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arXiv:1105.3115v1 [q-fin.TR] 16 May 2011

Dealing with the Inventory Risk

Olivier Gu´eant∗

, Charles-Albert Lehalle∗∗

, Joaquin Fernandez Tapia∗∗∗

2010

Abstract

Market makers have to continuously set bid and ask quotes for the stocks they have under

consideration. Hence they face a complex optimization problem in which their return, based

on the bid-ask spread they quote and the frequency they indeed provide liquidity, is chal-

lenged by the price risk they bear due to their inventory. In this paper, we provide optimal

bid and ask quotes and closed-form approximations are derived using spectral arguments.

Introduction

The optimization of the intra-day trading process on electronic markets was born with the

need to split large trades to make the balance between trading too fast (and possibly degrade

the obtained price via “market impact”) and trading too slow (and suﬀer from a too long

exposure to “market risk”). This “trade scheduling” viewpoint has been mainly formalized

in the late nineties by Bertsimas and Lo [8] and Almgren and Chriss [2]. More sophisticated

approaches involving the use of stochastic and impulse control have been proposed since

then (see for instance [9]). Another branch of proposals goes in the direction of modeling

the eﬀect of the “aggressive” (i.e. liquidity consuming) orders at the ﬁnest level, for instance

via a martingale model of the behavior market depth and of its resilience (see [1]).

From a quantitative viewpoint, market microstructure is a sequence of auction games be-

tween market participants. It implements the balance between supply and demand, forming

an equilibrium traded price to be used as reference for valuation of the listed assets. The

rule of each auction game (ﬁxing auction, continuous auction, etc), are ﬁxed by the ﬁrm

operating each trading venue. Nevertheless, most of all trading mechanisms on electronic

markets rely on market participants sending orders to a “queuing system” where their open

interests are consolidated as “liquidity provision” or form transactions [3]. The eﬃciency of

such a process relies on an adequate timing between buyers and sellers, to avoid too many

non-informative oscillations of the transaction price (for more details and modeling, see for

example [18]).

To take proﬁt of these oscillations, it is possible to provide liquidity to an impatient buyer

(respectively seller) and maintain an inventory until the arrival of the next impatient seller

(respectively buyer). Market participants focused on this kind of liquidity-providing activity

The authors wish to acknowledge the helpful conversations with Pierre-Louis Lions (Coll`ege de France), Jean-

Michel Lasry (Universit´e Paris-Dauphine), Yves Achdou (Universit´e Paris-Diderot), Vincent Millot (Universit´e

Paris-Diderot), Antoine Lemenant (Universit´e Paris-Diderot) and Vincent Fardeau (London School of Economics).

∗UFR de Math´ematiques, Laboratoire Jacques-Louis Lions, Universit´e Paris-Diderot. 175, rue du Chevaleret,

75013 Paris, France. olivier.gueant@ann.jussieu.fr

∗∗Head of Quantitative Research. Cr´edit Agricole Cheuvreux. 9, Quai du Pr´esident Paul Doumer, 92400

Courbevoie, France. clehalle@cheuvreux.com

∗∗∗PhD student, Universit´e Paris 6 - Pierre et Marie Curie. 4 place Jussieu, 75005 Paris, France.

1

are called “market makers”. On one hand they are buying at the bid price and selling at

the ask price they chose, taking proﬁt of this “bid-ask spread”. On the other hand, their

inventory is exposed to price ﬂuctuations mainly driven by the volatility of the market (see

[4, 7, 11, 13, 15, 21]).

The usual “market making problem” comes from the optimality of the quotes (i.e. the

bid and ask prices) that such agents provide to other market participants with respect to

the constraints on their inventory and their utility function as a proxy to their risk (see

[10, 16, 20, 23]).

The recent evolution of market microstructure and the ﬁnancial crisis reshaped the nature

of the interactions of the market participants during electronic auctions, one consequence

being the emergence of “high-frequency market makers” who are said to be part of 70% of

the electronic trades and have a massively passive (i.e. liquidity providing) behavior. A

typical balance between passive and aggressive orders for such market participants is around

80% of passive interactions (see [19]).

Avellaneda and Sto¨ıkov proposed in [5] an innovative framework for “market making in

an order book” and studied it using diﬀerent approximations. In such an approach, the “fair

price”Stis modeled via a Brownian motion with volatility σ, and the arrival of a buy or

sell liquidity-consuming order at a distance δof Stfollows a Poisson process with intensity

Aexp(−k δ). Our paper extends their proposal and provides results in two main directions:

•An explicit solution to the Hamilton-Jacobi-Bellman equation coming from the optimal

market making problem thanks to a non trivial change of variables and the resulting

expressions for the optimal quotes:

Main Result 1 (Theorems 1-2).The optimal quotes can be expressed as:

sb∗(t, q, s) = s−−1

kln vq+1(t)

vq(t)+1

γln 1 + γ

k

sa∗(t, q, s) = s+1

kln vq(t)

vq−1(t)+1

γln 1 + γ

k

where γis the risk aversion of the agent and where vis a family of strictly positive

functions (vq)q∈Zsolution of the linear system of ODEs (S)that follows:

∀q∈Z,˙vq(t) = αq2vq(t)−η(vq−1(t) + vq+1(t))

with vq(T) = 1, and α=k

2γσ2and η=A(1 + γ

k)−(1+ k

γ).

It means that to ﬁnd an exact solution to the generic high-frequency market making

problem, it is enough to solve on the ﬂy the companion ODEs in vq(t) provided by our

change of variables, and to plug the result in the upper equalities to obtain the optimal

quotes with respect to a given inventory and market state.

•Asymptotics of the solution that are numerically attained fast enough in most realistic

cases:

Main Result 2 (Theorem 3 (asymptotics) and the associated approximation equa-

tions).

lim

T→∞ s−sb∗(0, q, s) = δb∗

∞(q)≃1

γln 1 + γ

k+2q+ 1

2sσ2γ

2kA 1 + γ

k1+ k

γ

lim

T→∞ sa∗(0, q, s)−s=δa∗

∞(q)≃1

γln 1 + γ

k−2q−1

2sσ2γ

2kA 1 + γ

k1+ k

γ

2

These results open doors to new directions of research involving the modeling and con-

trol of passive interactions with electronic order books. If some attempts have been made

that did not rely on stochastic control but on forward optimization (see for instance [22]

for a stochastic algorithmic approach for optimal split of passive orders across competing

electronic order books), they should be complemented by backward ones.

This paper goes from the description of the model choices that had to be made (section

1), through the main change of variables (section 2), exposes the asymptotics of the obtained

dynamics (section 3), its comparative statics (section 4), extends the framework to trends

in prices and constraints on the inventory (section 5), ﬁnally discusses the model choices

that had to be made (section 6) and ends with an application to real data (section 7).

Adaptations of our results are already in use at Cr´edit Agricole Cheuvreux to optimize the

brokerage trading ﬂow.

In our framework, we follow Avellaneda and Sto¨ıkov in using a Poisson process model

pegged on a “fair price” diﬀusion (see section 1). As it is discussed in section 6, it is

an arguable choice since it does not capture “resistances” that can be built by huge passive

(i.e. liquidity-providing) orders preventing the market price to cross their prices. Our results

cannot be used as such for large orders, but are perfectly suited for high-frequency market

making as it is currently implemented in the market, using orders of small size (close to the

average trade size, see [19]).

Moreover, to our knowledge, no quantitative model of “implicit market impact” of such

small passive orders has never been proposed in the literature, despite very promising studies

linking updates of quantities in the order books to price changes (see [12]). Its combination

with recent applications of more general point processes to capture the process of arrival

of orders (like Hawkes models, see [6]) should give birth to such implicit market impact

models, specifying dependencies between the trend, the volatility and possible jumps in

the “fair price” semi-martingale process with the parameters of the multi-dimensional point

process of the market maker ﬁll rate. At this stage, the explicit injection of such path-

dependent approach (once they will be proposed in the literature) into our equations are

too complex to be handled, but numerical explorations around our explicit formulas will be

feasible. The outcomes of applications of our results to real data (section 7) show that they

are realistic enough so that no more that small perturbations should be needed.

