Sofiene El
email: sofiene.elaoud@ecp.fr

Frédéric Abergel
email: frederic.abergel@ecp.fr

A stochastic control approach to option market making

Keywords: JEL Classification: JEL: C51 Model construction and estimation, JEL: C52 Model evaluation and testing High frequency Market Making, Stochastic optimal control, Utility maximization, Mean-variance framework

This paper presents a model for the market making of options on a liquid stock. The stock price follows a generic stochastic volatility model under the real-world probability measure P. Market participants price options on this stock under a risk-neutral pricing measure Q, and they may misspecify the parameters controlling the dynamics of the volatility process. We first consider that there is a risk-neutral agent who is willing to make markets in an option on the stock, with the aim of maximizing the expected terminal wealth at maturity. Using standard tools in optimal stochastic control, we provide analytical expressions for the optimal bid and ask quotes of the market maker. We then assume that the agent is risk-averse, and perturb the linear utility function by adding a variance term. In this setting, analytic approximations of the optimal bid and ask quotes are obtained. In the case where the stock price process follows a Heston model, Monte Carlo simulations are used to compare the optimal strategy to a "zerointelligence" strategy, and to highlight the effects of some parameters' misspecification on the performance of the strategy.

Introduction

Market makers act on the market by providing liquidity on specific securities. Their role consists in continuously setting bid and ask quotes on instruments of their own choosing. Market makers play a fundamental role, in that they provide liquidity to other market participants, typically to impatient agents who are willing to cross the bid-ask spread. The profit made by a market making strategy comes from the alternation of buy and sell orders, and its targeted gain per trade should depend on the accumulated inventory, a market risk potentially causing significant losses. In the recent literature, several works focus on the problem of market making on stocks through the use of a stochastic optimization method. Inspired by the seminal paper [START_REF] Ho | Optimal dealer pricing under transactions and return uncertainty[END_REF], [START_REF] Avellaneda | High-frequency trading in a limit order book[END_REF] addresses the problem of high frequency market making of a stock. In this model framework, the stock price is modelled as a Wiener process at the intraday timescale, and the market maker seeks the optimal quotes in order to maximize an exponential utility function. More recently, [START_REF] Guéant | Optimal portfolio liquidation with limit orders[END_REF] deals with the problem of optimally using limit orders when liquidating a portfolio. In [START_REF] Guéant | Dealing with the inventory risk: a solution to the market making problem[END_REF], a similar approach is used to deal with the problem of market making with inventory risk. In [START_REF] Fodra | High-frequency market-making with inventory constraints and directional bets[END_REF] [START_REF] Fodra | High-frequency market-making for multidimensional markov processes[END_REF], the approach proposed in [START_REF] Avellaneda | High-frequency trading in a limit order book[END_REF] is generalized to the case where the process modelling the stock price has a drift term and the volatility is stochastic. The issue of model ambiguity has been addressed in [START_REF] Cartea | Robust market making. Available at[END_REF] or [START_REF] Nyström | Market making and portfolio liquidation under uncertainty[END_REF], to account for the fact that the dynamics followed by the stock price and/or the arrival rate of market orders may be misspecified. The conclusion is that, as the uncertainty increases, the market maker has to adapt her quotes in order to reduce her inventory. In this paper, we address the problem of market making in options. Despite its importance, this subject has received little theoretical attention. The only reference we are aware of is [START_REF] Stoikov | Option market making under inventory risk[END_REF], where a mean-variance framework is proposed for optimal market making in options in the case of a complete market. In our framework, the volatility of the stock is stochastic, and our goal is to determine the optimal market making strategy on options in the setting of a generic stochastic volatility model. The paper is organized as follows: In Section 2, we specify the joint dynamics of the spot and its instantaneous volatility under both the historical and pricing measures. Section ?? presents various realistic models for the order flow and market impact function. In Section 4, the market maker is risk neutral and aims to maximize the expectation of her wealth at the maturity date of the option, whereas Section 5 tackles the more challenging case of a risk-adverse market maker in a mean-variance framework. Finally, Section 6 presents a numerical study of the performance of the optimal strategy, and of the effects of model misspecification.

Model setup

Consider a financial market living on a stochastic basis (Ω, F, F, P), where the filtration F = {F t , t ∈ R + } satisfies the usual conditions, and where P denotes the real-world probability measure. Under P, the spot process S has the following dynamics:

dS t S t = µdt + σ(y t )dW (1)
t , (2.1)

dy t = a R (y t )dt + b R (y t )dW (2) t , (2.2)
where W (1) and W (2) are two (P, F) Wiener processes such that d W (1) , W (2) t = ρ R dt. The functions a R , b R satisfy sufficient conditions to ensure the existence of a strong solution to (2.2) satisfying, ∀T > 0:

E P T 0 σ(y t ) 2 dt < +∞, E P T 0 a 2 R (y t ) + b 2 R (y t ) dt < +∞.
Suppose that a european option with maturity T and payoff function h(S T ) is traded in the option market. Let the quantity C P be defined as:

C P (t, S t , y t ) = E P (h(S T )|F t ) . (2.3)
Although C P is not the option price -since P is not the pricing measure -introducing this notion will prove useful.

Market participants price options under the probability measure Q. To make the model as general as possible, we shall allow, as discussed e.g. in [START_REF] Rudra | Arbitrage opportunities in misspecified stochastic volatility models[END_REF], for misspecifications of the parameters characterizing the volatility dynamics.

Let r denote the risk-free rate. Under the pricing measure Q, the process S evolves as follows:

dS t S t = rdt + σ(y t )dW * , (1) t , (2.4
)

dy t = a I (y t )dt + b I (y t )dW * ,(2) t , (2.5)
where W * , (1) and W * , (2) are two (Q, F) Brownian motions such that d W * , (1) , W * ,(2) t = ρ I dt. Again, the equation (2.5) is supposed to have a unique strong solution satisfying ∀T > 0:

E Q T 0 σ(y t ) 2 dt < +∞, E Q T 0 a 2 R (y t ) + b 2 R (y t ) dt < +∞.
The consensus, and therefore, observed option price C Q is then given by C Q (t, S t , y t ) = e -r(T -t) E Q (h(S T )|F t ).

(2.6)

A market maker is an agent who publishes bid and ask quotes around the option mid-price C Q , meets market orders when they match his quotes, while trading continuously in the stock for delta-hedging purposes. At a given time t, this market maker sets an ask price C a t and a bid price C b t such that:

C a t = C Q (t, S t , y t ) + δ + t , C b t = C Q (t, S t , y t ) -δ - t
, where δ - t , δ + t > 0. The way the parameters δ + t , δ - t influence the probability of being hit by a market order and therefore, the inventory, is made precise in Section 3.

Below are some notations which will be used in the rest of the paper:

• q 1,t denotes the option inventory held by the market maker at time t.

• q 2,t is the stock inventory held by the market maker at time t.

• X t is the market value of the cash and stock position held by the market maker at time t.

• C Q (t, S t , y t ) is the mid-price of the option at time t.

• ∆ t = ∆(t, S t , y t ) is the option's delta at time t.

• W t denotes the wealth of the market maker at time t.

In order to ease notations, X will be also called the cash process. Thanks to the infinite liquidity assumption made on the stock, this abuse of notations is legitimate. The arrival of market orders matching the quotes of the market maker are modeled by two independent Poisson processes: N + for buy orders consuming the ask quotes, and N -for sell orders consuming the bid quotes. Therefore, the dynamics of the process q 1 can be described as follows:

dq 1,t = dN - t -dN + t . (2.7)
Due to the continuously adjusted inventory in stock, there holds q 2,t = -q 1,t ∆ t , and therefore:

dq 2,t = -∆ t dq 1,t -q 1,t d∆ t -d q 1 , ∆ t = -∆ t dN - t + ∆ t dN + t -q 1,t d∆ t . (2.8)
The cash process X evolves according to the arrival of market orders as well as continuous trading in the stock, and has the following dynamics:

dX t = (C Q (t, S t , y t ) + δ + t )dN + t -(C Q (t, S t , Y t ) -δ - t )dN - t + q 2,t dS t , (2.9)
so that the wealth of the market maker at time t is given by

W t = X t + q 1,t C Q (t, S t , y t ).
(2.10)

The market maker's strategy is to provide liquidity in order to maximize the expected utility of the terminal wealth. In order to formulate the optimization problem, a crucial step is to choose a model for the intensities of the processes N + and N -.

