswhoseintensitydependsontheliquida-
0
2 tion price. We set up a stochastic control problem in which the goal is to maximize the expected

n revenue from liquidating the entire position held. We solve this optimal liquidation problem for
a
power-law and exponential-decay order book models explicitly and discuss several extensions. We
J
6 alsoconsiderthecontinuousselling(orfluid)limit whenthetradingunitsareeversmallerandthe
2 intensity is everlarger. This limit provides an analytical approximation to the value function and

the optimal solution. Using techniques from viscosity solutions we show that the discrete state
]
R
problemanditsoptimalsolutionconvergetothecorrespondingquantitiesinthecontinuousselling
T
limit uniformly on compacts.
.
n
i
f
- 1. Introduction
q
[
Liquidation of large securities positions has emerged as an important problem in financial math-

5
v ematics, linking together models of market microstructure and control theory. In this paper we
7
consider an investor who liquidates a position through limit orders placed in a limit order book
4
2 (LOB). The investor does so by choosing the price of the limit order; the higher the price of the
0
. limit order, the smaller the probability that it would be filled. The objective of the investor is to
5
0 come up with an optimal limit order strategy that maximizes her expected revenue by date T.
1
1 Our model for the above problem is based on a point-process view of limit order books which
:
v treats liquidation as a sequence of discrete events, i.e. order matches. More precisely, we assume
i
X that the investor effectively controls the frequency of her trades by choosing the spread s above
r
a the current bid price Pt. The trade intensity is controlled as Λ(s) and when a trade occurs, the
investor generates a liquidation profit of s. Similar setups have been proposed in [5], [16, 15] and
rely essentially on a queueing system representation of LOB’s.
A crucial modeling difference is whether execution takes place through market or limit orders. If
investor trades via market orders, she necessarily encounters price impact through “eating away”
a portion of the LOB. The precise price impact depends on the shape of the LOB, as well as its
resilience. Conversely, there is no transactions or fill risk as market orders execute instantaneously.
This point of view is taken in e.g., [24, 1, 2]. On the other hand, if the investor trades through
Key words and phrases. Limit orderbooks, controlled intensity,optimal control of point processes, time toliqui-
dation, optimal control of queueing networks, fluid limit, viscosity solutions.
E.BayraktarissupportedinpartbytheNationalScienceFoundationunderanappliedmathematicsresearchgrant
andaCareergrant,DMS-0906257andDMS-0955463,respectively,andinpartbytheSusanM.SmithProfessorship.
1
2 ERHANBAYRAKTARANDMICHAELLUDKOVSKI
limit orders, liquidation depends on being “lifted” by a sufficiently large market order, leading to
substantial fill risk that again depends on the shape and depth of the LOB. The fill risk is related
to the concept of virtual price impact [28] and is the focus of our model here. Related approaches
to trading via limit orders can be found in [5, 14, 20, 21]. Also, in our previous work [12] we
considered the same LOB as here but with an uncontrolled trade intensity and temporary price
impact from order size.
Ideally, a fully specified model will reconcile the two approaches above, as well as consider the
underlying risk preferences of the investor. This is especially important for dealing with simulta-
neous trading on multiple exchanges, see the very recent preprints [23, 22], as well as Section 5.5
below. A full treatment of this problem will be the subject of a separate paper.
Ourstarting pointisadiscrete-state problemforaninvestor holdingnsharesofan illiquidasset,
n N. Since practically speaking n is often large (on the order of hundreds of thousands), we also
∈
investigate the fluid limit of our setup. On a technical level, the fluid limit provides asymptotic
results for the discrete-state problem (see Remark 3.3), which is the main focus of our paper.
On a formal level, our control problem is equivalent to a controlled death process and is closely
related to fluid approximations of some queueing problems. We refer to [8, 9, 10, 18, 25, 26] and
references therein for the most relevant strand of this rich literature. In contrast with the previous
literature, which uses probabilistic arguments, we utilize viscosity techniques to show convergence
(both of the value functions and the corresponding optimal controls) from the discrete- to the
continuous-state problems.
