This section solves the optimal liquidation problem that is central for this chapter.
The large investor is facing the task to sell Θ0− risky assets but has the possibility to split it into smaller orders to improve according to some performance criterion. Before Section 2.3, we will restrict ourselves to monotone control strategies that do not allow for intermediate buying. The analysis for this more restricted variant of control policies will be shown later in Section 2.3 to carry over to an alternative problem with a wider set of controls, being of finite variation, admitting also intermediate buy orders.
For an initial position of Θ0− shares, the set of admissible trading strategies is Amon(Θ0−) :={
Θ⏐
⏐Θ is decreasing, c`adl`ag, predictable, with Θ0−≥Θt≥0}
. (2.6)
Here, the quantity Θt represents the number of shares held at timet. Any admissible strategy Θ∈ Amon(Θ0−) decomposes into a continuous and a discontinuous part
Θt= Θct+ ∑
0≤s≤t
∆Θs, (2.7)
16
2.2 The problem case for monotone strategies where Θct is continuous (and decreasing) and ∆Θs:= Θs−Θs− ≤0. Aiming for an explicit analytic solution, we consider trading on the infinite time horizon [0,∞) with discounting. Theγ-discounted proceeds from strategy Θ up to time T <∞are
LT(y; Θ) :=−
∫ T 0
e−γtf(Yt)StdΘct− ∑
0≤t≤T
∆Θt̸=0
e−γtSt
∫ ∆Θt
0
f(Yt−+x) dx, (2.8)
where y = Y0− is the initial state of process Y. Clearly, Y0− and Θ determine Y by (2.2).
Remark 2.2.1. The (possibly) infinite sum in (2.8) has finite expectation. Indeed, for any Θ∈ Amon(Θ0−) one has supt≤T|Yt| <∞. Hence, the mean value theorem and properties off imply fort∈[0, T] that
0≤ −
∫ ∆Θt 0
f(Yt−+x) dx≤ −∆Θt sup
x∈(∆Θt,0)
f(Yt−+x)≤ −∆Θt·sup
t≤T
f(Yt). Thus, by finite variation of Θ the infinite sum in (2.8) a.s. converges absolutely. For Θ∈ Amon(Θ0−) the sum is bounded in expectation, becauseY and hence supt≤Tf(Yt) are bounded, and we haveE[supt∈[0,T]St]<∞and 0≤∑
t∈[0,T](−∆Θs)≤Θ0−. Note that the monotone limit L∞(y; Θ) := limT↗∞LT(y; Θ) always exists. We consider the control problem to find the optimal strategy that maximizes the expected (discounted) liquidation proceeds over an open (infinite) time horizon
Θ∈Amaxmon(Θ0−)J(y; Θ) for J(y; Θ) :=E[L∞(y; Θ)], (2.9) with value function v(y, θ) := sup
Θ∈Amon(θ)
J(y; Θ). (2.10)
For this problem maximizing over deterministic strategies turns out to be sufficient (see Remark 2.2.6 below). Since expectations E[exp(−γt)St] =S0exp(−t(γ−µ)), t ≥0, depend onµ, γ only throughδ:=γ−µ, for our optimization problem just the difference δmatters which needs to be positive to havev(y, θ)<∞forθ >0. Thus, regardingγ andµ, only the difference δwill be needed, and it might be interpreted as impatience parameter chosen by the large investor (when choosingγ), specifying her preferences to liquidate earlier rather than later, as a drift rate of the risky asset returnsdS/S, or as a combination thereof. The following conditions onδ, f, h are assumed for the remaining Sections 2.2 and 2.3 of this chapter.
Assumption 2.2.2. The mapt↦→E[e−γtSt],t≥0, is decreasing, i.e. δ:=γ−µ >0.
The price impact function f : R→ (0,∞) satisfies f(0) = 1, f ∈C2 and is strictly increasing such thatλ(y) :=f′(y)/f(y)>0 everywhere.