Intensity of arrivals of market orders

Recent researches have studied the market order flow in order to estimate the probability of execution of limit orders. In [START_REF] Omura | Execution probability of limit orders on the tokyo stock exchange[END_REF], the authors point out that the execution probability of limit orders is affected by the liquidity on the opposite side of the order book, and also the bid-ask spread. [START_REF] Andrew W Lo | Econometric models of limit-order executions[END_REF] uses survival analysis in order to estimate the conditional distribution of limit-order execution times as a function of different variables such as the limit price, the order size, and current market conditions (volatility, bid-ask spread). In particular, the authors proved empirically the seemingly natural fact that, the larger the distance between the mid-quote price and the limit price, the longer the expected time-to-execution.

For the sake of simplicity, we will assume that the probability of execution of a limit order depends only on its distance to the mid-price, in such a way that the instantaneous intensities λ + t and λ - t can be expressed as decreasing functions of δ + t and δ - t respectively. More precisely, the following hypothesis will be enforced throughout the rest of this paper:

∀δ + , δ -≥ 0, λ + (δ + ) = A B + (δ + ) 1 β γ , λ -(δ -) = A B + (δ -) 1 β γ ,
where A, B > 0, γ > 1, β is a positive parameter characterizing the market impact function, and λ + (δ + ) (resp. λ -(δ -)) denotes, with a slight abuse of notation, the intensity of N + (resp. N -) conditional on δ + t = δ + (resp. δ - t = δ -). A heuristic derivation of this functional form is provided in Appendix 8.1.

Optimization problem for a risk-neutral market maker

Generally speaking, the market maker seeks to maximize the expected utility of the terminal wealth at maturity T , and the optimal distances (δ + L, * ,t , δ - L, * ,t ) solve the following problem:

(δ + L, * ,t , δ - L, * ,t ) = ArgSup {δ + t ,δ - t } E P (U (X T + q 1,T h(S T ))|S t = s, y t = y, q 1,t = q 1 , X t = x).
where U is the utility function. In this section, the market maker is risk-neutral, hence the utility function U is linear:

U (T, S T , y T , q 1,T , X T ) = X T + q 1,T h(S T ), (4.1)
where

X T = T 0 (C Q (t, S t , y t ) + δ + t )dN + t - T 0 (C Q (t, S t , Y t ) -δ - t )dN - t + T 0 q 2,t dS t .
In this section, the problem associated with the linear utility function is addressed. The cases corresponding to β = 1 2 (square root market impact) and β = 1 (linear market impact) are studied separately, and the optimal bid and ask quotes are provided analytically in each case.

In order to solve the optimization problem, a stochastic control approach is used. The value function of the market maker can be defined in the following way:

u(t, s, y, q 1 , x) = Sup {(δ + t ,δ - t )∈A} E P (X T + q 1,T h(S T ))|S t = s, y t = y, q 1,t = q 1 , X t = x),
where A = R + × R + denotes the set of admissible values for the controls.

Introducing the differential operators L 1 , L 2 L 1 = µS t ∂ ∂S + 1 2 σ 2 (y t )S 2 t ∂ 2 ∂S 2 + a R (y t ) ∂ ∂y + 1 2 b 2 R (y t ) ∂ 2 ∂y 2 + ρ R b R (y t )σ(y t )S t ∂ 2 ∂S∂y , L 2 = ∂ ∂x q 2,t µS t + 1 2 ∂ 2 ∂x 2 q 2 2,t σ(y t ) 2 S 2 t + ∂ 2 ∂x∂S q 2,t σ 2 (y t )S 2 t + ∂ 2 ∂x∂y q 2,t σ(y t )S t b R (y t )ρ R ,
we let u 0 be the solution of the Hamilton-Jacobi-Bellman (HJB) equation:

(∂ t + L 1 + L 2 )u 0 + sup {(δ + ,δ -)∈A} J + δ + + J -δ - = 0, (4.2)
where the functions J + and J -are defined as follows:

J + (δ + ) = λ + (δ + ) u 0 (t, s, y, q 1 -1, x + (C Q + δ + )) -u 0 (t, s, y, q 1 , x) , J -(δ -) = λ -(δ -) u 0 (t, s, y, q 1 + 1, x -(C Q -δ -)) -u 0 (t, s, y, q 1 , x) ,
and u 0 satisfies the terminal condition u 0 (T, s, y, q 1 , x) = x + q 1 h(s).

A deep result in [START_REF] Pham | Optimisation et contrôle stochastique appliqués à la finance[END_REF] implies that, if u 0 is smooth, finite and has polynomial growth at infinity, then the value function u is equal to u 0 . Hence, our approach will be to solve the HJB equation, and prove that it satisfies the finiteness, smoothness and growth conditions of [START_REF] Pham | Optimisation et contrôle stochastique appliqués à la finance[END_REF].

4.1 The case β = 1 2 (square root market impact)

The intensities of arrivals of market orders are as follows:

λ + (δ + ) = A (B+(δ + ) 2 ) γ and λ -(δ -) = A (B+(δ -) 2 ) γ ,
where A, B ≥ 0 and γ > 1. Moreover, the value function is given by: u 0 (t, s, y, q 1 , x) = x + θ 0 (t, s, y) + q 1 C P (t, s, y)

-µE P t,s,y T t ∆(u, S u , y u )S u du , (4.5)
where θ 0 (t, s, y) = E t,s,y T t J 0 (u, S u , y u )du and:

J 0 (t, s, y) = λ + (δ + L, * ,t ) γ 2 M 2 0 + B(2γ -1) + M 0 (γ -1) 2γ -1 + λ -(δ - L, * ,t ) γ 2 M 2 0 + B(2γ -1) + M 0 (1 -γ) 2γ -1 .
Proof. Since the utility function is linear, the solution is sought under the form: u(t, s, y, q 1 , x) = x + θ 0 (t, s, y) + q 1 θ 1 (t, s, y). (4.6)

Let then f + 0 = J + . Using (4.6), the function f + 0 can be rewritten as:

f + 0 (δ + ) = λ + (δ + )(δ + + M 0 (t, s, y)),
where M 0 (t, s, y) = C Q (t, s, y) -θ 1 (t, s, y).

In order to determine δ + L, * ,t = ArgM ax {x>0} f + 0 (x), the derivative of f + 0 is computed:

(f + 0 ) (δ + ) = λ + (δ + ) B + (δ + ) 2 δ + 2 (1 -2γ) -2γM 0 (t, s, y) δ + + B .
The function (f + 0 ) vanishes at the points x + 1 and x + 2 :

x

+ 1 = -γM 0 -γ 2 M 2 0 + B(2γ -1) 2γ -1 , x + 2 = -γM 0 + γ 2 M 2 0 + B(2γ -1) 2γ -1 .
Since γ ≥ 1 and B > 0, there holds:

γ|M 0 | ≤ γ 2 M 2 0 + B(2γ -1),
which implies that x + 1 < 0 and x + 2 > 0.