An investor holds n shares of an asset. Let (P ) be the bid price process for the underlying
t t≥0
asset. Let r 0 be the risk-free rate. We assume that e−rtP is a martingale with respect to the
t
≥
optimization measure P on a filtered probability space (Ω, ,( )). This assumption is consistent
t
F G
with standard market microstructure models, see e.g. [2]. Let Λ be the (controlled) intensity of
t
order fill, and let s 0 be the spread between the bid price and the limit order of the investor.
t
≥
Denote by N the -adapted counting process of order fills and τ the corresponding arrival times,
t k
G
N = 1 .
t {τk≤t}
k
X
t
Then N Λ ds is a martingale and expected revenue is
t− 0 s
R n
(1.1) E e−rτi(P +s 1 ) .
τi τi {τi≤T}
" #
i=1
X
We assume that the investor has a deadline date T + by which all trades must be completed.
≤ ∞
Remaining shares are liquidated at zero profit at T.
To introduce the liquidation control, we assume that Λ = Λ(s ), so that the intensity of order
t t
fills is a function of the offered spread above the bid price. Moreover, we assume that the bid price
P is unaffected by the limit orders created via (s ). Since e−rtP is a martingale, the first term in
t t
(1.1) is independent of τ . Indeed, E[ n e−rτiP ] =nP and we may ignore P in the subsequent
i i=1 τi 0
analysis.
P
LIQUIDATION IN LIMIT ORDER BOOKS WITH CONTROLLED INTENSITY 3
We define
n
(1.2) V(n,T):= sup E e−rτis 1
τi {τi≤T}
(st)∈ST "i=1 #
X
T∧τ(X) T∧τ(X)
(1.3) = sup E e−rts dN = sup E e−rts Λ(s )dt ,
t t t t
(st)∈ST "Z0 # (st)∈ST "Z0 #
where
τ(X) := inf t 0: X = 0
t
{ ≥ }
is the time of liquidation. Here, X := X N , with X = n, is a “death” (or inventory) process
t 0 t 0
−
with intensity Λ(s ). Note that T in (1.2) represents time-to-maturity and is the collection
t T
S
of -adapted controls, s 0 with := σ(N : s t). The boundary conditions on V are
t t s
F ≥ F ≤
V(n,0) = 0 n (terminal condition in time) and V(0,T) = 0 T (exhaustion).
∀ ∀
Remark 1.1. Our model is related to the limit order book setup of [5], which assumes that limit
orders are “lifted” through sufficiently large market buy orders. Namely, a market buy order of
size q, hits all limit sell orders that are within I(q) of the best bid. Assuming that buy market
orders arrive in the form of a Poisson random measure on R R with arrival intensity λ¯dt
+ +
×
and volume (mark) distribution f(dq), q 0, a sell limit order at a given spread u is lifted with
≥
probability P(I(q) > u). By the thinning lemma on Poisson processes, such matching buy orders
form a Poisson process with intensity λ¯ ∞ f(dq). Empirical studies suggest a power-law depth
I−1(u)
function f(dq) q−1−adq [5] and therefore if Λ(0) < , we can view our model within the [5]
R
∝ ∞
framework, Λ(s) [I−1(s)]−a, with I−1 the virtual price impact function [28].
∝
As in [11], the above control problem can be transformed into a discrete-time Markov decision
problem and the classical results from [13, Ch. 8] can be used to prove a dynamic programming
principle. Using the latter result one can show that the value function is a viscosity solution of
(1.4) V +supΛ(s) V(n 1,T) V(n,T)+s rV(n,T)= 0,
T
− − − −
s≥0
(cid:2) (cid:3)
with boundary conditions V(0,T) = V(n,0) = 0 and V denoting partial derivative with respect
T
to time-to-expiration. Standard results also imply that an optimal control can be taken of Markov
feedback type, s∗ = s(X∗,T t). However, in most of the examples below we will obtain explicit
t t −
solutions to this dynamicprogramming equation. Then a verification lemma can beused to justify
that the solution of (1.4) is indeed the value function.
The optimization problem described in (1.2) is simplified but highly tractable. In most of the
examples below, we are able to obtain closed-form solutions which provide direct insight into the
relationship between the LOBmodel and its depth function and the investor’s liquidation strategy.
In Section 2 we give an explicit solution for (1.2) in the case of a power-law intensity control Λ(s).
Section 3 then studies convergence of the discrete problem (1.4) to its continuous-state fluid limit.