The resilience function h:R→Rfrom (2.2) isC2withh(0) = 0 andh′>0.
Resilience and market impact satisfy (hλ)′ >0 and (hλ+h′)′>0.
There exist solutions y0 toh(y0)λ(y0) +δ= 0 and y∞ toh(y∞)λ(y∞) +h′(y∞) +δ= 0.
(Uniqueness ofy0andy∞holds by the other conditions.)
Remark 2.2.3 (Interplay of impact and resilience functions). The two assumptions (hλ)′>0 and (hλ+h′)′>0 are technical requirements for our verification of optimality.
2 A deterministic price impact model for optimal liquidation
Already from the shadow limit order book (LOB) perspective (cf. Remark 2.1.1), some sort of condition connecting both resilience speedhand price impactf, thus LOB shape, appears natural. Examples satisfying Assumption 2.2.2 withf(y) :=eλy for constant λ >0 are e.g. linear resilience speed,h(y) =βywith β >0 and any discountingδ >0;
or for instance bounded resilience,h(y) =αarctan(βy), forα, β >0, withβ < λand not too large discounting 0< δ < 12αλπ(a largerδwould give that the trivial strategy to sell everything initially at time 0 is optimal).
The main results Theorems 2.2.4 and 2.3.2 of this chapter solve the optimal liquidation problem for one- respectively two- sided limit order books in infinite time horizon. The proof of Theorem 2.2.4 is developed in [BBF18b, Sect. 4] using smooth pasting and calculus of variations approaches to obtain a candidate solution, together with direct verification of the variational (in-)equalities (2.17)–(2.20).
Theorem 2.2.4. Let the model parametersh,λ,δsatisfy Assumption 2.2.2 andΘ0−≥0 be given. Define y∞< y0<0 as the unique solutions ofh(y∞)λ(y∞) +h′(y∞) +δ= 0 andh(y0)λ(y0) +δ= 0, respectively, and let
τ(y) :=−1 δlog
(f(y) f(y0)
h(y)λ(y) +h′(y) +δ h′(y)
)
, (2.11)
for y∈(y∞, y0] with inverse function τ↦→y(τ) : [0,∞)→(y∞, y0]. Moreover, consider the decreasing functionθ: (y∞, y0]→[0,∞)given by
θ(y) :=
∫ y y0
(
1 + h(z)λ(z)
δ −h(z)h′′(z)
δh′(z) +h(z)(
hλ+h′+δ)′ (z) δ(
hλ+h′+δ) (z)
)
dz (2.12)
and denote its inverse byθ↦→y(θ),θ≥0. For givenΘ0− ≥0 andY0−∈R
∆0:= inf{d∈[0,Θ0−]|Y0−−d≤y(Θ0−−d)} ∧Θ0−, Tw:= inf{t >0|yw(t)>y(Θ0−)},
T :=Tw+τ(
y(Θ0−−∆0)) ,
whereyw ∈C1([0,∞))solves yw′ (t) =−h(yw(t)), for t≥0, withyw(0) =Y0−. Define the processΘopt by
Θoptt := (Θ0−−∆0)1[0,Tw)(t) +θ( y(
T−t))
1[Tw,T)(t) fort≥0. (2.13) Then the strategyΘoptis the unique maximizer to the problem (2.9)of optimal liquidation maxΘ∈Amon(Θ0−)E[L∞(y; Θ)] forΘ0− assets with initial market impact beingY0−=y.
Note that the optimal liquidation strategy does not depend on the particular form of the martingaleM (what has been noted as a robust property in related literature).
SinceT <∞is finite, the open horizon control from Theorem 2.2.4 is clearly optimal for the problem on any finite horizonT′ ≥T; cf. Remark 2.2.10 and Chapter 3 forT′< T. Remark 2.2.5(The optimal sell-only strategy). The strategy Θoptfrom Theorem 2.2.4 acts as follows.