Recall here that 1 -2γ < 0. Therefore, f + 0 reaches its maximum on R + at x + 2 . It follows that:

δ + L, * ,t = γ 2 M 2 0 + B(2γ -1) -γM 0 2γ -1
and:

f + 0 (δ + L, * ,t ) = λ + (δ + L, * ,t ) γ 2 M 2 0 + B(2γ -1) + M 0 (γ -1) 2γ -1 . (4.7)
Let now f - 0 = J -. As above, f - 0 can be rewritten as:

f - 0 (δ -) = λ -(δ -)(δ --M 0 (t, s, y)).
and its derivative is given by:

(f - 0 ) (δ -) = λ -(δ -) B + (δ -) 2 (1 -2γ) (δ -) 2 + 2γM 0 δ -+ B .
The function (f - 0 ) vanishes at the two points:

x - 1 = γM 0 -γ 2 M 2 0 + B(2γ -1) 2γ -1 , x - 2 = γM 0 + γ 2 M 2 0 + B(2γ -1) 2γ -1 ,
and, using the same reasoning as before, it can be proved that x - 1 < 0 and x - 2 > 0. Hence, there holds:

δ - L, * ,t = γM 0 + γ 2 M 2 0 + B(2γ -1) 2γ -1 . and f - 0 (δ - L, * ,t ) = λ -(δ - L, * ,t ) γ 2 M 2 0 + B(2γ -1) + M 0 (1 -γ) 2γ -1 . (4.8)
Finally, using (4.7) and (4.8), J 0 (t, s, y) = f + 0 (δ + L, * ,t ) + f - 0 (δ - L, * ,t ) can be written as follows:

J 0 (t, s, y) = λ + (δ + L, * ,t ) γ 2 M 2 0 + B(2γ -1) + M 0 (γ -1) 2γ -1 + λ -(δ - L, * ,t ) γ 2 M 2 0 + B(2γ -1) + M 0 (1 -γ) 2γ -1 .
and it is straightforward to see that J 0 is independent of q 1 . Equation (4.2) becomes:

(∂ t + L 1 + L 2 )u + J 0 (t, s, y) = 0.
This equation is solved by separately cancelling its constant and linear part in terms of q 1 , leading to the following equations:

(0) : (∂ t + L 1 )θ 0 + J 0 (t, s, y) = 0, (1) :

(∂ t + L 1 )θ 1 -∆ t µS t = 0.
The function θ 1 , satisfying the terminal condition θ 1 (T, s, y) = h(s), is given by the the Feynman-Kac formula

θ 1 (t, s, y) = E P t,s,y (h(S T )) -µE P t,s,y ( T t ∆ u S u du).
The quantity M 0 (t, s, y) can now be computed:

M 0 (t, s, y) = C Q (t, s, y) -C P (t, s, y) + µE P t,s,y ( T t ∆ u S u du).
As for θ 0 , given the terminal condition θ 0 (T, s, y) = 0, it is given using Feynman-Kac formula, by:

θ 0 (t, s, y) = E t,s,y T t J 0 (u, S u , y u )du .
As a conclusion, u 0 (t, s, y, q 1 , x) = x + θ 0 (t, s, y) + q 1 C P (t, s, y) -µE P t,s,y ( T t ∆ u S u du) is the solution of the HJB equation (4.2). Finally, proceeding as for Theorem 3.5.2 in [START_REF] Pham | Optimisation et contrôle stochastique appliqués à la finance[END_REF], we prove that the verification theorem holds (see Appendix 8.4 for details), so that u 0 is indeed equal to the value function.

Interpretation of the strategy

The quantity C P (t, s, y) represents the expected payoff of the option under the historical probability measure P. In addition, the quantity:

µE P t,s,y T t ∆(u, S u , y u )S u du ≡ E P t,s,y T t ∆(u, S u , y u )dS u
represents the cost of delta-hedging the option when there is a trend in the dynamics of the underlying. Thus, the indifference price for the option under the probability measure P is C P (t, s, y) -µE P t,s,y T t ∆(u, S u , y u )S u du , and the quantity:

M 0 (t, s, y) = C Q (t, s, y) -C P (t, s, y) -µE P t,s,y T t ∆(u, S u , y u )S u du
is naturally interpreted as the difference between the option price under the risk-neutral probability Q and its indifference price under the historic probability P.

Let the functions f, g be defined by:

f (x) = γx + γ 2 x 2 + B(2γ -1) 2γ -1 , g(x) = γ 2 x 2 + B(2γ -1) -γx 2γ -1 .
By simple differentiation, it can be shown:

∂δ - L, * ,t ∂M 0 ≡ f (M 0 ) = γ 2γ -1 γ 2 M 2 0 + B(2γ -1) + γM 0 γ 2 M 2 0 + B(2γ -1) ≥ 0, ∂δ + L, * ,t ∂M 0 ≡ g (M 0 ) = γ 2γ -1 γM 0 -γ 2 M 2 0 + B(2γ -1) γ 2 M 2 0 + B(2γ -1) ≤ 0. It follows that the bid distance δ - L, * ,t = f (M 0 ) is an increasing function of M 0 while the ask distance δ + L, * ,t = g(M 0
) is a decreasing function of M 0 . Indeed, if M 0 increases, it becomes more rational for the market maker to sell the option. Thus, she decreases both her bid quote (δ - L, * ,t increases) and her ask quote (δ + L, * ,t decreases). In this way, the ask quote is more likely, and the bid quote, less likely to be executed. Following the same reasoning, if M 0 decreases, it becomes more profitable to buy the option. Thus, the market maker increases both her bid quote (δ - L, * ,t decreases) and her ask quote (δ + L, * ,t increases). In the particular case where M 0 = 0, the market price of the option is equal to its indifference price under P.

In this case δ - L, * ,t = δ + L, * ,t = B √ 2γ-1 .
This means that the bid quote C b t and the ask quote C a t are symmetric around the mid-price C Q , and the market maker makes no directional bets.

It is also interesting to study the impact of the parameters on the bid-ask spread. Recall that the bid-ask spread S L, * ,t is simply δ - L, * ,t + δ + L, * ,t , so that S L, * ,t = 2

√ γ 2 M 2 0 +B(2γ-1) 2γ-1
. Differentiating with respect to the variable γ yields:

∂S L, * ,t ∂γ = - 2 (B (2γ -1) + γM 2 0 ) (2γ -1) 2 γ 2 M 2 0 + B (2γ -1) < 0.
This means that the bid-ask spread S L, * ,t is a decreasing function of the parameter γ. Indeed when γ increases, the intensity of arrivals of market orders decreases and the probability of execution of a quote at a distance δ from the mid-price decreases. Consequently, the market maker contracts her bid-ask spread and places her quotes closer to the mid price. Moreover, the spread S L, * ,t increases with the parameter B, and does not depend on A. It can also be noticed that S L, * ,t is an increasing function of |M 0 |, which means that the market maker widens her spread when the gap between the indifference price and the market price becomes significant.

The case β = 1 (linear market impact)

The intensities of arrivals of market orders are now given by:

λ + (δ + ) = A (B+δ + ) γ and λ -(δ -) = A (B+δ -) γ .
The optimization problem is solved similarly to the previous case. The optimal controls (δ + L, * , δ - L, * ) are:

δ + L, * ,t = B -γM 0 (t, s, y) γ -1 + , (4.9)
δ - L, * ,t = B + γM 0 (t, s, y) γ -1 + , (4.10)
and the value function is:

u 0 (t, s, y, q 1 , x) = x + θ 0 (t, s, y) + q 1 C P (t, s, y) -µE P t,s,y T t ∆ u S u du , (4.11)
where

θ 0 (t, s, y) = E t,s,y T t J 0 (u, S u , y u )du
and:

J 0 (t, s, y) =        A(γ-1) γ-1 γ γ (B-M 0 (t,s,y)) γ-1 + A (B-C) γ (-M 0 (t, s, y)) if M 0 (t, s, y) ∈] -∞, -S], A(γ-1) γ-1 γ γ (B-M 0 (t,s,y)) γ-1 - A(1-γ) γ-1 (-γ) γ (B+M 0 (t,s,y)) γ-1 if M 0 (t, s, y) ∈ [-S, S], - A(1-γ) γ-1 (-γ) γ (B+M 0 (t,s,y)) γ-1 + A B γ (M 0 (t, s, y)) if M 0 (t, s, y) ∈ [S, +∞[.
The proof of Proposition 4.2 is essentially identical to that of Proposition 4.1. For the sake of completeness, it is given in Appendix (8.2).