Our key Theorem 3.1, complemented by Proposition 3.2 and Corollary 3.3, gives a full account of
thisconvergenceusingtechniquesfromviscositysolutionsofnonlinearpartialdifferentialequations.
4 ERHANBAYRAKTARANDMICHAELLUDKOVSKI
In Section 4 we return to (1.2) for the case whereΛ(s) is of exponential shape; we are again able to
provide several closed-form solutions. Finally, Section 5 considers several extensions and numerical
illustrations of (1.2), including generic Λ(s), which shed additional light on the problem structure.
2. Power-Law Limit Order Books
Inthis section weassumethatincomingbuyordershave apower-law distributionfor thespread,
Λ(s)= λ for some α> 1. It can be observed from the computations below that if α 1, then no
sα ≤
optimal control exists. Similar assumption was made (and justified empirically) by [5] who write
that in realistic markets α [1.5,3].
∈
Proposition 2.1. Assume that Λ(s) = λs−α with boundary conditions V(0,T) = V(n,0) = 0 for
all n. Then the solution of (1.4) and the optimal spread are respectively
1/(α−1)
(2.1) V(n,T)= c 1 e−rαT 1/α, s∗(n,T) = λ 1 e−rαT 1/α,
n
− αrc · −
(cid:18) n(cid:19)
(cid:0) (cid:1) (cid:0) (cid:1)
with c satisfying the recursion
n
(2.2) rc = A λ(c c )1−α, n 1, c = 0,
n α n n−1 0
− ≥
where
(α 1)α−1
(2.3) A := − .
α αα
Remark 2.1. Note that V is “concave” in n in the sense that
V(n+1,T) V(n,T) V(n,T) V(n 1,T),
− ≤ − −
i.e., its linear interpolation in n is concave in the usual sense. This follows immediately from (2.2)
since
1/(1−α) 1/(1−α)
rc rc
n+1 n
c c = = c c ,
n+1 n n n−1
− λA ≤ λA −
(cid:18) α (cid:19) (cid:18) α(cid:19)
wheretheinequality follows from thefact that c (or V(n,T)) is increasing in n. Thelatter follows
n
directly from (1.2).
Also observe from (2.1) and (2.2) that
α
s∗(n,T)= (V(n,T) V(n 1,T)),
α 1 − −
−
which implies that n s∗(n,T) is a decreasing function, because n V(n,T) is “concave”. One
7→ 7→
can also think of s∗ as the derivative of the linear interpolation of V in n.
Proof. With the power law assumption (1.4) reduces to
λ
(2.4) V +sup (V(n 1,T) V(n,T)+s) rV = 0,
− T sα − − −
s
LIQUIDATION IN LIMIT ORDER BOOKS WITH CONTROLLED INTENSITY 5
and therefore the candidate optimal policy is t s∗(X ,T t) in which s∗(n,T)= α (V(n,T)
7→ t − α−1 −
V(n 1,T)). To begin solving this equation, we start with n = 1. Since V(0,T) = 0 for all T, we
−
obtain for V = V(1,T)
V +A λV1−α rV = 0.
T α
− −
This is a separable ordinary differential equation (ODE) which simplifies to
· Vα−1
T +C = dV.
A λ rVα
α
Z −
Using the boundary condition V(1,0) = 0 we integrate to obtain
1/α
A λ
log(A λ rVα) = rα(T +C) V(1,T) = α (1 e−rαT) .
α
− − ⇐⇒ r −
(cid:26) (cid:27)
Consideringtheequationforgeneraln> 1wethereforemaketheansatzV(n,T) = c (1 e−rαT)1/α
n
−
and plugging into (2.4) the relation (2.2) follows. (cid:3)
Remark 2.2. Ifnodiscountingispresentr = 0,onecanverifythatthesolutionof (1.4)isV(n,T) =
d T1/α, where the sequence (d ) satisfies the recursion
n n
α−1
α 1
d = λ − (d d )1−α, d =0.
n n n−1 0
α −
(cid:18) (cid:19)
This result can also be obtained by taking the limit r 0 in (2.1), (2.2).