1 Setup of the model

1.1 Prices and Orders

We consider a market maker operating on a single stock and whose size is small enough to

consider price dynamics exogenous. For the sake of simplicity and since we will basically

only consider short horizon problems we suppose that the mid-price of the stock moves as a

brownian motion:

dSt=σdWt

The market maker under consideration will continuously propose bid and ask quotes denoted

respectively Sb

tand Sa

tand will hence buy and sell stocks according to the rate of arrival of

aggressive orders at the quoted prices.

His inventory q, that is the (signed) quantity of stocks he holds, is given by qt=Nb

t−Na

t

where Nband Naare the jump processes giving the number of stocks the market maker

respectively bought and sold. These jump processes are supposed to be Poisson processes

and to simplify the exposition (although this may be important, see the discussion part)

we consider that jumps are of size 1. Arrival rates obviously depend on the prices Sb

tand

Sa

tquoted by the market maker and we assume, in accordance with most datasets, that

3

intensities λband λaassociated to Nband Naare of the following form1:

λb(sb, s) = Aexp(−k(s−sb)) λa(sa, s) = Aexp(−k(sa−s))

This means that the closer to the mid-price an order is quoted, the faster it will be executed.

As a consequence of his trades, the market maker has an amount of cash whose dynamics

is given by:

dXt=Sa

tdNa

t−Sb

tdNb

t

1.2 The optimization problem

As we said above, the market maker has a time horizon Tand his goal is to optimize the

expected utility of his P&L at time T. In line with [5], we will focus on CARA utility

functions and we suppose that the market maker optimizes:

sup

Sa,Sb

E[−exp (−γ(XT+qTST))]

where γis the absolute risk aversion characterizing the market maker, where XTis the

amount of cash at time Tand where qTSTis the mid-price evaluation of the (signed) re-

maining quantity of stocks in the inventory at time T(liquidation at mid-price2).

2 Resolution

2.1 Hamilton-Jacobi-Bellman equation

The optimization problem set up in the preceding section can be solved using classical

Bellman tools. To this purpose, we introduce a Bellman function udeﬁned as:

u(t, x, q, s) = sup

Sa,Sb

E[−exp (−γ(XT+qTST))|Xt=x, St=s, qt=q]

The Hamilton-Jacobi-Bellman equation associated to the optimization problem is then

given by the following proposition:

Proposition 1 (HJB).The Hamilton-Jacobi-Bellman equation for uis:

(HJB) 0 = ∂tu(t, x, q, s) + 1

2σ2∂2

ssu(t, x, q, s)

+ sup

sb

λb(sb, s)hu(t, x −sb, q + 1, s)−u(t, x, q, s)i

+ sup

saλa(sa, s) [u(t, x +sa, q −1, s)−u(t, x, q, s)]

with the ﬁnal condition:

u(T, x, q, s) = −exp (−γ(x+qs))

This equation is not a simple 4-variable PDE. Rather, because the inventory is discrete,

it is an inﬁnite system of 3-variable PDEs. To solve it, we will use a change of variables that

is diﬀerent from the one used in [5] and transforms the system of PDEs in a system of linear

ODEs.

1Although this form is in accordance with real data, some authors used a linear form for the intensity functions

– see [17] for instance.

2We will discuss other hypotheses below.

4

2.2 Reduction to a system of linear ODEs

In [5], the authors proposed a change of variables to factor out wealth. Here we go further

and propose a rather non-intuitive change of variables that allows to write the problem in a

linear way.

Proposition 2 (A system of linear ODEs).Let’s consider a family of strictly positive func-

tions (vq)q∈Zsolution of the linear system of ODEs (S)that follows:

∀q∈Z,˙vq(t) = αq2vq(t)−η(vq−1(t) + vq+1(t))

with vq(T) = 1, where α=k

2γσ2and η=A(1 + γ

k)−(1+ k

γ).

Then u(t, x, q, s) = −exp(−γ(x+qs))vq(t)−γ

kis solution of (HJB).

Theorem 1 (Well-posedness of the system (S)).There exists a unique solution of (S)in

C∞([0, T ), ℓ2(Z)) and this solution consists in strictly positive functions.

2.3 Optimal quotes characterization

Theorem 2 (Optimal quotes and bid-ask spread).Let’s consider the solution vof the system

(S)as in Theorem 1. Then optimal quotes can be expressed as:

sb∗(t, q, s) = s−−1

kln vq+1(t)

vq(t)+1

γln 1 + γ

k

sa∗(t, q, s) = s+1

kln vq(t)

vq−1(t)+1

γln 1 + γ

k

Moreover, the bid-ask spread quoted by the market maker, that is ψ∗=sa∗(t, q, s)−sb∗(t, q, s),

is given by:

ψ∗(t, q) = −1

kln vq+1(t)vq−1(t)

vq(t)2+2

γln 1 + γ

k

We see that the diﬀerence between each quoted price and the mid-price has two compo-

nents. If we consider the case of the bid quote – the same analysis would be true in the case

of the ask quote –, we need to separate the term −1

kln vq+1(t)

vq(t)from the term 1

γln 1 + γ

k.

If σ= 0, then vq(t) = exp(2η(T−t)) deﬁnes a solution of the system (S). Hence, the

relations s−sb∗=sa∗−s=1

γln 1 + γ

kdeﬁne the optimal quotes in the “no-volatility”

benchmark case3. Consequently, in the expression that deﬁnes the optimal quotes, the sec-

ond term corresponds to the “no-volatility” benchmark while the ﬁrst one takes account of

the inﬂuence of volatility.

3 Examples and asymptotics

To motivate the asymptotic approximation we provide, and before discussing the way to

solve the problem numerically, let us present some graphs to understand the behavior in

time and inventory of both the optimal quotes and the bid-ask spread.

3Smaller quotes would lead to trade more often with less revenue per trade in a way that is not in favor of the

market maker. Symmetrically, larger quotes would lead to more revenue per trade but less trades and the welfare

of the market maker would also be reduced.

5

3.1 Numerical examples

−30 −20 −10 010 20 30

0

100

200

300

400

500

600

0.5

1

1.5

2

2.5

3

Inventory

Time [Sec]

s − sb [Tick]

−30 −20 −10 0 10 20 30

0.5

1

1.5

2

2.5

3

Inventory

s − sb [Tick]

−30 −20 −10 010 20 30 0

100

200

300

400

500

600

0.5

1

1.5

2

2.5

3

Time [Sec]

Inventory

sa − s [Tick]

−30 −20 −10 0 10 20 30

0.5

1

1.5

2

2.5

3

Inventory

sa − s [Tick]

−30 −20 −10 010 20 30 0

100

200

300

400

500

600

3.275

3.28

3.285

3.29

3.295

3.3

3.305

3.31

3.315

3.32

Time [Sec]

Inventory

ψ [Tick]

−30 −20 −10 0 10 20 30

3.3114

3.3116

3.3118

3.312

3.3122

3.3124

3.3126

3.3128

3.313

3.3132

Inventory

ψ [ Tick]

Figure 1: Left: Behavior of the optimal quotes and bid-ask spread with time and inventory.

Right: Behavior of the optimal quotes and bid-ask spread with inventory, at time t= 0. σ=

0.3 Tick ·s−1/2,A= 0.9 s−1,k= 0.3 Tick−1,γ= 0.01 Tick−1,T= 600 s.

3.2 Asymptotics

In [5], the authors propose a heuristic approximation for the bid-ask spread. Namely they

propose to approximate ψ∗(t, q) by γσ2(T−t) + 2

γln 1 + γ

k. However, as suggested by the

graphs exhibited above, the predominant feature is that both the bid-ask spread and the

distance between the quotes and the mid-price are rather constant, except near the time

horizon T(and in numerical examples, a few minutes are enough to be near the asymptotic

values), and certainly not linearly decreasing with time.