Interpretation of the strategy

Similarly to the approach used in 4.2, the following derivatives are computed:

∂δ + L, * ,t ∂M 0 = - γ γ -1 1 {M 0 ≤S} ≤ 0, ∂δ - L, * ,t ∂M 0 = γ γ -1 1 {M 0 ≥-S} ≥ 0.
The results are qualitatively identical to the case β = 1 2 , that is, the distance δ + L, * ,t of the ask-quote to the mid-price is a decreasing function of the mispricing term M 0 , and the distance δ - L, * ,t of the bid quote to the mid-price is an increasing function of the mispricing term M 0 . Now, the expression for the spread S L, * ,t = δ + L, * ,t + δ - L, * ,t is:

S L, * ,t =      B-γM 0 (t,s,y) γ-1 if M 0 (t, s, y) ≤ -S, 2B γ-1 if M 0 (t, s, y) ∈ [-S, S], B+γM 0 (t,s,y) γ-1 if M 0 (t, s, y) ≥ S, so that ∂S L, * ,t ∂γ =      M 0 (t,s,y)-B (γ-1) 2 if M 0 (t, s, y) ≤ -S, -2B (γ-1) 2 if M 0 (t, s, y) ∈ [-S, S], -M 0 (t,s,y)+B (γ-1) 2 if M 0 (t, s, y) ≥ S.
Again, S L, * ,t is a decreasing function of the parameter γ, as in the case β = 1 2 .

Optimization problem for a risk-averse market maker

In order to risk-manage her strategy, the market maker may want to solve a different optimization problem for the optimal distances δ - t, * , δ + t, * . Typically, a mean-variance approach would seek to maximize the expected wealth at maturity while keeping its variance under control. In this section, the variance of final wealth is going to be approximated in such a way that the HJB equation retain its analytical tractability. Under the assumption that δ + , δ - C Q , we can write:

X T ∼ T t C Q (u, S u , y u )dN + u - T t C Q (u, S u , Y u )dN - u + T 0 q 2,u dS u .
so that the conditional variance at time t of X T is:

V (X T |F t ) ∼ E P T t C 2 Q (u, S u , y u ) λ + u + λ - u du + T t q 2 1,u ∆ 2 u σ 2 (y u )S 2 u du|F t .
Moreover, there holds (using similar notations)

V (q 1,T h(S T )|F t ) ∼ E P T t E P h 2 (S T )|F u λ + u + λ - u du|F t ,
and:

Cov(X T , q 1,T h(S T )|F t ) ∼ -E P T t C Q (u, S u , y u )C P (u, S u , y u ) λ + u + λ - u du|F t .
As a consequence, V ar (X T + q 1,T h(S T )|F t ) can be approximated by

E P T t q 2 1,u V u + Z u du|F t where Z t = C 2 Q + E P h 2 (S T )|F t -2C Q C P λ + L,t + λ - L,t , (5.1)
V t = ∆ 2 t σ 2 (y t )S 2 t . (5.2)
A final approximation consists in replacing the quantities λ + u and λ - u , which depend on δ + u and δ - u , by λ + L,u = λ + (δ + L, * ,u ) and λ - L,u = λ -(δ - L, * ,u ). The stochastic control problem we consider is now introduced. The value function u of the market maker is defined as follows:

u (t, s, y, q 1 , x) = Sup {(δ - t ,δ + t )∈A} E P t,s,y,q 1 ,x (H (t, T, S T , y T , q 1,T , X T )) ,
with H given by:

H (t, T, S T , y T , q 1,T , X T ) = X T + q 1,T h(S T ) - T t q 2 1,u V u + Z u du.
Let u 0 be the solution of the following HJB equation:

(∂ t + L) u 0 + sup {(δ -,δ + )∈A} J (δ -, δ + ) = q 2 1 V + Z , (5.3)
where as in Section 4 L = L 1 + L 2 and J = J -, + J +, with

J +, (δ + ) = λ + (δ + ) u 0 (t, s, y, q 1 -1, x + (C Q + δ + )) -u 0 (t, s, y, q 1 , x) , J -, (δ -) = λ -(δ -) u 0 (t, s, y, q 1 + 1, x -(C Q -δ -)) -u 0 (t, s, y, q 1 , x) .
Also note that in the RHS of Equation ( 5.3), we have denoted by V, Z the function of the state variables defiend by (5.1)(5.2). Again, if u 0 is smooth, finite and has polynomial growth, it coincides with the value function u .

The optimal spreads (δ + * ,t , δ - * ,t ) solve the optimal control problem:

(δ + * ,t , δ - * ,t ) = Argsup {δ + t ,δ - t } E P (H (t, T, S T , y T , q 1,T , X T )|S t = s, y t = y, q 1,t = q 1 , X t = x) ,
and the cases β = 1 2 and β = 1 are now studied.

5.1

The case β = 1 2 (square-root market impact)

Analytic approximation

We state and prove the Proposition 5.1. The optimal controls (δ + * ,t , δ - * ,t ) can be approximated at the first order in the penalization parameter by ( δ- * ,t , δ+ * ,t ) defined as

δ+ * ,t = -γM + + γ 2 (M + ) 2 + B(2γ -1) 2γ -1 , (5.4) δ- * ,t = γM -+ γ 2 (M -) 2 + B(2γ -1) 2γ -1 , (5.5)
where the quantities M + (t, s, y, q 1 ) and M -(t, s, y, q 1 ) are given by M + (t, s, y, q 1 ) = M 0 (t, s, y) + M 1 (t, s, y, q 1 ), M -(t, s, y, q 1 ) = M 0 (t, s, y) + M 2 (t, s, y, q 1 ), and

M 1 (t, s, y, q 1 ) = -θ 3 (t, s, y) + (1 -2q 1 )θ 4 (t, s, y), M 2 (t, s, y, q 1 ) = -θ 3 (t, s, y) -(1 + 2q 1 )θ 4 (t, s, y), with θ 4 (t, s, y) = -E P t,s,y T t V u du , θ 3 (t, s, y) = -2E P t,s,y T t θ 4 (u, s u , y u ) λ + (δ + L, * ,u ) -λ -(δ - L, * ,u ) du .
Moreover, there holds

|δ + * ,t -δ+ * ,t | = O 2 , |δ - * ,t -δ- * ,t | = O 2 .
Proof. Let u 0 be the solution of the HJB equation (5.3). Under the assumption that ∼ 0, a singular perturbation technique can be performed with respect to the parameter :

u 0 (t, s, y, q 1 , x) = x + +∞ k=0
k v k (t, s, y, q 1 ).

(5.6) If = 0, then the utility is obviously linear. This implies that u 0 0 (t, s, y, q 1 , x) = x + v 0 (t, s, y, q 1 ) = u 0 (t, s, y, q 1 , x) where u 0 is the function defined in (4.5). Therefore, it is assumed that v 0 has the following form: v 0 (t, s, y, q 1 ) = θ 0 (t, s, y) + q 1 θ 1 (t, s, y).

Since the utility function contains a constraint on the square of the option inventory q 1 , v 1 is assumed to have the following form: v 1 (t, s, y, q 1 ) = θ 2 (t, s, y) + q 1 θ 3 (t, s, y) + q 2 1 θ 4 (t, s, y).

In order to solve the HJB equation, the jump terms J +, and J -, have to be calculated.