1. IfY0−≥y0+ Θ0−, sell all assets at once: Θ0= 0.
18
2.2 The problem case for monotone strategies
Remark 2.2.6(On deterministic optimal controls). The optimal liquidation strategy is deterministic and the value function turns out to be continuous (even differentiable).
More precisely, the value function is V(y, θ) = solution of Theorem 2.2.4, cf. [BBF18b, Lemma 4.2]. This is shown in [BBF18b, Sect. 4]
while proving Theorem 2.2.4. Here, we show directly why non-deterministic strategies are suboptimal for (2.9) and optimizing over deterministic admissible controls is sufficient.
Yet, finding explicit solutions here still requires to construct candidate solutions and prove optimality, as in the sequel.
If one considers optimization just over strategies that are to be executed until a time T <∞, then the value function will be the same as if we were optimizing over the subset of deterministic strategies. Indeed, by optional projection (see [DM82, VI.57]) we have
E[LT(y; Θ)] =−E[ thatℓis a deterministic functional of Θ, and that the measureP˜ does not depend on Θ.
Thus, optimization for any finite horizon T can be doneω-wise, i.e. for the finite-horizon problem optimizing over the subset of deterministic strategies gives the same value function. Note that this is similar to [Løk12, Prop. 7.2]. Using monotonicity ofLT inT, we have E[L∞(y; Θ)] = supT∈[0,∞)E[LT(y; Θ)], hence the change of measure argument
Moreover, one can check that any deterministic maximizer Θ∗∈ Amon(θ) to (2.15) is also optimal for the original problem (2.9), wherev(y, θ)<∞thanks toδ <0.
2 A deterministic price impact model for optimal liquidation
Remark 2.2.7. In [PSS11], Predoiu, Shaikhet and Shreve consider a similar optimal execution problem, with an additive price impact ψ such that St= St+ψ(Yt) with volume effect process Ytas in (2.2). They study the case of martingale St on a finite time horizon [0, T]. The execution costs, which they seek to minimize in expectation, are equal to the negative liquidation proceeds−LT in our model (for γ, µ= 0) with fixedY0−:= 0. See also Remark 2.2.10 below.
The next result provides sufficient conditions for optimality to the problem (2.9) for each possible initial stateY0− = y ∈R of the impact process, by the martingale optimality principle, for proof see [BBF18b, Prop. 3.6]. In contrast, in the related additive model in [PSS11] the optimal buying strategy for finite time horizon without drift (δ= 0), and impact process starting at zero was characterized using an elegant convexity argument; cf. Remark 2.2.10.
Proposition 2.2.8. LetV : R×[0,∞)→[0,∞) be a continuous function such that Gt(y; Θ) :=Lt(y; Θ)+e−γtSt·V(Yt,Θt), withY =YΘandy=Y0−, is a supermartingale for each Θ ∈ Amon(Θ0−) and additionally G0(y; Θ)≤G0−(y; Θ) :=S0·V(Y0−,Θ0−).
Then
S0·V(y, θ)≥v(y, θ)
with θ = Θ0−. Moreover, if there exists Θ∗ ∈ Amon(Θ0−) such that G(y; Θ∗) is a martingale and it holds G0(y; Θ∗) = G0−(y; Θ∗), then S0 ·V(y, θ) = v(y, θ) and v(y, θ) =J(y; Θ∗).
Remark 2.2.9. The additional condition onG0andG0− can be regarded as extending the (super-)martingale property from time intervals [0, T] to time
”0−“.