6

In fact, we can prove the existence of an asymptotic behavior and provide semi-explicit

expressions for the asymptotic values of the bid-ask spread and the quotes:

Theorem 3 (Asymptotic quotes and bid-ask spread).

∀q∈Z,∃δb∗

∞(q), δa∗

∞(q), ψ∗

∞(q)∈R

lim

T→∞ s−sb∗(0, q, s) = δb∗

∞(q)

lim

T→∞ sa∗(0, q, s)−s=δa∗

∞(q)

lim

T→∞ ψ∗(0, q) = ψ∗

∞(q)

Moreover,

δb∗

∞(q) = 1

γln 1 + γ

k−1

kln f0

q+1

f0

q!δa∗

∞(q) = 1

γln 1 + γ

k+1

kln f0

q

f0

q−1!

and

ψ∗

∞(q) = −1

kln f0

q+1f0

q−1

f0

q2!+2

γln 1 + γ

k

where f0∈ℓ2(Z)is characterized by:

f0∈argmin

kfkℓ2(Z)=1 X

q∈Z

αq2f2

q+ηX

q∈Z

(fq+1 −fq)2

As we have seen in the above numerical examples, only these asymptotic values seem

to be relevant in practice. Consequently, we provide an approximation for f0that happens

to ﬁt the actual ﬁgures. This approximation is based on the continuous counterpart of f0,

namely ˜

f0∈L2(R), a function that veriﬁes:

˜

f0∈argmin

k˜

fkL2(R)=1 Z∞

−∞ αx2˜

f(x)2+η˜

f′(x)2dx

It can be proved4that such a function ˜

f0must be proportional to the probability distri-

bution function of a normal variable with mean 0 and variance qη

α. Hence, we expect f0

q

to behave as exp −1

2qα

ηq2.

This heuristic viewpoint induces an approximation for the optimal quotes and bid-ask

spread if we replace f0

qby exp −1

2qα

ηq2:

δb∗

∞(q)≃1

γln 1 + γ

k+1

2krα

η(2q+ 1)

≃1

γln 1 + γ

k+2q+ 1

2sσ2γ

2kA 1 + γ

k1+ k

γ

δa∗

∞(q)≃1

γln 1 + γ

k−1

2krα

η(2q−1)

≃1

γln 1 + γ

k−2q−1

2sσ2γ

2kA 1 + γ

k1+ k

γ

4To prove this we need to proceed as in the proof of Theorems 1 and 3. In a few words, we introduce the positive,

compact and self-adjoint operator Lcdeﬁned for f∈L2(R) as the unique weak solution vof αx2v−ηv′′ =f

with R∞

−∞ αx2v(x)2+ηv′(x)2dx < +∞.Lccan be diagonalized and largest eigenvalue of Lccan be shown to

be associated to the eigenvector f(x) = exp −1

2qα

ηx2.

7

ψ∗

∞(q)≃2

γln 1 + γ

k+1

krα

η

≃2

γln 1 + γ

k+sσ2γ

2kA 1 + γ

k1+ k

γ

We exhibit below the values of the optimal quotes and the bid-ask spread, both with their

associated approximations. Empirically, these approximations for the quotes are satisfactory

in most cases and are always good for small values of the inventory q. The apparent diﬃculty

to approximate the bid-ask spread comes from the chosen scale (the bid-ask spread being

almost uniform across values of the inventory).

−30 −20 −10 0 10 20 30

0

0.5

1

1.5

2

2.5

3

3.5

Inventory

s − sb [Tick]

−30 −20 −10 0 10 20 30

−6

−4

−2

0

2

4

6

8

10

Inventory

s − sb [Tick]

−30 −20 −10 0 10 20 30

0

0.5

1

1.5

2

2.5

3

3.5

Inventory

sa − s [Tick]

−30 −20 −10 0 10 20 30

−6

−4

−2

0

2

4

6

8

10

Inventory

sa − s [Tick]

−30 −20 −10 0 10 20 30

3.321

3.3215

3.322

3.3225

3.323

3.3235

3.324

3.3245

Inventory

ψ [Tick]

−30 −20 −10 0 10 20 30

3.36

3.38

3.4

3.42

3.44

3.46

3.48

3.5

3.52

3.54

3.56

Inventory

ψ [Tick]

Figure 2: Asymptotic behavior of optimal quotes and the bid-ask spread (bold line). Approxima-

tion (dotted line). Left: σ= 0.4 Tick ·s−1/2,A= 0.9 s−1,k= 0.3 Tick−1,γ= 0.01 Tick−1,

T= 600 s. Right: σ= 1.0 Tick ·s−1/2,A= 0.2 s−1,k= 0.3 Tick−1,γ= 0.01 Tick−1,

T= 600 s.

8

4 Comparative statics

Before starting with the comparative statics, we rewrite the approximations done in the

previous section to be able to have some intuition about the behavior of the optimal quotes

and bid-ask spread with respect to the parameters:

δb∗

∞(q)≃1

γln 1 + γ

k+2q+ 1

2sσ2γ

2kA 1 + γ

k1+ k

γ

δa∗

∞(q)≃1

γln 1 + γ

k−2q−1

2sσ2γ

2kA 1 + γ

k1+ k

γ

ψ∗

∞(q)≃2

γln 1 + γ

k+sσ2γ

2kA 1 + γ

k1+ k

γ

Now, from these approximations, we can “deduce” the behavior of the optimal quotes

and the bid-ask spread with respect to price volatility, trading intensity and risk aversion.

4.1 Dependence on σ2

From the above approximations we expect the dependence of optimal quotes on σ2to be a

function of the inventory. More precisely, we expect:

∂δb∗

∞

∂σ2<0,∂ δa∗

∞

∂σ2>0,if q < 0

∂δb∗

∞

∂σ2>0,∂ δa∗

∞

∂σ2>0,if q= 0

∂δb∗

∞

∂σ2>0,∂ δa∗

∞

∂σ2<0,if q > 0

For the bid-ask spread we expect it to be increasing with respect to σ2:

∂ψ∗

∞

∂σ2>0

The rationale behind this is that a rise in σ2increases the inventory risk. Hence, to

reduce this risk, a market maker that has a long position will try to reduce his exposure and

hence ask less for his stocks (to get rid of some of them) and accept to buy at a cheaper

price (to avoid buying new stocks). Similarly, a market maker with a short position tries to

buy stocks, and hence increases its bid quote, while avoiding short selling new stocks, and

he increases its ask quote to that purpose. Overall, due to the increase in risk, the bid-ask

spread widens as it is well instanced in the case of a market maker with a ﬂat position (this

one wants indeed to earn more per trade to compensate the increase in inventory risk.

These intuitions can be veriﬁed numerically on Figure 3.

4.2 Dependence on A

Because of the above approximations, and in accordance with the form of the system (S),

we expect the dependence on Ato be the exact opposite of the dependence on σ2, namely

∂δb∗

∞

∂A >0,∂δa∗

∞

∂A <0,if q < 0;

∂δb∗

∞

∂A <0,∂δa∗

∞

∂A <0,if q= 0

∂δb∗

∞

∂A <0,∂δa∗

∞

∂A >0,if q > 0

For the same reason, we expect the bid-ask spread to be decreasing with respect to A.

∂ψ∗

∞

∂A <0

9

The rationale behind these expectations is that an increase in Areduces the inventory

risk. An increase in Aindeed increases the frequency of trades and hence reduces the risk

of being stuck with a large inventory (either positive or negative). For this reason, a rise in

Ashould have the same eﬀect as a decrease in σ2.

These intuitions can be veriﬁed numerically on Figure 4.