Let f + = J +, . The function f + writes

f + (δ + ) = λ + (δ + )(u(t, s, y, q 1 -1, x + (c + δ + )) -u(t, s, y, q 1 , x)), = λ + (δ + ) δ + + M 0 (t, s, y) + M 1 (t, s, y, q 1 ) + 2 R + (t, s, y, q 1 )
, where:

M 1 (t, s, y, q 1 ) = v 1 (t, s, y, q 1 -1) -v 1 (t, s, y, q 1 ), = -θ 3 (t, s, y) + (1 -2q 1 )θ 4 (t, s, y), and:

R + (t, s, y, q 1 ) = +∞ k=2 k-2 (v k (t, s, y, q 1 -1) -v k (t, s, y, q 1 )) .
Let M + (t, s, y, q 1 ) = M 0 (t, s, y)+ M 1 (t, s, y, q 1 ). In order to determine δ + * ,t = ArgSup {x≥0} f + (x), the derivative (f + ) is computed:

(f + ) (δ + ) = λ + (δ + ) B + (δ + ) 2 δ + 2 (1 -2γ) -2γ M + (t, s, y) + 2 R + (t, s, y, q 1 ) δ + + B .
The function (f + ) has two zeros x + 1 and x + 2 :

x

+ 1 = -γ (M + + 2 R + ) -γ 2 (M + + 2 R + ) 2 + B(2γ -1) 2γ -1 < 0, x + 2 = -γ (M + + 2 R + ) + γ 2 (M + + 2 R + ) 2 + B(2γ -1) 2γ -1 > 0.
Hence, δ + * ,t = x + 2 , and a Taylor expansion yields:

δ + * ,t = -γM + + γ 2 (M + ) 2 + B(2γ -1) 2γ -1 + O 2 or else δ + * ,t = δ + L, * ,t + - γ 2γ -1 M 1 + γ 2 M 0 M 1 γ 2 M 2 0 + B(2γ -1) + O 2 .
It will be useful to write f + (δ + * ,t ) as the sum of f + 0 (δ + L, * ,t ) plus a perturbation term: noticing that f + (x) = f + 0 (x)+ λ + (x)M 1 (t, s, y, q 1 )+O( 2) and using a Taylor expansion, the following obtains:

f + (δ + * ,t ) = f + (δ + L, * ,t ) + (f + ) (δ + L, * ,t )(δ + * ,t -δ + L, * ,t ) + O (δ + * ,t -δ + L, * ,t ) 2 , = f + 0 (δ + L, * ,t ) + λ + (δ + L, * ,t )M 1 + - γ 2γ -1 M 1 + γ 2 M 0 M 1 γ 2 M 2 0 + B(2γ -1) (f + ) (δ + L, * ,t ) + O 2 .
Using that (f + ) (x) = (f + 0 ) (x) + O( ) and (f + 0 ) (δ + L, * ,t ) = 0, there holds:

f + (δ + * ,t ) = f + 0 (δ + L, * ,t ) + λ + (δ + L, * ,t )M 1 + O 2 .
Similarly, let now f -= J -, . Using the form of the value function suggested in (5.6), the function f -can be written as

f -(δ -) = λ -(δ -)(u(t, s, y, q 1 + 1, x -(c -δ -)) -u(t, s, y, q 1 , x)), = λ -(δ -) δ --M 0 (t, s, y) + M 2 (t, s, y, q 1 ) + 2 R -(t, s, y, q 1 )
where

M 2 (t, s, y, q 1 ) = -(v 1 (t, s, y, q 1 + 1) -v 1 (t, s, y, q 1 )) , = -θ 3 (t, s, y) -(1 + 2q 1 )θ 4 (t, s, y), and
R -(t, s, y, q 1 ) = - +∞ k=2 k-2 (v k (t, s, y, q 1 + 1) -v k (t, s, y, q 1 )) .
The quantity M -(t, s, y, q 1 ) = M 0 (t, s, y) + M 2 (t, s, y, q 1 ) is introduced to simplify the notations. Standard computations yield that

(f -) (δ -) = λ -(δ -) B + (δ -) 2 (1 -2γ) (δ -) 2 + 2γ M -+ 2 R -δ -+ B .
Thus, (f -) vanishes at the points

x - 1 = γ (M -+ 2 R -) -γ 2 (M -+ 2 R -) 2 + B(2γ -1) 2γ -1 , x - 2 = γ (M -+ 2 R -) + γ 2 (M -+ 2 R -) 2 + B(2γ -1) 2γ -1 . Since x - 1 < 0, x - 2 > 0 and γ > 1, then δ - * ,t = ArgSup {x≥0} f -(x) = x - 2 .
It follows that:

δ - * ,t = γM -+ γ 2 (M -) 2 + B(2γ -1) 2γ -1 + O 2 .
Using again a Taylor expansion with respect to , the last relation yields:

δ - * ,t = δ - L, * ,t + γ 2γ -1 M 2 + γ 2 M 0 M 2 γ 2 M 2 0 + B(2γ -1) + O 2 .
The quantity f -(δ - * ,t ) can be written as the sum of f - 0 (δ - L, * ,t ) plus a perturbation term. Indeed, since:

f -(x) = f - 0 (x) -λ -(x)M 2 (t, s, y, q 1 ) + O 2 , it follows that f -(δ - * ,t ) = f -(δ - L, * ,t ) + (f -) (δ - L, * ,t )(δ - * ,t -δ - L, * ,t ) + O (δ - * ,t -δ - L, * ,t ) 2 , = f - 0 (δ - L, * ,t ) -λ -(δ - L, * ,t )M 2 + γ 2γ -1 M 2 + γ 2 M 0 M 2 γ 2 M 2 0 + B(2γ -1) (f -) (δ - L, * ,t ) + O 2 .
Recalling that (f -) (x) = (f - 0 ) (x) + O( ) and (f - 0 ) (δ - L, * ,t ) = 0, the previous equation becomes:

f -(δ - * ,t ) = f - 0 (δ - L, * ,t ) -λ -(δ - L, * ,t )M 2 + O 2 .
Using the above approximations for f + (δ + * ,t ) and f -(δ - * ,t ), the quantity

J (δ - * ,t , δ + * ,t ) ≡ f -(δ - * ,t )+ f + (δ + * ,t
) can be written as

J (δ - * ,t , δ + * ,t ) = f + 0 (δ + L, * ,t ) + M 1 λ + (δ + L, * ,t ) + f - 0 (δ - L, * ,t ) -M 2 λ -(δ - L, * ,t ) + O 2 , = J 0 (δ - L, * ,t , δ + L, * ,t ) + M 1 λ + (δ + L, * ,t ) -M 2 λ -(δ - L, * ,t ) + O 2 .
Rearranging the expression for J (δ - * ,t , δ + * ,t ) in powers of yields

J (δ - * ,t , δ + * ,t ) = J 0 (t, s, y) + J 1 (t, s, y, q 1 ) + O 2 ,
where J 1 (t, s, y, q 1 ) = J 1,0 (t, s, y) + q 1 J 1,1 (t, s, y) and

J 1,0 (t, s, y) = λ + (δ + L, * ,t )(-θ 3 + θ 4 ) -λ -(δ - L, * ,t )(-θ 3 -θ 4 ), J 1,1 (t, s, y) = -2θ 4 λ + (δ + L, * ,t ) -λ -(δ - L, * ,t ) .
The HJB equation (5.3) is now rewritten in increasing powers of . The zero th order term leads to the equation:

(∂ t + L 1 + L 2 )(x + θ 0 + q 1 θ 1 ) + J 0 (t, s, y) = 0
with the final conditions

θ 0 (T, s, y) = 0, θ 1 (T, s, y) = h(s).
The functions θ 0 , θ 1 are then easily calculated:

θ 1 (t, s, y) = C P (t, s, y) -µE P t,s,y T t ∆(u, S u , y u )S u du , θ 0 (t, s, y) = E t,s,y T t J 0 (u, S u , y u )du .
The first order term yields the following equation:

(∂ t + L)(θ 2 + q 1 θ 3 + q 2 1 θ 4 ) + J 1 (t, s, y) = q 2 V + Z,
and the terms of this equation are now sorted by their orders in q 1 . The second order term in q 1 provides the following equation for θ 4 :

(∂ t + L 1 )θ 4 (t, s, y) = V.
Using the final condition θ 4 (T, s, y) = 0, it can be deduced that:

θ 4 (t, s, y) = -E P t,s,y T t V u du .
The function θ 3 is calculated using the first order term in q 1 :

(∂ t + L 1 )θ 3 (t, s, y) + J 1,1 (t, s, y) = 0,
and since it also satisfies the final condition θ 3 (T, s, y) = 0, it follows that:

θ 3 (t, s, y) = E P t,s,y T t J 1,1 (u, s u , y u )du .
Finally, the function θ 2 solves:

(∂ t + L 1 )θ 2 (t, s, y) + J 1,0 (t, s, y) = Z
subject to the final condition θ 2 (T, s, y) = 0. Using again the Feynman-Kac formula, the following is obtained:

θ 2 (t, s, y) = E P t,s,y T t (J 1,0 -T ) (u, s u , y u )du .
As before, a verification theorem can be used, once it is proven that u 0 is smooth, finite and satisfies a polynomial growth condition.