In order to make use of Proposition 2.2.8, one applies Itˆo’s formula to G, assuming thatV is smooth enough and using the fact that [S·, e−γ·V(Y·,Θ·)] = 0 becauseS is quasi-left-continuous ande−γ·V(Y·,Θ·) is predictable and of bounded variation, to get
dGt=e−δtV(Yt−,Θt−) dMt
+e−δtMt−
(
(−δV −hVy
)(Yt−,Θt−) dt +(
Vy+Vθ−f)
(Yt−,Θt−) dΘct +
∫ ∆Θt 0
(Vy+Vθ−f)
(Yt−+x,Θt−+x) dx )
(2.16)
with the abbreviating conventions (
−δV −hVy
)(a, b) :=−δV(a, b)−h(a)Vy(a, b) and (Vy+Vθ−f)
(a, b) :=Vy(a, b) +Vθ(a, b)−f(a). The martingale optimality principle now suggests equations for regions where the optimal strategy should sell or wait, in that the dΘ-integrands should be zero when there is selling and the dt-integrand must vanish when only time passes (waiting). We will construct a classical solution to thevariational inequalitymax{−δV−hVy, f−Vy−Vθ}= 0, that is a functionV inC1,1(R×[0,∞),R)
20
2.2 The problem case for monotone strategies
and a strictly decreasingfree boundary functiony(·)∈C2([0,∞),R), such that
−δV −h(y)Vy = 0 inW (2.17)
−δV −h(y)Vy <0 in S (2.18)
Vy+Vθ=f(y) in S (2.19)
Vy+Vθ> f(y) inW (2.20)
V(y,0) = 0 ∀y∈R (2.21)
for wait regionW and sell regionS (cf. Figure 2.2) defined as W :={(y, θ)∈R×[0,∞)|y <y(θ)},
S :={(y, θ)∈R×[0,∞)|y >y(θ)}. (2.22) The optimal liquidation studied here belongs to the class of finite-fuel control problems,
Figure 2.2: The division of the state space, forδ= 0.5,h(y) =y andλ(y)≡1.
which often lead to free boundary problems similar to the one derived above. See [KS86]
for an explicit solution of the finite-fuel monotone follower problem, and [JJZ08] for further examples and an extensive list of references. The proof of Theorem 2.2.4 consists of an explicit construction ofy(θ) and the value functionV as in (2.14) by means of smooth pasting (or alternatively calculus of variations) and direct verification of the variational (in-)equalities (2.17)–(2.21). For details, see [BBF18b, Sect. 4].
Remark 2.2.10 (A first look on the finite time horizon problem). For a given finite horizonT <∞, the execution problem with general order book shape has been solved by [PSS11] for additive price impact and no drift (δ= 0). The problem with multiplicative impact could be transformed to the additive situation using intricate state-dependent order book shapes, cf. [Løk12]. Let us show how a convexity argument as in [PSS11] can be applied also directly to solve the finite horizon case in the multiplicative setup when the driftδis zero, but not forδ̸= 0. We will solve the general caseδ∈Rin Chapter 3 using calculus of variations.
2 A deterministic price impact model for optimal liquidation
By Remark 2.2.6 it suffices to consider deterministic strategies Θ∈ Amon(Θ0−). Let F(y) =∫y
(h−1(x)), the functiong obtains a global minimum at h(y∞) and is decreasing on the left and increasing on the right. So its convex hull ˆg(x) = sup{ℓ(x)|ℓis affine withℓ≤g}exists. With Cδ,T :=∫T and we have equality in (2.24) if and only ifh(Yt) stays constant in the interval where ˆg andg coincide for almost allt∈(0, T). Such impact-fixing strategies are analogous to thetype A strategiesof [PSS11].
The integral in (2.24) can be solved in general only for δ = 0. Now letδ = 0 but keep the other parts of Assumption 2.2.2. Note thaty0= 0 subsequently. In that case Cδ,T =T and∫T In order to identify the convex hull ˆg explicitly forδ= 0, consider the second derivative
g′′(x) = 2f′( convex on an open interval covering [
h(y∞), h(y0)]
. Moreover, by y0 = 0 we have g′(0) =g′(h(y0)) =f(0) = 1 and for everyx >0,g′(x)> f(h−1(x))≥f(0) = 1. Hence, we found a convex function that is dominated byg and therefore also by ˆg:
g(x)≥g(x)ˆ ≥
2.3 The problem case for non-monotone strategies