−30 −20 −10 010 20 30 0

0.1

0.2

0.3

0.4

0.5

0

0.5

1

1.5

2

2.5

3

3.5

σ[Tick/√Sec]

Inventory

s−sb[Tick]

−30 −20 −10 010 20 30 0

0.1

0.2

0.3

0.4

0.5

0

0.5

1

1.5

2

2.5

3

3.5

σ[Tick/√Sec]

Inventory

sa−s[Tick]

−30

−20

−10

0

10

20

30

0

0.1

0.2

0.3

0.4

0.5

3.28

3.29

3.3

3.31

3.32

3.33

3.34

3.35

σ[Tick/√Sec]

Inventory

Spread [Tick]

Figure 3: Asymptotic optimal quotes and bid-ask spread for diﬀerent inventories and diﬀerent

values for the volatility σ.A= 0.9 s−1,k= 0.3 Tick−1,γ= 0.01 Tick−1,T= 600 s.

−30

−20

−10

0

10

20

30

0

0.5

1

1.5

−4

−2

0

2

4

6

8

A[Sec−1]

Inventory

s−sb[Tick]

−30 −20 −10 010 20 30 0

0.5

1

1.5

−4

−2

0

2

4

6

8

A[Sec−1]

Inventory

sa−s[Tick]

10

−30 −20 −10 010 20 30 0

0.5

1

1.5

3.3

3.35

3.4

3.45

3.5

3.55

A[Sec−1]

Inventory

Spread [Tick]

Figure 4: Asymptotic optimal quotes and bid-ask spread for diﬀerent inventories and diﬀerent

values of A.σ= 0.3 Tick ·s−1/2,k= 0.3 Tick−1,γ= 0.01 Tick−1,T= 600 s.

4.3 Dependence on k

From the above approximations we expect δb∗

∞to be decreasing in kfor qgreater than some

negative threshold. Below this threshold we expect it to be increasing. Similarly we expect

δa∗

∞to be decreasing in kfor qsmaller than some positive threshold. Above this threshold

we expect it to be increasing.

Eventually, as far as the bid-ask spread is concerned, the above approximation indicates

that the bid-ask spread should be a decreasing function of k.

∂ψ∗

∞

∂k <0

In fact several eﬀects are in interaction. On one hand, there is a “no-volatility” eﬀect

that is completely orthogonal to any reasoning on the inventory risk: when kincreases,

trades occur closer to the mid price. For this reason, and in absence of inventory risk, the

optimal quotes have to get closer to the mid-price. However, an increase in kalso aﬀects the

inventory risk since it decreases the probability to be executed (for δb, δa>0). Hence, an

increase in kis also, in some aspects, similar to a decrease in A. These two eﬀects explain

the expected behavior.

Numerically, one of two eﬀects dominates for the values of the inventory under consider-

ation:

−30

−20

−10

0

10

20

30

0

0.5

1

1.5

2

−1

0

1

2

3

4

5

Inventory

k[Tick−1]

s−sb[Tick]

−30

−20

−10

0

10

20

30 0

0.5

1

1.5

2

−1

0

1

2

3

4

5

k[Tick−1]

Inventory

sa−s[Tick]

11

−30

−20

−10

0

10

20

30 0

0.5

1

1.5

2

0

1

2

3

4

5

6

7

k[Tick−1]

Inventory

Spread [Tick]

Figure 5: Asymptotic optimal quotes and bid-ask spread for diﬀerent inventories and diﬀerent

values of k.σ= 0.3 Tick ·s−1/2,A= 0.9 s−1,γ= 0.01 Tick−1,T= 600 s.

4.4 Dependence on γ

Using the above approximations, we see that the dependence on γis ambiguous. The

market maker faces two diﬀerent risks that contribute to the inventory risk: (i) trades occur

at random times and (ii) the mid price is stochastic. But if risk aversion increases, the

market maker will mitigate the two risks: (i) he may set his quotes closer to one another to

reduce the randomness in execution (as in the “no-volatility” benchmark) and (ii) he may

enlarge his spread to reduce price risk. The tension between these two roles played by γ

explains the diﬀerent behaviors we may observe, as in the ﬁgures below:

−30

−20

−10

0

10

20

30

0

0.1

0.2

0.3

0.4

0.5

−6

−4

−2

0

2

4

6

8

γ

Inventory

s−sb[Tick]

−30

−20

−10

0

10

20

30

0

0.1

0.2

0.3

0.4

0.5

−3

−2

−1

0

1

2

3

4

γ

Inventory

s−sb[Tick]

−30 −20 −10 010 20 30 0

0.1

0.2

0.3

0.4

0.5

−6

−4

−2

0

2

4

6

8

γ

Inventory

sa−s[Tick]

−30 −20 −10 010 20 30 0

0.1

0.2

0.3

0.4

0.5

−3

−2

−1

0

1

2

3

4

γ

Inventory

sa−s[Tick]

12

−30

−20

−10

0

10

20

30

0

0.1

0.2

0.3

0.4

0.5

2

2.5

3

3.5

γ

Inventory

Spread [Tick]

−30

−20

−10

0

10

20

30

0

0.1

0.2

0.3

0.4

0.5

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

γ

Inventory

Spread [Tick]

Figure 6: Asymptotic optimal quotes and bid-ask spread for diﬀerent inventories and diﬀerent val-

ues for the risk aversion parameter γ. Left: σ= 0.3 Tick ·s−1/2,A= 0.9 s−1,k= 0.3 Tick−1,

T= 600 s. Right: σ= 0.6 Tick ·s−1/2,A= 0.9 s−1,k= 0.9 Tick−1,T= 600 s

5 Diﬀerent settings

In what follows we provide the settings of several variants of the initial model. We will

alternatively consider a model with a trend in prices, a model with a penalization term for

not having cleared one’s inventory and a model with inventory constraints from which all

the ﬁgures have been drawn.

For each model, we enounce the associated results and some speciﬁc points are proved in

the appendix. However, the general proofs are not repeated since they can be derived from

adaptations of the proofs of the initial model.

5.1 Trend in prices

In the preceding setting, we supposed that the mid-price of the stock followed a brownian

motion. However, we can also build a model in presence of a trend:

dSt=µdt +σdWt

In that case we have the following proposition:

Proposition 3 (Resolution with drift).Let’s consider a family of functions (vq)q∈Zsolution

of the linear system of ODEs that follows:

∀q∈Z,˙vq(t) = (αq2−βq)vq(t)−η(vq−1(t) + vq+1(t))

with vq(T) = 1, where α=k

2γσ2,β=kµ and η=A(1 + γ

k)−(1+ k

γ).

Then, optimal quotes can be expressed as:

sb∗(t, q, s) = s−−1

kln vq+1(t)

vq(t)+1

γln 1 + γ

k

sa∗(t, q, s) = s+1

kln vq(t)

vq−1(t)+1

γln 1 + γ

k

and the bid-ask spread quoted by the market maker is :

ψ∗(t, q) = −1

kln vq+1(t)vq−1(t)

vq(t)2+2

γln 1 + γ

k

13

Moreover,

lim

T→∞ s−sb∗(0, q, s) = 1

γln 1 + γ

k−1

kln f0

q+1

f0

q!

lim

T→∞ sa∗(0, q, s)−s=1

γln 1 + γ

k+1

kln f0

q

f0

q−1!

lim

T→∞ ψ∗(0, q) = −1

kln f0

q+1f0

q−1

f0

q2!+2

γln 1 + γ

k

where f0∈ℓ2(Z)is characterized by:

f0∈argmin

kfkℓ2(Z)=1 X

q∈Z

αq−β

2α2

f2

q+ηX

q∈Z

(fq+1 −fq)2

Using the same continuous approximation as in the initial model we ﬁnd the following

approximations for the optimal quotes and the bid-ask spread:

δb∗

∞(q)≃1

γln 1 + γ

k+−µ

γσ2+2q+ 1

2sσ2γ

2kA 1 + γ

k1+ k

γ

δa∗

∞(q)≃1

γln 1 + γ

k+µ

γσ2−2q−1

2sσ2γ

2kA 1 + γ

k1+ k

γ

ψ∗

∞(q)≃2

γln 1 + γ

k+sσ2γ

2kA 1 + γ

k1+ k

γ

5.2 Inventory liquidation below mid price

In the initial model we imposed a terminal condition based on the assumption that the

market maker liquidates his inventory at mid-price at time t=T. This hypothesis is ques-

tionable and we propose to introduce an additional term to model liquidation cost that can

also be interpreted as a penalization term for having a non-zero inventory at time T.