Interpretation of the strategy

The effect of the mispricing term M 0 on the optimal bid and ask distances remains the same as in the case without inventory constraints ( = 0), since

∂ δ+ * ,t ∂M 0 = ∂ δ+ * ,t ∂M + ∂M + ∂M 0 = g (M + ) < 0, ∂ δ- * ,t ∂M 0 = ∂ δ- * ,t ∂M - ∂M - ∂M 0 = f (M -) > 0.
The new feature here is the dependence of the distances δ- * ,t and δ+ * ,t on the inventory q 1 . In order to study this dependence, the partial derivatives of δ- * ,t and δ+ * ,t with respect to the variable q 1 are computed:

∂ δ+ * ,t ∂q 1 = -2 θ 4 (t, s, y) γ 2γ -1 g (M + ) < 0, ∂ δ- * ,t ∂q 1 = -2 θ 4 (t, s, y) γ 2γ -1 f (M -) > 0,
It can be deduced from the expressions above that, when the option inventory q 1 increases, the market maker posts more aggressive ask quotes and more conservative bid quotes in order to reduce her inventory. Thus, the market maker, being risk-averse, adjusts her quoting policy in order not to accumulate a large inventory, thereby avoiding being too exposed to market moves.

The bid-ask spread of the market maker is S * ,t = δ- * ,t + δ+ * ,t . Using a Taylor expansion, S * ,t can be approximated as follows:

S * ,t = 2 γ 2 M 2 0 + B(2γ -1) 2γ -1 + γ 2γ -1 (M 2 -M 1 ) + M 0 γ γ 2 M 2 0 + B(2γ -1) (M 1 + M 2 ) = S L, * ,t + γ 2γ -1 -2θ 4 + 2 M 0 γ γ 2 M 2 0 + B(2γ -1) (-θ 3 -2q 1 θ 4 ) .
The term (M 2 -M 1 ) is a positive quantity that increases the bid-ask spread. Indeed, having a constraint on the variance of her final wealth, the risk-averse market maker requires a bigger margin in order to compensate the risk. Therefore, the bid-ask spread is widened. The next term depends on q 1 and M 0 , its effect on S * ,t is not straightforward to interpret. It is easier to see the effects of q 1 directly on δ- * ,t and δ+ * ,t .

5.2

The case β = 1 (linear market impact)

Analytic approximation

Proposition 5.2. The optimal controls (δ + * ,t , δ - * ,t ) of the market maker can be approximated at the first order in by ( δ- * ,t , δ+ * ,t ) defined as

δ+ * ,t = B-γM + (t,s,y,q 1 ) γ-1 if M + (t, s, y) ≤ S, 0 if M + (t, s, y) ≥ S, (5.7)
δ-

* ,t = 0 if M -(t, s, y) ≤ -S, B+γM -(t,s,y,q 1 ) γ-1 if M -(t, s, y) ≥ -S (5.8)
and the approximation error is of order 2 in :

| δ+ * ,t -δ + * ,t | = O 2 , | δ- * ,t -δ - * ,t | = O 2 .
The proof, similar to that of Proposition 5.1, is omitted.

Interpretation of the strategy

The effect of the mispricing term M 0 on the optimal bid and ask distances remains the same as in the case without inventory constraints ( = 0), indeed:

∂ δ+ * ,t ∂M 0 = - γ γ -1 1 {M + ≤S} ≤ 0, ∂ δ- * ,t ∂M 0 = γ γ -1 1 {M -≥-S} ≥ 0.
The proxies δ- * ,t and δ+ * ,t of the optimal distances depend on the inventory q 1 . Their first derivatives with respect to the variable q 1 can be computed explicitly:

∂ δ+ * ,t ∂q 1 = 2 θ 4 (t, s, y) γ γ -1 1 {M + ≤S} ≤ 0, ∂ δ- * ,t ∂q 1 = -2 θ 4 (t, s, y) γ γ -1 1 {M -≥-S} ≥ 0.
The first derivatives highlights the effect of the option inventory q 1 on the distances δ- * ,t and δ+ * ,t . Indeed, if q 1 increases, the market maker lowers her bid and ask quotes, seeking to cut down her option inventory.

Numerical Simulations

In this section, Monte Carlo simulations are performed in order to test the performance of the market making strategies previously characterized. It is supposed in this section that the spot process follows a Heston model under P:

dS t S t = µdt + √ y t dW (1) t dy t = k R (θ R -y t )dt + η R √ y t dW (2) t
where d W (1) , W (2) t = ρ R dt.

Market participant evaluate options using Q, under which the spot process S has the following dynamics:

dS t S t = rdt + √ y t dW * ,(1) t dy t = k I (θ I -y t )dt + η I √ y t dW * ,(2) t
where d W * , (1) , W * ,(2) t = ρ I dt. The functions a R , b R and σ are:

a R (y t ) = k R (θ R -y t ) , b R (y t ) = η R √ y t , σ(y t ) = √ y t ,
The term C Q (t, s, y) is the option price in the Heston model (see [START_REF] Steven L Heston | A closed-form solution for options with stochastic volatility with applications to bond and currency options[END_REF]). The term C P = E P ((S T -K) + |F t ) can be computed explicitly as explained in Appendix (??).

The first part of the numerical study is devoted to the comparison between the optimal strategy and a zero-intelligence strategy, in the case of a linear utility function. In the second part, the effect of the misspecification of the parameters is studied.

The numerical simulations are specified as follows: the traded option has a maturity equal to 3 Months (T = 0.25) and a strike equal to 100 (K = 100). We simulate 1000 Monte Carlo paths of the spot and instantaneous variance (S t , v t ) {0≤t≤T } starting from S 0 = 100 and v 0 = 0.04. It is assumed that there are 6 trading hours per day, and that the market maker refreshes the quotes every 5 minutes. This means that there are 12 × 6 = 72 points per day. Since there are approximately 64 business days in a 3 months period, this amounts to 64 × 72 = 4608 points per simulated path. At each point, the quantities C Q (t, s, y) and C P (t, s, y) are computed using a Fast Fourier Transform method. This simulation task is numerically consuming and was performed using the computing cluster1 at École Centrale Paris.

For each simulated path, the optimal market making strategy is used from the inception date (t = 0) until the maturity date of the option (t = T ).

Comparison with a zero-intelligence agent

The implied volatility Σ t (K, T ) of the call option is defined as usual by C Q (t, S t , y t ) = P BS (t, S t , Σ t (K, T )), where P BS denotes the Black-Scholes call price formula. In addition, let ϑ BS = ∂P BS (t,St,Σt(K,T )) ∂Σt be the Black-Scholes vega of the option. In this subsection, we suppose that there is an agent who places equidistant bid and ask quotes denoted by C b ZI,t and C a ZI,t respectively:

C a ZI,t = C Q (t, S t , y t ) + δ ZI,t , C b ZI,t = C Q (t, S t , y t ) -δ ZI,t ,
where δ ZI,t = 0.005 × ϑ BS . In other words, the "zero-intelligence" agent attempts to earn 0.005ϑ BS for each trade.

The assumption here is that (k R , θ R , η R , ρ R ) = (k I , θ I , η I , ρ I ) = (4, 0.04, 0.5, -0.4) (no misspecification). Besides, the historical drift µ and risk-free rate r are both zero. The results of the numerical simulations are summarized below: The numerical simulations show that the strategy using the optimal quotes performs better than the zero-intelligence strategy.

Effect of the misspecification of parameters

In this section, the effect of parameter misspecification in the case of a linear utility function is addressed.