→
Fixing X =n, the inter-trade intervals σ := τ τ , i n have survival functions given by
0 i i i−1
− ≤
t
P(σ > t τ )= exp Λ(s∗(n i+1,T τ s))ds .
i i−1 i−1
| − − − −
(cid:18) Z0 (cid:19)
NotingthatΛ(s∗(n,T)) = C(n) forsomeconstantC(n),itfollows that εΛ(s∗(n,T))dT = +
1−e−rαT 0 ∞
forallnandεandthereforeP(σ T τ ) = 1foralli n. We concludethateven thoughthere
i i−1 R
≤ − ≤
isnodirectpenalty ifsomeordersremainatT,withprobabilityone, thefullinventory isliquidated
by T, X∗ = 0 P-a.s. In particular, the problem with a hard liquidation constraint V(x,0) =
T
M1 for any liquidation penalty M 0 will have the same solution as in Proposition 2.1.
{x≥0}
− ≥
2.1. Infinite Horizon. As the execution horizon T grows, the investor faces a weaker liquidation
constraint. Nevertheless, she still prefers to sell earlier than later due to the discount parameter
r that incentivizes faster liquidation. For the limit T we obtain an infinite-horizon model
→ ∞
whereby strategies are time-homogenous.
Taking T in (2.1) we find that V(n) = c and s∗(n) = λ1/(α−1)(αrc )1/(1−α). To under-
n n
→ ∞
stand how quickly execution takes place let us introduce expected time to liquidate S(n) which is
defined to be S(n) := E[τ(X∗)X∗ = n], in which X∗ is the death process whose intensity at time
| 0
t is Λ(s∗(X∗)) (representing optimally controlled inventory at t). When the inventory is X∗ = n,
t t
6 ERHANBAYRAKTARANDMICHAELLUDKOVSKI
liquidation occurs at rate Λ(s∗(n)), so that the interval until the next trade has an Exponential
distribution with mean 1/Λ(s∗(n)). It follows that
n n
(2.5) S(n) = Λ(s∗(j))−1 = λ1/(α−1) (αrc )−α/(α−1).
j
j=1 j=1
X X
3. Continuous selling limit
To better understand the results of Proposition 2.1 we consider a limiting continuous model.
Let us denote the number of shares initially held by x.
We firstintroduceasequenceofdiscretecontrolproblemsthatconverge tothecontinuous selling
limit. For 0 < ∆ 1, consider theproblemwheresharesaresoldat∆ incrementsandtheintensity
≤
of order fills is Λ∆(s) := Λ(s)/∆. We will denote by X∆ the “death” process with this intensity
and decrements of size ∆. Then the resulting value function
x/∆ T∧τ(X∆)
(3.1) V∆(x,T) := sup E e−rτi∆ s 1 = sup E e−rts Λ(s )dt ,
 · τi {τi≤T} t t
(st)∈ST i=1 (st)∈ST "Z0 #
X
 
x 0,∆,2∆, , T R would satisfy
+
∈ { ···} ∈
λ
(3.2) V∆ +sup (V∆(x ∆,T) V∆(x,T)+s∆) rV∆ =0
− T sα∆ − − −
s≥0
in viscosity sense.
Let us consider the first order partial differential equation (PDE)
s v
x
(3.3) v +supλ − rv = 0,
− T sα −
s≥0
which can be written as
v +A λv1−α rv = 0,
− T α x −
with boundary conditions v(x,0) = v(0,T) = 0. The solution of (3.3) has the following determin-
istic control representation
T∧τ(X(0),x) λ
(3.4) v(x,T) = sup e−rtdt,
sα−1
(st)∈ST Z0 t
where dX(0),x = λs−αdt, X(0),x = x. In fact, the solution of (3.3) is explicitly given by
t − t 0
1/α
(3.5) v(x,T) = λ x(α−1)/α 1 e−rαT 1/α.
rα −
(cid:18) (cid:19)
(cid:0) (cid:1)
We denote the optimizer in (3.3) by s(0)(x,T), which is explicitly given by
1/α
s(0)(x,T) = λ 1 1 e−rαT 1/α.
αr x1/α −
(cid:18) (cid:19)
(cid:0) (cid:1)
LIQUIDATION IN LIMIT ORDER BOOKS WITH CONTROLLED INTENSITY 7
Remark 3.1. Plugging the optimizer back into the dynamics for X(0),x we obtain that
(0),x
αrX
dX(0),x = t dt,
t −1 e−rα(T−t)
−
which can be explicitly solved as
t αr
(0),x
(3.6) X = xexp du .
t − 1 e−αr(T−u)
(cid:18) Z0 − (cid:19)
Let x R be fixed and let us consider all the collections 0,∆,2∆, of grids that contain x
+
∈ { ···}
as an element. In the next result, will show that as ∆ 0 then V∆(x) v(x). In fact, the next
→ →
result shows that this convergence is uniform on compacts.