From a mathematical perspective it means that the control problem is now:

sup

Sa,Sb

E[−exp (−γ(XT+qTST−φ(|qT|)))]

where φ(·)≥0 is an increasing function with φ(0) = 0 that represents the penalization term

modeling the incurred cost at the end of the period for not having cleared the inventory5.

The analysis can then be done in the same way as in the initial model and we get the

following result:

Proposition 4 (Resolution with inventory liquidation cost).Let’s consider a family of

functions (vq)q∈Zsolution of the linear system of ODEs that follows:

∀q∈Z,˙vq(t) = αq2vq(t)−η(vq−1(t) + vq+1(t))

5For analytical reasons we supposed that this penalization term does not depend on ST. However, nothing

prevents this penalization term to depend on price through S0for instance, a rather acceptable hypothesis in a

short horizon perspective.

14

with vq(T) = e−kφ(|q|), where α=k

2γσ2and η=A(1 + γ

k)−(1+ k

γ).

Then, optimal quotes can be expressed as:

sb∗(t, q, s) = s−−1

kln vq+1(t)

vq(t)+1

γln 1 + γ

k

sa∗(t, q, s) = s+1

kln vq(t)

vq−1(t)+1

γln 1 + γ

k

and the bid-ask spread quoted by the market maker is :

ψ∗(t, q) = −1

kln vq+1(t)vq−1(t)

vq(t)2+2

γln 1 + γ

k

Moreover, the asymptotic behavior of the optimal quotes and the bid-ask spread does not

depend on the liquidation cost term φ(|q|)and is the same as in the initial model.

5.3 Introduction of inventory constraints

Another possible setting is to consider explicitly in the model that the market maker cannot

have too large an inventory. This is interesting by itself but it also provides numerical

methods to solve the problem and all the graphs presented above have been made using this

model that approximates the general one when the inventory limits are large.

5.3.1 The model

In this model, we introduce limits on the inventory. This means that once the agent holds

a certain amount Qof shares, he does not propose an ask quote until he sells some of his

shares. Symmetrically, once the agent is short of Qshares, he does not short sell anymore

before he buys a share.

In modeling terms, it means that the Hamilton-Jacobi-Bellman equation of the problem

is the following:

∀q∈ {−(Q−1),...,0,...,Q−1},

0 = ∂tu(t, x, q, s) + 1

2σ2∂2

ssu(t, x, q, s)

+ sup

sb

λb(sb, s)hu(t, x −sb, q + 1, s)−u(t, x, q, s)i

+ sup

saλa(sa, s) [u(t, x +sa, q −1, s)−u(t, x, q, s)]

for q=Qwe have:

0 = ∂tu(t, x, Q, s) + 1

2σ2∂2

ssu(t, x, Q, s)

+ sup

saλa(sa, s) [u(t, x +sa, Q −1, s)−u(t, x, Q, s)]

and symmetrically, for q=−Qwe have:

0 = ∂tu(t, x, −Q, s) + 1

2σ2∂2

ssu(t, x, −Q, s)

+ sup

sb

λb(sb, s)hu(t, x −sb,−Q+ 1, s)−u(t, x, −Q, s)i

15

with the ﬁnal condition:

∀q∈ {−Q, . . . , 0,...,Q}, u(T , x, q, s) = −exp (−γ(x+qs))

As in the initial model we can reduce it to a linear system of ODEs. However the linear

system associated to this model will be simpler since it involves 2Q+ 1 equations only.

Proposition 5 (Resolution with inventory limits).Let’s introduce the matrix Mdeﬁned by:

M=

αQ2−η0··· ··· ··· 0

−η α(Q−1)2−η0.......

.

.

0................

.

.

.

.

.................

.

.

.

.

................0

.

.

.......0−η α(Q−1)2−η

0··· ··· ··· 0−η αQ2

where α=k

2γσ2and η=A(1 + γ

k)−(1+ k

γ).

Then, if

v(t) = (v−Q(t), v−Q+1(t),...,v0(t),...,vQ−1(t), vQ(t))′= exp(−M(T−t)) ×(1,...,1)′

the optimal quotes are:

∀q∈ {−Q, . . . , 0,...,Q−1}, sb∗(t, q, s) = s−−1

kln vq+1(t)

vq(t)+1

γln 1 + γ

k

∀q∈ {−(Q−1),...,0,...,Q}, sa∗(t, q, s) = s+1

kln vq(t)

vq−1(t)+1

γln 1 + γ

k

and the bid-ask spread quoted by the market maker is given by:

∀q∈ {−(Q−1),...,0,...,Q−1}, ψ∗(t, q) = −1

kln vq+1(t)vq−1(t)

vq(t)2+2

γln 1 + γ

k

Moreover, the asymptotic quotes and bid-ask spread can be expressed as:

δb∗

∞(q) = 1

γln 1 + γ

k−1

kln f0

q+1

f0

q!δa∗

∞(q) = 1

γln 1 + γ

k+1

kln f0

q

f0

q−1!

and

ψ∗

∞(q) = −1

kln f0

q+1f0

q−1

f0

q2!+2

γln 1 + γ

k

where f0∈R2Q+1 is an eigenvector corresponding to the smallest eigenvalue of M.

5.3.2 Application to numerical resolution

This model based on a slight modiﬁcation of the initial one leads to a system of linear ODEs

whose associated matrix is solely tridiagonal. Hence, for all numerical resolutions we con-

sidered this modiﬁed problem with the inventory limit Qlarge enough and the numerical

resolution simply boiled down to exponentiate a tridiagonal matrix. The rationale behind

this is that the variant of the model under consideration imposes vq(t) = 0 for |q|> Q and

16

for all times. Since the solution of the initial problem is in ℓ2(Z) for t < T , this approxima-

tion will be valid when Qis large as long as tis far enough from the terminal time Tand q

not too close to Q.

Another possible method is to compute an eigenvector associated to the smallest eigen-

value of M. As we noticed before, we expect f0

qto behave as exp −1

2qα

ηq2.

Hence it’s a better idea to look for g0instead of f0where g0

q=f0

qexp 1

2qα

ηq2. To this

purpose, we replace the spectral analysis of Mby the spectral analysis of the tridiagonal

matrix DM D−1where Dis a diagonal matrix whose terms are exp 1

2qα

ηq2q∈{−Q,...,Q}

and g0will be an eigenvector associated to the smallest eigenvalue of DM D−1.

Now, once g0has been calculated, the asymptotic values of the optimal quotes and the

bid-ask spread are:

δb∗

∞(q) = 1

γln 1 + γ

k−1

kln g0

q+1

g0

q!+1

2krα

η(2q+ 1)

δa∗

∞(q) = 1

γln 1 + γ

k+1

kln g0

q

g0

q−1!−1

2krα

η(2q−1)

and

ψ∗

∞(q) = 2

γln 1 + γ

k−1

kln g0

q+1g0

q−1

g0

q2!+1

krα

η

6 Discussion on the model

6.1 Exogenous nature of prices

In our model, the mid-price is modeled by a brownian motion independent of the behavior of

the agent. Since we are modeling a single market maker who operates through limit orders,

it seems natural to consider the price process exogenous in the medium run. However, even

if we neglect the impact of our market maker on the market, the very notion of mid-price

must be clariﬁed. Indeed, one may consider that, although it has little impact on the mar-

ket, the market maker can put an order inside the bid-ask spread of the market order book

and hence change the mid-price. This would be a misunderstanding of the model since the

mid-price is to be considered in the model before any insertion of an order. Hence, the

mid-price in this model must be understood as the mid-price of an order book in which our

market maker’s orders would be removed. More generally, it may be viewed as a generalized

mid-price calculated across trading facilities or any reference price for which the hypothesis

on orders arrival is a good approximation of reality.