Misspecification of the parameter ρ

Monte Carlo simulations are performed using the parameters: (µ, k R , θ R , η R , ρ R ) = (0, 4, 0.04, 0.5, -0.4) and (r, k I , θ I , η I , ρ I ) = (0, 4, 0.04, 0.5, -0.9). The statistics of 1000 simulations are given below: In order to interpret these results, it is convenient to use [START_REF] Forde | Small-time asymptotics for implied volatility under the heston model[END_REF], where an approximation of the implied volatility for the Heston model is derived:

Σ(K, T ) = √ y 0 1 + 1 4 ρη y 0 log( K S 0 ) + 1 24 - 5 48 ρ 2 η 2 y 2 0 log( K S 0 ) 2 + O log( K S 0 ) 3 . (6.1)
The approximation (6.1), valid for small maturities and log-moneyness, points out the effect of the parameter ρ on the option price when T ∼ 0. Indeed, at the first order in log( K St ), the implied volatility Σ t (K, T ) is an increasing function of ρ if K > S t and a decreasing function if K < S t . Consequently, in each simulation, the evolution of the moneyness log( K St ) of the option determines the quantity C P (t, s, y) -C Q (t, s, y) and influences the aggressiveness of the quotes on either the bid or the ask side. It can be noticed that the distribution of the option inventory q 1 (T ) at the maturity date T is more spread out in the case where ρ R = ρ I . The statistics show that the final wealth in the case where ρ R = -0.4 and ρ I = -0.9 is on average higher than in the case where ρ R = ρ I = -0.4. Indeed, the quoting policy of the market maker is adapted in order to benefit from the misspecification of the parameter ρ.

Cases

Misspecification of the parameter θ

Monte Carlo simulations are performed using the following parameters: (µ, k R , θ R , η R , ρ R ) = (0, 4, 0.04, 0.5, -0.4) and (r, k I , θ I , η I , ρ I ) = (0, 4, 0.0625, 0.5, -0.4). The final wealth in the case where the parameter θ does not have the same values under P and Q (θ R = 0.04, θ I = 0.0625) is in average higher than in the default case (θ R = θ I = 0.04). The numerical simulations support the results of the theoretical study and show experimentally how the optimal strategy enables the market maker to take advantage from the parameter misspecification.

Misspecification of the parameter η

In this subsection, Monte Carlo simulations are performed using the following parameters: (µ, k R , θ R , η R , ρ R ) = (0, 4, 0.04, 0.5, -0.4) and (r, k I , θ I , η I , ρ I ) = (0, 4, 0.04, 0.7, -0.4). The statistics of 1000 simulations are given below: It can be seen through the approximation (6.1) that the effect of the parameter η on the implied volatility Σ(K, T ) of the option depends on its moneyness. Thus, at the first order in log( K St ), if ρ < 0, an increase of the parameter η increases the option price if K < S t , and decreases it if K > S t . Therefore, the aggressiveness of the quotes of the market maker on either the bid or the ask side of the order book depends on the path followed by S in each simulation. The interesting point here is that the final wealth W T in the case η R = η I has a lower average and a higher standard deviation than in the case η R = η I (for both β = 1 2 and β = 1). This observation may seem, at first glance, counter-intuitive, since the misspecification of a parameter should give the opportunity to benefit from price inefficiency. However, there is a simple explanation to this result: increasing the parameter η under the pricing measure also increases the volatility risk of the option. Since this risk is not hedged out, the variance of the final wealth W T increases, thereby curbing the profit of the market maker.

Cases

Conclusion

In this paper, we have proposed a framework for the study of option market making. Using a stochastic control approach, we have derived analytic expressions for optimal bid and ask quotes in the case of a risk-neutral market maker. The risk-adverse case is also considered, in a mean-variance framework, and we have used a singular perturbation technique in order to provide approximations for the optimal quotes. Finally, using Monte Carlo simulations, we have provided some numerical evidence for our theoretical findings, and studied the impact of parameters misspecification.

Appendices

Intensity of arrivals of market orders

In order to determine the form of the functions λ + and λ -, we need to specify the distribution function of the size of market orders and also the market impact following the execution of a market order.

Let f V denote the density distribution of the size of market orders in absolute value of their cash amount. Several studies proved that this density decays as a power law (see [START_REF] Avellaneda | High-frequency trading in a limit order book[END_REF]). We will suppose here that f V can be well fitted by a power law density:

∀x > 0, f V (x) = γL γ (L + x) γ+1 . (8.1)
From a practical point of view, there is a strictly positive lower bound for x which corresponds to the option price x M in . Nevertheless, it is supposed here that the density f V is positive for 0 ≤ x ≤ x M in .

On the other hand, market impact has been studied by different authors in the econophysics literature, and it is widely accepted that the change in price ∆P following a market order of size V can be written as follows:

∆P = KV β . (8.2)
There are two values of β which are supported by different researchers: β = 1 which corresponds to a linear market impact and β = 1 2 which corresponds to a square root market impact.

The probability that a bid quote (respectively ask quote) placed at a distance δ -(respectively δ + ) from the mid price gets executed is equal to the probability that a sell market order (respectively buy market order) triggers a market impact which is higher or equal to δ - (respectively δ + ). Suppose as in [START_REF] Avellaneda | High-frequency trading in a limit order book[END_REF] that the arrival rate of market orders is constant and equal to F . Then, for δ ≥ 0:

λ(δ) = F × P (∆P ≥ δ), = F × P (V β ≥ δ K ), = F × P V ≥ δ K 1 β , = F +∞ ( δ K ) 1 β f V (x)dx, = F L γ (L + δ K 1 β ) γ .
If M 0 (t, s, y) ≤ -B γ , then ∀ δ -≥ 0, (f - 0 ) (δ -) < 0 and therefore δ - L, * ,t = 0. On the other hand, if M 0 (t, s, y) ≥ -B γ , it can be proved that δ - L, * ,t = x -= B+γM 0 (t,s,y) γ-1 . Consequently, δ - L, * ,t writes:

δ - L, * ,t = B + γM 0 (t, s, y) γ -1 + ,
and:

f - 0 (δ - L, * ,t ) = - A(1-γ) γ-1 (-γ) γ (B+M 0 (t,s,y)) γ-1 if M 0 (t, s, y) ≥ -B γ -A B γ M 0 (t, s, y) if M 0 (t, s, y) ≤ -B γ (8.5)
In order to simplify the notations, the following quantities are introduced: S = B γ and J 0 (t, s, y) = f + 0 (δ + L, * ,t ) + f - 0 (δ - L, * ,t ). The equation (4.2) becomes:

H : (∂ t + L 1 + L 2 ) (x + θ 0 (t, s, y) + q 1 θ 1 (t, s, y)) + J 0 (t, s, y) = 0
The terms of the HJB equation are sorted by powers of q 1 : (0) : (∂ t + L 1 )θ 0 + J 0 (t, s, y) = 0, (1) : (∂ t + L 1 )θ 1 -µ∆s = 0, Using the final conditions and applying the Feynman-Kac formula yields: Based on these results, the quantity M 0 (t, s, y) can be deduced: M 0 (t, s, y) = C Q (t, s, y) -C P (t, s, y) + µE P t,s,y T t ∆(u, S u , y u )S u du , and u 0 (t, s, y, q 1 , x) = x + θ 0 (t, s, y) + q 1 C P (t, s, y) -µE P t,s,y T t ∆(u, S u , y u )S u du is the solution of the HJB equation (4.2).

The function u 0 coincides with the value function if it is smooth, finite and has a quadratic growth. In order to prove the quadratic growth of u 0 , we start by studying the function θ 0 :

• If M 0 (t, s, y) ∈ [-S, S], then:

( γ + 1 γ B) -(γ-1) ≤ (B -M 0 (t, s, y)) -(γ-1) ≤ ( γ -1 γ B) -(γ-1) , ( γ + 1 γ B) -(γ-1) ≤ (B + M 0 (t, s, y)) -(γ-1) ≤ ( γ -1 γ B) -(γ-1) ,
Then ∃M > 0 such that |J 0 (t, s, y)| ≤ M .

• If M 0 (t, s, y) ≥ S, then:

E P t,s,y (|J 0 (u, S u , y u )|) ≤ A(γ -1) γ-1 γ γ (B + S) γ-1 + A B γ E P t,s,y (|M 0 (u, S u , y u )|) ,
• If M 0 (t, s, y) ≤ -S then:

E P t,s,y (|J 0 (u, S u , y u )|) ≤ A(γ -1) γ-1 γ γ (B -S) γ-1 + A B γ E (|M 0 (u, S u , y u )|) ,
We assume here that the traded option is a call or a put, so that there exists

C
, x)| ≤ C 2 1 + x 2 + s 2 + y 2 + q 2
1 , which implies that u 0 is finite and has a quadratic growth. Recall here that θ 0 is the solution of the equation (∂ t + L 1 )θ 0 + J 0 (t, s, y) = 0 with the final condition θ 0 (T, s, y) = 0. Since the function J 0 is continuous (J 0 is at least C 0,0,0 ), then θ 0 is smooth. Consequently, the function u 0 is also smooth and it coincides with the value function.