Theorem 3.1. As ∆ 0, V∆ v uniformly on compact sets.
→ →
Proof. Let us consider the regularized stochastic control problem
x/∆
(3.7) V∆,k(x,T) := sup E e−rτi∆ s 1 , x 0,∆,2∆, ,
 · τi {τi≤T} ∈ { ···}
(st)∈STk Xi=1
 
where k := s : s [1/k,k] , k > 1. Using a representation similar to the one in (3.1) and
ST { ∈ ST t ∈ }
using the lower bound on the controls s k, it can be seen that
∈ST
λ
(3.8) V∆,k(x,T) kα−1.
≤ r
We will follow the arguments of [7] in the proof of their Theorem 2.1 (also see Theorem 4.1
on page 334 of [19]) to show that V∆,k converges uniformly on compacts to the unique viscosity
solution of
s vk
(3.9) vk + sup λ − x rvk = 0, vk(x,0) = 0.
− T sα −
s∈[1/k,k]
Let v¯k and vk be defined by:
v¯k(x,T) := limsuplimsupsup V∆,k(y,S) : x y + T S δ, y 0,∆, ,
| − | | − |≤ ∈ { ···}
δ→0 ∆→0
n o
vk(x,T) := liminfliminfinf V∆,k(y,S) : x y + T S δ, y 0,∆, .
δ→0 ∆→0 | − | | − |≤ ∈ { ···}
n o
By definition we have that vk vk v¯k and that vk is lower semi-continuous, and v¯k is upper
≤ ≤
semi-continuous; see e.g. Proposition 5.2.1 of [6]. We will show that v¯k is a subsolution and that
vk is a supersolution of (3.9). It follows from Theorem 5.4.20 in [6] that a comparison result holds
for this PDE (the compactness of the control space is required in order to apply this result). This
comparison theorem would then imply that v¯k vk. As a result, vk = v¯k = vk is the unique
≤
continuous viscosity solution of (3.9). This fact together with the way the functions v¯k and vk are
defined also imply the local uniform convergence of V∆,k to vk. (For a similar argument see page
35 of [17].)
We now prove that v¯k is a viscosity subsolution of (3.9); the fact that vk is a viscosity su-
persolution follows similarly. Let (x ,T ) be a local maximum of v¯k φ for some test function
0 0
−
8 ERHANBAYRAKTARANDMICHAELLUDKOVSKI
φ C1,1. Without loss of generality, we will assume that (x ,T ) is a strict local maximum and
0 0
∈
that v¯k(x ,T )= φ(x ,T ), and φ 2λkα−1 outside the ball B(x ,T ;r), where r > 0 is chosen so
0 0 0 0 ≥ r 0 0
that (x ,T ) is the maximum of v¯k φ on B(x ,T ;r). Thanks to the choice of the test function
0 0 0 0
−
outside this ball, (x ,T ) is in fact a global maximum of the function v¯k φ and it is attained on
0 0
−
B(x ,T ;r). (This is where the uniform boundedness assumption in (3.8) is used.)
0 0
Let (x∆,T∆) 0,∆, R be a point at which V∆,k φ attains its (global) maximum. It
+
∈ { ···}× −
follows from the definition of v¯k and the fact that (x ,T ) is a strict global maximum of v¯k φ
0 0
−
that there exists a sequence ∆ 0 such that (x∆n,T∆n) (x ,T ), V∆n,k φ attains its global
n 0 0
→ → −
maximum at that point and V∆n,k(x∆n,T∆n) v¯k(x ,T ). From the global maximality
0 0
→
V∆n,k(x,T) V∆n,k(x∆n,T∆n) φ(x,T) φ(x∆n,T∆n).