6.2 Dependence on price

What may be counterintuitive at ﬁrst sight is that the bid-ask spread or any of the two

spreads between quoted prices and market mid-price seems not to depend on the mid-price

itself. In fact, in our model, the bid-ask spread and the gap between quoted prices and

the market mid-price depends on price, though indirectly, through parameters. Prices are

indeed hidden in the trading intensity λ, and more speciﬁcally into the parameter k. We

indeed considered a trading intensity depending on the distance between the quoted prices

and the mid-price. Thus, kmust depend indirectly on prices to normalize prices diﬀerences.

17

6.3 Constant size of orders

Another apparent issue of the model is that market makers set orders of size 1 at all times.

A ﬁrst remark is that, if this is an issue, it is only limited to the fact that orders are of

constant size since we can consider that the unitary orders stand for orders of constant size

δq or equivalently, though more abstractly, orders of size 1 on a bunch of δq stocks.

If all the orders are of size δq then the stochastic process representing cash is:

d˜

Xt=δq(Sa

tdNa

t−Sb

tdNb

t) = δq ×dXt

where the jump processes model the event of being hit by an aggressive order (of size δq).

Then, if we consider that an order of size δq is a unitary order on a bunch of δq stocks, we

can write ˜qt=δq ×qtand the optimization criterion becomes:

sup

Sa,Sb

Eh−exp −γ(˜

XT+ ˜qTST)i= sup

Sa,Sb

E[−exp (−γδq(XT+qTST))]

Hence, we can solve the problem for orders of size δq using a modiﬁed risk aversion,

namely solving the problem for unitary orders, with γmultiplied by δq.

However, if we can transform the problem with orders of constant size δq into the ini-

tial problem where δq = 1, the parameters must be adjusted in accordance with the fact

that orders are of size δq. As such, it must be noticed that Ahas to be estimated to take

account of the expected proportion of an order of size δq ﬁlled by a single trade and approx-

imations must be made to take account of market making with orders of constant size. In

fact, we can consider that, after each trade that partially ﬁlled the order, the market maker

sends a new order so that the total size of his orders is δq, using a convex combination of the

model recommendations for the price6, since in that case the inventory is not a multiple of δq.

In our view, this issue is important but it should not be considered a problem to describe

qualitative market maker’s behavior and appropriate approximation on Aallows us to believe

that the error made, as far as quantitative modeling results are concerned, is relatively small.

6.4 Constant parameters

Another issue is that the parameters σ,Aand kare constant. While models can be devel-

oped to take account of deterministic or stochastic variations of the parameters, the most

important point is to take account of the links between the diﬀerent parameters. σ,Aand

kshould not indeed be considered independent of one another since, for instance, an in-

crease in Ashould induce an increase in the number of trades and hence an increase in price

volatility.

Some attempts have been made in this direction to model the link between volatility and

trades intensity. Hawkes processes (see [6]) for instance may provide good modeling per-

spectives to link the parameters but this has been left aside for future work.

7 Applications

In spite of the limitations discussed above we used this model to backtest the strategy on

real data. We rapidly discuss the change that have to be made to the model and the way

backtests have been carried out. Then we present the result on the French stock AXA.

6If the optimal quote changed, the remaining part of the order is canceled before a new order is inserted.

Otherwise, the market maker may just insert a new order so that the cumulated size of his orders is δq.

18

7.1 Empirical use

Before using the above model in reality, we need to discuss some features of the model that

need to be adapted before any backtest is possible.

First of all, the model is continuous in both time and space while the real control problem

under scrutiny is intrinsically discrete in space, because of the tick size, and in time, because

orders have a certain priority and changing position too often reduces the actual chance to

be reached by a market order. Hence, the model has to be reinterpreted in a discrete way.

In terms of prices, quotes must not be between two ticks and we decided to ceil or ﬂoor the

optimal quotes with probabilities that depend on the respective proximity to the neighboring

quotes. In terms of time, an order is sent to the market and is not canceled nor modiﬁed

for a given period ∆t(say 20 or 60 seconds), unless a trade occurs and, though perhaps

partially, ﬁlls one of the market maker’s orders. Now, when a trade occurs and changes the

inventory or when an order stayed in the order book for longer than ∆t, then the optimal

quotes on both sides are updated and, if necessary, new orders are inserted.

Now, concerning the parameters, σ,Aand kcan be calibrated easily on trade-by-trade limit

order book data while γhas to be chosen. However, it is well known by practitioners that

Aand khave to depend at least on the actual market bid-ask spread. Since we do not

explicitly take into account the underlying market, there is no market bid-ask spread in

the model. Thus, we simply chose to calibrate kand Aas functions of the market bid-ask

spread, making then an oﬀ-model hypothesis.

Turning to the backtests, they were carried out with trade-by-trade data and we assumed

that our orders were entirely ﬁlled when a trade occurred above (resp. below) the ask (resp.

bid) price quoted by the market maker.

7.2 Results

To present the results, we chose to illustrate the case of the French stock AXA on November

2nd 2010.

We ﬁrst show the evolution of the inventory and we see that this inventory mean-reverts

around 0.

10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00 15:30 16:00

−4

−3

−2

−1

0

1

2

3

4

5

Figure 7: Inventory when the strategy is used on AXA (02/11/2010) from 10:00 to 16:00 with

γ= 0.05 Tick−1

19

Now, to better understand the very nature of the strategy, we focused on a subperiod

of 20 minutes and we plotted the state of the market both with the quotes of the market

maker. Trades occurrences involving the market maker are signalled and we can see on the

following plot the corresponding evolution of the inventory.

12:30 12:32 12:34 12:36 12:38 12:40 12:42 12:44 12:46 12:48 12:50

13.2

13.21

13.22

13.23

13.24

13.25

13.26

13.27

13.28

Figure 8: Details for the quotes and trades when the strategy is used on AXA (02/11/2010) with

γ= 0.05 Tick−1. Thin lines represent the market while bold lines represent the quotes of the

market maker. Dotted lines are associated to the bid side while plain lines are associated to the

ask side. Black points represent trades in which the market maker is involved.

12:30 12:32 12:34 12:36 12:38 12:40 12:42 12:44 12:46 12:48 12:50

−4

−3

−2

−1

0

1

2

3

Figure 9: Details for the inventory when the strategy is used on AXA (02/11/2010) with γ=

0.05 Tick−1

20

Conclusion

In this paper we presented a model for the optimal quotes of a market maker. Starting with

the model by Avellaneda ans Stoikov [5] we introduced a change of variables7that allows

to ﬁnd semi-explicit expressions for the quotes. Then, we exhibited the asymptotic value of

the optimal quotes and argued that the asymptotic values were very good approximations

for the quotes even for rather small times. Closed-form approximations were then obtained

using spectral arguments. The model is ﬁnally backtested on real data and the results are

promising.

7In a companion paper (see [14]) we used a change of variables similar to the one introduced above to solve

the Hamilton-Jacobi-Bellman equation associated to an optimal execution problem with passive orders.

21

Appendix

Proof of Proposition 1:

This is the classical PDE representation of a stochastic control problem with jump pro-

cesses.

Proof of Proposition 2 and Theorem 2:

Let’s consider a solution (vq)qof (S) and introduce u(t, x, q, s) = −exp (−γ(x+qs)) vq(t)−γ

k.

Then:

∂tu+1

2σ2∂2

ssu=−γ

k

˙vq(t)

vq(t)u+γ2σ2

2q2u

Now, concerning the bid quote, we have:

sup

sb

λb(sb, s)hu(t, x −sb, q + 1, s)−u(t, x, q, s)i

= sup

sb

Ae−k(s−sb)u(t, x, q, s)"exp γ(sb−s)vq+1(t)

vq(t)−γ

k

−1#

The ﬁrst order condition of this problem corresponds to a maximum (because uis nega-

tive) and writes:

(k+γ) exp γ(sb∗−s)vq+1(t)

vq(t)−γ

k

=k

Hence:

s−sb∗=−1

kln vq+1(t)

vq(t)+1

γln 1 + γ

k

and

sup

sb

λb(sb, s)hu(t, x −sb, q + 1, s)−u(t, x, q, s)i

=−γ

k+γAexp(−k(s−sb∗))u(t, x, q, s)

=−γA

k+γ1 + γ

k−k

γvq+1(t)

vq(t)u(t, x, q, s)

Similarly,

sa∗−s=1

kln vq(t)

vq−1(t)+1

γln 1 + γ

k

and

sup

saλa(sa, s) [u(t, x +sa, q −1, s)−u(t, x, q, s)]

=−γ

k+γAexp(−k(sa∗−s))u(t, x, q, s)

=−γA

k+γ1 + γ

k−k

γvq−1(t)

vq(t)u(t, x, q, s)

22

Hence, putting the three terms together we get:

∂tu(t, x, q, s) + 1

2σ2∂2

ssu(t, x, q, s)

+ sup

sb

λb(sb, s)hu(t, x −sb, q + 1, s)−u(t, x, q, s)i

+ sup

saλa(sa, s) [u(t, x +sa, q −1, s)−u(t, x, q, s)]

=−γ

k

˙vq(t)

vq(t)u+γ2σ2

2q2u−γA

k+γ1 + γ

kk

γvq+1(t)

vq(t)+vq−1(t)

vq(t)u

=−γ

k

u

vq(t)"˙vq(t)−kγσ2

2q2vq(t) + A1 + γ

k−1+ k

γ(vq+1(t) + vq−1(t))#= 0

Now, noticing that the terminal condition for vqis consistent with the terminal condition

for u, we get that uveriﬁes (HJB).

Proof of Theorem 1 and Theorem 3:

Before starting the very proof, let’s introduce the necessary functional framework.

Let’s introduce H=nu∈ℓ2(Z),Pq∈Zαq2u2

q+η(uq+1 −uq)2<+∞o.His a Hilbert

space equipped with the scalar product:

hv, wiH=X

q∈Z

αq2vqwq+η(vq+1 −vq)(wq+1 −wq),∀v, w ∈H

The ﬁrst preliminary lemma indicates that the ℓ2(Z)-norm can be controlled by the H-

norm.

Lemma 1.

∃C > 0,∀w∈H, kwkℓ2(Z)≤CkwkH

Proof:

∀w∈H, kwk2

ℓ2(Z)=|w0|2+X

q∈Z∗

|wq|2≤ |w0|2+1

αkwk2

H

≤(|w1|+|w0−w1|)2+1

αkwk2

H≤

r1

α+r1

η!2

+1

α

kwk2

H

A second result that is central in the proof of our results is that His compactly embedded

in ℓ2(Z).

Lemma 2. His compactly embedded in ℓ2(Z)

23

Proof:

To prove the result we consider a bounded sequence (sk)k∈Nin HN. For each k∈Nwe

introduce ˆskdeﬁned by:

ˆsk

0=sk

0,∀q∈Z∗,ˆsk

q=qsk

q

For (sk)k∈Nis a bounded sequence in HN, (ˆsk)k∈Nis a bounded sequence in ℓ2(Z)N. Hence,

we can ﬁnd ˆs∞∈ℓ2(Z) so that there exists a subsequence indexed by (kj)j∈Nsuch that

(ˆskj)j∈Nweakly converges toward ˆs∞in ℓ2(Z).

Now, we can deﬁne s∞∈Hby the inverse transformation:

s∞

0= ˆs∞

0,∀q∈Z∗, s∞

q=1

qˆs∞

q

and it is easy to check that (skj)Nconverges in the ℓ2(Z) sense toward s∞. Indeed,

kskj−s∞k2

ℓ2(Z)≤ |skj

0−s∞

0|2+X

q∈Z∗

1

q2|ˆskj

q−ˆs∞

q|2

=|ˆskj

0−ˆs∞

0|2+X

q6=0,|q|≤N

1

q2|ˆskj

q−ˆs∞

q|2+X

|q|>N

1

q2|ˆskj

q−ˆs∞

q|2

The ﬁrst two terms tend to 0 because (ˆskj)Nweakly converges in ℓ2(Z) towards ˆs∞. The

last one can be made smaller than any ǫ > 0 as Nbecomes large because P|q|>N 1

q2tends

to zero as Ntends to inﬁnity and (kˆskj−ˆs∞kℓ2(N))j∈Nis a bounded sequence.

Now, we are going to consider a linear operator Lthat is linked to the system (S). Lis

deﬁned by

L:f∈ℓ2(Z)7→ v∈H⊂ℓ2(Z)

where

∀q∈Z, αq2vq−η(vq+1 −2vq+vq−1) = fq

We need to prove that Lis well-deﬁned and we use the weak formulation of the equation.

Lemma 3. Lis a well-deﬁned linear (continuous) operator.

Moreover ∀f∈ℓ2(Z),∀w∈H, hLf , wiH=hf, wiℓ2(Z)

Proof:

Let’s consider f∈ℓ2(Z).

Because of Lemma 1, w∈H7→ hf, wiℓ2(Z)is continuous. Hence, by Riesz representation

Theorem there exists a unique v∈Hsuch that ∀w∈H, hv, wiH=hf , wiℓ2(Z).

This equation writes

∀w∈H, X

q∈Z

fqwq=X

q∈Z

αq2vqwq+η(vq+1 −vq)(wq+1 −wq)

=αX

q∈Z

q2vqwq+ηX

q∈Z

(vq+1 −vq)wq+1 −ηX

q∈Z

(vq+1 −vq)wq

=αX

q∈Z

q2vqwq+ηX

q∈Z

(vq−vq−1)wq−ηX

q∈Z

(vq+1 −vq)wq

24

This proves ∀q∈Z, αq2vq−η(vq+1 −2vq+vq−1) = fq.

Conversely, if ∀q∈Z, αq2vq−η(vq+1 −2vq+vq−1) = fq, then we have by the same

manipulations as before that:

∀w∈H, hv, wiH=hf, wiℓ2(Z)

Hence Lis well-deﬁned, obviously linear and Lf ∈H.

Now, if we take w=Lf we get

hLf, Lf iH=hf, Lf iℓ2(Z)≤ kfkℓ2(Z)kLfkℓ2(Z)≤Ckfkℓ2(Z)kLf kH

so that kLfkH≤Ckfkℓ2(Z)and Lis hence continuous.

Now, we are able to prove important properties about L.

Lemma 4. Lis a positive, compact, self-adjoint operator

Proof:

As far as the positiveness of the operator is concerned we just need to notice that, by

deﬁnition:

∀f∈ℓ2(Z),hLf, f iℓ2(Z)=kLfk2

H≥0

For compactness, we know that kLf kH≤Ckfkℓ2(Z)and Lemma 2 allows to conclude.

Now, we prove that the operator Lis self-adjoint.

Consider f1, f 2∈ℓ2(Z), we have:

hf1, Lf 2iℓ2(Z)=hLf1, Lf 2iH=hLf 2, Lf 1iH=hf2, Lf 1iℓ2(Z)=hLf1, f 2iℓ2(Z)

Now, we can go to the very proof of Theorem 1 and Theorem 3.

Step 1: Spectral decomposition and building of a solution when the terminal condition

is in ℓ2(Z).