− ≤ −
Moreover, it can be argued as in [11] using the discrete dynamic programming principle (see [13])
that V∆n,k satisfies
λ
V∆n,k + sup V∆n,k(x∆n ∆ ,T∆n) V∆n,k(x∆n,T∆n)+s∆ rV∆n,k = 0
− T sα∆ − n − n −
s∈[1/k,k] n
(cid:16) (cid:17)
in the viscosity sense. Then
λ
φ + sup φ(x∆n ∆ ,T∆n) φ(x∆n,T∆n)+s∆ rφ+r(φ V∆n,k) 0.
− T sα∆ − n − n − − ≥
s∈[1/k,k] n
(cid:0) (cid:1)
Taking the limit as ∆ 0 we obtain from this equation that
n
→
λ
φ (x ,T )+ sup (s φ (x ,T )) rφ(x ,T ) 0,
− T 0 0 sα − x 0 0 − 0 0 ≥
s∈[1/k,k]
which proves the subsolution property of v¯k. Here, we exchange the limit in ∆ and the supremum
n
with respect to s using Proposition 7.32 in [13] which we can apply thanks to the compactness of
the control space.
It follows again from Theorem 5.4.20 in [6] that the unique solution of (3.9) is given by
T∧τ(X(k),x) λe−rt
(3.10) vk(x,T) = sup dt,
((s 1/k) k)α−1
(st)∈ST Z0 t∨ ∧
where dX(k),x = λ/((s 1/k) k)αdt, X(k),x = x. We will show that vk converges pointwise to
t − t∨ ∧ 0
v:
T∧τ(X(k),x) λe−rt
lim vk(x,T) = sup sup dt
k→∞ k (st)∈ST Z0 ((st∨1/k)∧k)α−1
T∧τ(X(k),x) λe−rt T∧τ(X)λe−rt
= sup sup dt sup dt = v(x,T),
(st)∈ST k Z0 ((st ∨1/k)∧k)α−1 ≥(st)∈ST Z0 sαt−1
wherethe inequality follows from the lower semi-continuity of the map X τ(X); see Lemma 5 in
7→
[18]. On the other hand, since v is a supersolution of (3.9), the comparison result Theorem 5.4.20
in [6] implies that vk v for each k, and as a result
≤
lim vk(x,T) v(x,T).
k→∞ ≤
LIQUIDATION IN LIMIT ORDER BOOKS WITH CONTROLLED INTENSITY 9
Combining the last two inequalities, we obtain the pointwise convergence of vk to v. Pointwise
convergence, on the other hand, implies uniform convergence on compacts due to Dini’s theorem,
since we already know that v is a continuous function of its arguments, and that vk is an increasing
sequence of functions. The latter fact follows from the fact that vk+1 is a supersolution of the PDE
vk satisfies. (cid:3)
Remark 3.2. Results somewhat similar to Theorem 3.1 appeared in [8, 9, 10, 18, 25, 26] which are
on the optimal control of queueing networks. (Among these papers only [18] considered optimal
time-to-empty queueing control problems.) To prove Theorem 3.1 we used a completely different
approach than the above literature, which had relied on probabilistic arguments. Our approach
relies in contrast on the analytical approximation ideas of [7]. We see the prelimit control problem
as the discretization (only in the space variable but not in the time variable) of the “fluid limit”
first order non-linear PDE (3.3) and rely on convergence of the approximation schemes to the
viscosity solutions of such non-linear PDEs. This approach could be fruitful in general in proving
“fluid limit” results associated to controlled queueing networks.
The following is a strengthening of Theorem 3.1 which is an interesting result in its own right.
Proposition 3.2. For any sequence (∆ ) with ∆ = δ2−k, we have V∆k v as k .
k k
↑ → ∞
Proof. We show that for any ∆ > 0, V2∆ V∆. Due to the factoring of T and x in Proposition
≤
2.1, it suffices to establish this result on the infinite horizon where strategies are constant between
trading times.
Fix ε > 0 and let s2∆ be an ε-optimal strategy for V2∆. This policy is defined over x
∈
0,2∆,4∆, . We will recursively construct a policy s∆ over the domain x 0,∆,2∆,
{ ···} ∈ { ···}
that outperforms s2∆. The dynamic programming principle implies that
V2∆(2n∆) E e−rτ1[2s2∆(2n∆)∆+V2∆((2n 2)∆) +ε
≤ −
hΛ(s2∆(2n∆)) i
= 2s2∆(2n∆)∆+V2∆((2n 2)∆) +ε.
Λ(s2∆(2n∆))+r −
(cid:8) (cid:9)
Similarly, given the liquidation strategy s∆ and corresponding trading times τ˜, the resulting ex-
i
pected profits denoted as V˜∆(x), x 0,∆,2∆, satisfy for y = 2n∆,
∈ { ···}
V˜∆(y) = E e−rτ˜1s∆(y)∆+e−rτ˜2 s∆(y ∆)∆+V˜∆(y 2∆)
− −
2Λh(s∆(y)) n 2Λ(s∆(y ∆)) oi
= s∆(y)∆+ − [s∆(y ∆)∆+V˜∆(y 2∆)] .
2Λ(s∆(y))+r 2Λ(s∆(y ∆))+r − −
(cid:26) − (cid:27)
Given s s2∆(2n∆) we prove below that there exists u R , such that
2n +
≡ ∈
Λ(s ) 2Λ(u) 2Λ(u)
2n
(3.11) (2s ∆+V) u∆+ (u∆+V) ,
2n
Λ(s )+r ≤ 2Λ(u)+r 2Λ(u)+r
2n (cid:26) (cid:27)
foranyV 0. ThiswouldestablishV∆(2n∆) V˜∆(2n∆) V2∆(2n∆) εbyinductiononnafter
≥ ≥ ≥ −
setting s∆(2n∆) = s∆((2n 1)∆) = u. Since ε is arbitrary, the statement of the proposition would
−
10 ERHANBAYRAKTARANDMICHAELLUDKOVSKI
then follow. Note that in the above construction, the ∆-investor trading in smaller increments and
twice as much, uses the same spread u to trade when her inventory is 2n∆ or (2n 1)∆.
−
Let z2 := Λ(s2n) and define u implicitly through 2Λ(u) := z < 1. Solving for s and u in
Λ(s2n)+r 2Λ(u)+r 2n
terms of z and using Λ(s) =λs−α we obtain
λ(1 z2) α−1 2λ(1 z) α−1
s = − > − = u.
2n rz2 rz
(cid:18) (cid:19) (cid:18) (cid:19)
Observe that by construction
Λ(s ) 4Λ(u)2
2n
= ,
Λ(s )+r (2Λ(u)+r)2
2n
so that the Laplace transform at r of the duration to execute two trades by the ∆-investor is equal
to the Laplace transform at r of the duration to execute one trade by the 2∆-investor. Using this
fact, (3.11) is equivalent to
2Λ(s ) 2Λ(u) 2Λ(u)
2n
s u+ u
2n
Λ(s )+r ≤ 2Λ(u)+r 2Λ(u)+r
2n (cid:26) (cid:27)
2z2s z(1+z)u.
2n
⇐⇒ ≤
Since α > 1 and the terms on both sides of the above inequality are positive, we may raise both
sides to the α-power and plug-in the expressions for s and u to find
2n
λ(1 z2) 2λ(1 z)
(2z2)αsα zα(1+z)αuα = (2z2)α − zα(1+z)α −
2n− rz2 − rz
= 2λr−1(1 z)(1+z)zα−1[(2z)α−1 (1+z)α−1] < 0,
− −
wherethelastinequality follows sincez < 1andα > 1. Thisshowsthat (3.11)holdsandconcludes
the proof of the proposition. (cid:3)
Next, we show that the strategies also converge thanks to the concavity of all the functions
involved.
Corollary 3.3. Let us denote by s(∆) the pointwise optimizer in (3.2). Then we have that
s(∆)(x,T) s(0)(x,T). (Here, x is fixed and we take the limit over the grids that pass through x.)
→
Proof. On the one hand, as in Remark 2.1 we can think of s(∆) as something proportional to the
left derivative of the linear interpolation of V∆, denoted by Vˆ∆, which is increasing and concave.
On the other hand, s(0) is proportional to the derivative of the concave differentiable function v.
By Theorem 24.5 on page 233 of [27] it however follows that for any x > 0,
D−Vˆ∆(x,T) v (x,T) ε
| x − x | ≤
for small enough ∆, where D− denotes the left derivative operator with respect to x. (cid:3)
x
Remark 3.3. Theorem 3.1 tells us about the asymptotics of c in (2.2):
n
1/α
λ
c n(α−1)/α as n .
n
∼ rα → ∞
(cid:18) (cid:19)