We know that there exists an orthogonal basis (fk)k∈Nof ℓ2(Z) made of eigenvectors of

L(that in fact belongs to Hand we can take for instance ||fk||H= 1) and we denote λk>0

the eigenvalue8associated to fk(we suppose that the eigenvalues are ordered, λ0being the

largest one). We have:

αq2fk

q−η(fk

q+1 −2fk

q+fk

q−1) = 1

λkfk

q

Hence, if we want to solve (S′) that is similar to (S) but with a terminal condition

v(T)∈ℓ2(Z) instead of v(T) = 1 (where 1 stands for the sequence equal to 1 for all indices),

classical argument shows that we can search for a solution of the form v(t) = Pk∈Nµk(t)fk.

80 cannot be an eigenvalue. If indeed λk= 0 then ∀w∈H, hfk, wiℓ2(Z)=hLf k, wiH= 0. But because His

dense in ℓ2(Z), fk= 0.

25

Since ∀q∈Z

˙vq(t) = αq2vq(t)−η(vq−1(t) + vq+1(t))

=αq2vq(t)−η(vq+1(t)−2vq(t) + vq−1(t)) −2ηvq(t)

We must have dµk

dt (t) = ( 1

λk−2η)µk(t) and hence, since λk→0 we can easily deﬁne a

solution of (S′) by:

v(t) = X

k∈N

hv(T), f kiℓ2(Z)exp 2η−1

λk(T−t)fk

and the solution is in fact in C∞([0, T ], ℓ2(Z))

Step 2: Building of a solution when v(T) = 1

The ﬁrst thing to notice is that H⊂ℓ1(Z) (indeed, w∈H⇒(qwq)q∈ℓ2(Z)⇒(wq)q∈

ℓ1(Z) by Cauchy-Schwarz inequality). Hence, the sequence v(T) that equals 1 at each index

is in H′, the dual of H. As a consequence, to build a solution of (S), we can consider a

similar formula:

v(t) = X

k∈N

h1, f kiH′,H exp 2η−1

λk(T−t)fk

Step 3: Uniqueness

Uniqueness follows easily from the ℓ2(Z) analysis. If indeed v(T) = 0 we see that we

must have that

∀k∈Z,hfk,˙v(t)iℓ2(Z)=dhfk, v(t)iℓ2(Z)

dt = ( 1

λk−2η)hfk, v(t)iℓ2(Z)

Hence, ∀k∈Z,hfk, v(T)iℓ2(Z)= 0 =⇒ ∀k∈Z,∀t∈[0, T ],hfk, v(t)iℓ2(Z)= 0 and v= 0.

Step 4: Asymptotics

To prove Theorem 3, we will show that the largest eigenvalue λ0of Lis simple and that

the associated eigenvector f0can be chosen so as to be a strictly positive sequence.

If this is true then we have ∀q∈Z, vq(0) ∼

T→∞ h1, f 0iH′,Hf0

qexp (2η−1

λ0)Tso that the

result is proved with

δb∗

∞(q) = 1

γln 1 + γ

k−1

kln f0

q+1

f0

q!δa∗

∞(q) = 1

γln 1 + γ

k+1

kln f0

q

f0

q−1!

and

ψ∗

∞(q) = −1

kln f0

q+1f0

q−1

f0

q2!+2

γln 1 + γ

k

Hence we just need to prove the following lemma:

Lemma 5. The eigenvalue λ0is simple and any associated eigenvector is of constant sign

(in a strict sense).

26

Proof:

Let’s consider the following characterization of λ0and ofthe associated eigenvectors (this

characterization follows from the spectral decomposition):

1

λ0= inf

f∈ℓ2(Z)

kfk2

H

kfk2

ℓ2(Z)

= inf

kfkℓ2(Z)=1 kfk2

H= inf

kfkℓ2(Z)=1 X

q∈Z

αq2f2

q+η(fq+1 −fq)2

Let’s consider fan eigenvector associated to λ0. We have that:

X

q∈Z

αq2|fq|2+η(|fq+1| − |fq|)2≤X

q∈Z

αq2f2

q+η(fq+1 −fq)2

Hence, since |f|has the same ℓ2(Z)-norm as f, we know that |f|is an eigenvector asso-

ciated to λ0.

Now, since αq2|fq| − η(|fq+1| − 2|fq|+|fq−1|) = 1

λ0|fq|, if |fq|= 0 at some point q, we

have −η(|fq+1|+|fq−1|) = 0 and this induces |fq+1|=|fq−1|= 0, and |f|= 0 by immediate

induction. Since f6= 0, we must have therefore |f|>0.

This proves that there exists a strictly positive eigenvector associated to λ0.

Now, if the eigenvalue λ0were not simple, there would exist an eigenvector gassociated

to λ0with h|f|, giℓ2(Z)= 0. Hence, there would exist both positive and negative values in

the sequence g. But, in that case, since |g|must also be an eigenvector associated to λ0, we

must have equality in the following inequality:

X

q∈Z

αq2|gq|2+η(|gq+1| − |gq|)2≤X

q∈Z

αq2g2

q+η(gq+1 −gq)2

In particular, we must have that ||gq+1| − |gq|| =|gq+1 −gq|,∀q. This implies that ∀q,

either gqand gq+1 are of the same sign or at least one of the two terms is equal to 0. Thus,

since gcannot be of constant sign, there must exist ˆqso that gˆq= 0. But then, because

|g|is also an eigenvector associated to λ0we have by immediate induction, as above, that

g= 0.

This proves that there is no such gand that the eigenvalue is simple.

Step 5: Positiveness

So far, we did not prove that v, the solution of (S) was strictly positive. A rapid way to

prove that point is to use a Feynman-Kac-like representation of v. If (qs)sis a continuous

Markov chain on Zwith intensities ηto jump to immediate neighbors, then we have the

following representation for v:

vq(t) = Eexp −ZT

t

(αq2

s−2η)ds

qt=q

This representation guarantees that v > 0.

This ends the proof of Theorems 1 and 3

Proof of Proposition 3:

If dSt=µdt +σdWtthen the (HJB) equation becomes:

27

(HJB) 0 = ∂tu(t, x, q, s) + µ∂su(t, x, q, s) + 1

2σ2∂2

ssu(t, x, q, s)

+ sup

sb

λb(sb, s)hu(t, x −sb, q + 1, s)−u(t, x, q, s)i

+ sup

sa

λa(sa, s) [u(t, x +sa, q −1, s)−u(t, x, q, s)]

Using the same change of variables as in the proof of Proposition 2, we can consider a

family of strictly positive functions (vq)q∈Zis solution of the linear system of ODEs that

follows:

∀q∈Z,˙vq(t) = (αq2−βq)vq(t)−η(vq−1(t) + vq+1(t))

with vq(T) = 1, α=k

2γσ2,β=kµ and η=A(1 + γ

k)−(1+ k

γ).

Then u(t, x, q, s) = −exp (−γ(x+qs)) vq(t)−γ

kis a solution of (HJB) and the ﬁnal condition

is satisﬁed.

Also, as in the proof of Proposition 2, the optimal quotes are given by:

sb∗(t, q, s) = s−−1

kln vq+1(t)

vq(t)+1

γln 1 + γ

k

sa∗(t, q, s) = s+1

kln vq(t)

vq−1(t)+1

γln 1 + γ

k

and the bid-ask spread follows straightforwardly.

To prove the counterparts of Theorem 1 and Theorem 3, we have to write:

˙vq(t) = (αq2−βq)vq(t)−η(vq−1(t) + vq+1(t))

="αq−β

2α2

vq(t)−η(vq−1(t)−2vq(t) + vq+1(t))#−2η+β2

4αvq(t)

Hence the operator Land the Hilbert space Hare modiﬁed but the results are the same

mutatis mutandis.

Proof of Proposition 4:

In this setting, the Bellman function is deﬁned by:

u(t, x, q, s) = sup

Sa,Sb

E[−exp (−γ(XT+qTST−φ(qT)))|Xt=x, qt=q, St=s]

The (HJB) equation is:

(HJB) 0 = ∂tu(t, x, q, s) + 1

2σ2∂2

ssu(t, x, q, s)

+ sup

sb

λb(sb