Solution of the HJB equation with linear impact in the meanvariance framework

Let u be the solution of the HJB equation (5.3). Under the assumption that ∼ 0, a singular perturbation technique is performed with respect to the parameter :

u (t, s, y, q 1 , x) = x + +∞ k=0 k v k (t, s, y, q 1 ),
Given the form of the utility function, the following Ansatz on v 0 and v 1 is made: v 0 (t, s, y, q 1 ) = θ 0 (t, s, y) + q 1 θ 1 (t, s, y), v 1 (t, s, y, q 1 ) = θ 2 (t, s, y) + q 1 θ 3 (t, s, y) + q 2 1 θ 4 (t, s, y), In order to solve the HJB equation, the jump terms J +, and J -, have to be calculated. Let f + = J +, , the function f + writes:

f + (δ + ) = λ + (δ + )(u(t, s, y, q 1 -1, x + (c + δ + )) -u(t, s, y, q 1 , x)), = λ + (δ + ) δ + + M 0 (t, s, y) + M 1 (t, s, y, q 1 ) + 2 R + (t, s, y, q 1 ) ,
Let M + (t, s, y, q 1 ) = M 0 (t, s, y, q 1 ) + M 1 (t, s, y, q 1 ). By differentiating f + , it can be shown that:

(f + ) (δ + ) = λ + (δ + ) B + δ + δ + (1 -γ) + B -γM + (t, s, y, q 1 ) -γ 2 R + (t, s, y, q 1 ) ,
In order to determine δ + * ,t = ArgM ax {x≥0} f + (x), two cases should be distinguished:

• M + (t, s, y, q 1 ) + 2 R + (t, s, y, q 1 ) ≥ S: in this case, ∀ δ + ≥ 0, (f + ) (δ + ) ≤ 0, then the function f + is decreasing on the interval [0, +∞[ and δ + * ,t = 0.

• M + (t, s, y, q 1 ) + 2 R + (t, s, y, q 1 ) ≤ S: the function (f + ) changes its sign on [0, +∞[ and vanishes at x + :

x + = B -γ (M + (t, s, y, q 1 ) + 2 R + (t, s, y, q 1 )) γ -1 .

Since γ > 1 then δ + * ,t = x + .

In conclusion, δ + * ,t writes:

δ + * ,t = B -γ (M + (t, s, y, q 1 ) + 2 R + (t, s, y, q 1 )) γ -1 + .
Using a Taylor expansion, it follows:

δ + * ,t = δ + L, * ,t - γ γ -1 M 1 (t, s, y, q 1 )1 {M + + 2 R + ≤S} + O 2 .
In order to solve the HJB equation, it is useful to write f + (δ + * ,t ) as the sum of f + 0 (δ + L, * ,t ) plus a correction that depends on the parameter . Indeed, we have:

f + (x) = f + 0 (x) + λ + (x)M 1 (t, s, y, q 1 ) + O 2 ,
and therefore:

f + (δ + * ,t ) = f + (δ + L, * ,t ) + (f + ) (δ + L, * ,t )(δ + * ,t -δ + L, * ,t ) + O (δ + * ,t -δ + L, * ,t ) 2 , = f + 0 (δ + L, * ,t ) + λ + (δ + L, * ,t )M 1 -(f + ) (δ + L, * ,t ) γ γ -1 M 1 1 {M + + 2 R + ≤S} + O 2 .
Since (f + ) (x) = (f + 0 ) (x) + O( ), the last equation becomes:

f + (δ + * ,t ) = f + 0 (δ + L, * ,t ) + M 1 λ + (δ + L, * ,t ) - γ γ -1 (f + 0 ) (δ + L, * ,t )1 {M + + 2 R + ≤S} + O 2 ,
It can be recalled at this stage that (f

+ 0 ) (δ + L, * ,t ) = B-γM 0 B λ + (0)1 {M 0 >S} . Notice that if M 0 ∈ [min (S, S -M 1 -2 R + ) , max (S, S -M 1 -2 R + )], then | B-γM 0 B λ + (0)| = O( ). This means: (f + 0 ) (δ + L, * ,t ) = B -γM 0 B λ + (0)1 {M + + 2 R + >S} + O( ) ,
and then:

f + (δ + * ,t ) = f + 0 (δ + L, * ,t ) + M 1 (t, s, y, q 1 )λ + (δ + L, * ,t ) + O 2 .
On the other hand, the optimal bid distance δ - * ,t can be determined similarly. Let f -= J -, , then the function f -writes:

f -(δ -) = λ -(δ -)(u(t, s, y, q 1 + 1, x -(c -δ -)) -u(t, s, y, q 1 , x)), = λ -(δ -) δ --M 0 (t, s, y) + M 2 (t, s, y, q 1 ) + 2 R -(t, s, y, q 1 ) .
Let M -(t, s, y, q 1 ) = M 0 (t, s, y) + M 2 (t, s, y, q 1 ). Differentiating f -yields:

(f -) (δ -) = λ -(δ -) B + δ -δ -(1 -γ) + B + γM -(t, s, y, q 1 ) + γ 2 R -(t, s, y, q 1 ) .
Afterward, it can be seen that if M -+ 2 R -< -S, then δ - * ,t = 0, whereas if M -+ 2 R -≥ -S, then (f -) changes its sign on [0, +∞[ and gets null in x -= B+γM -(t,s,y,q 1 )+γ 2 R -(t,s,y,q 1 ) γ-1 = δ - * ,t . So, the quantity δ - * ,t writes:

δ - * ,t = B + γ (M -(t, s, y, q 1 ) + 2 R -(t, s, y, q 1 )) γ -1 + .
Using again the Taylor expansion, we obtain:

δ - * ,t = δ - L, * ,t + γ γ -1 M 2 (t, s, y, q 1 )1 {M -+ 2 R -≥-S} + O 2 .
Once again, we aim to write the quantity f -(δ - * ,t ) as the sum of f - 0 (δ - L, * ,t ) plus a correction term due to the parameter . First, we use the relation:

f -(x) = f - 0 (x) -λ -(x)M 2 (t, s, y, q 1 ) + O 2 ,
and then, based on Taylor's expansion, we obtain:

f -(δ - * ,t ) = f -(δ - L, * ,t ) + (f -) (δ - L, * ,t )(δ - * ,t -δ - L, * ,t ) + O (δ - * ,t -δ - L, * ,t ) 2 , = f - 0 (δ - L, * ,t ) -λ -(δ - L, * ,t )M 2 + (f -) (δ - L, * ,t ) γ γ -1 M 2 1 {M -+ 2 R -≥-S} + O 2 . The relation (f -) (x) = (f - 0 ) (x) + O( ) implies: f -(δ - * ,t ) = f - 0 (δ - L, * ,t ) + M 2 (t, s, y, q 1 ) -λ -(δ - L, * ,t ) + γ γ -1 (f - 0 ) (δ - L, * ,t )1 {M -+ 2 R -≥-S} + O 2 .
We have also

(f - 0 ) (δ - L, * ,t ) = B+γM 0 B λ -(0) 1 {M 0 <-S} . Following the same method, it can be shown that if M 0 ∈ [min(-S, -S -M 2 -2 R -), max(-S, -S -M 2 -2 R -)], then | B+γM 0 B λ -(0)| = O( ).
Therefore, it can be deduced that:

(f - 0 ) (δ - L, * ,t ) = B + γM 0 B λ -(0)1 {M -+ 2 R -<-S} + O( ) .
and:

f -(δ - * ,t ) = f - 0 (δ - L, * ,t ) -M 2 (t, s, y, q 1 )λ -(δ - L, * ,t ) + O 2 ,
Now that the terms f + (δ + * ,t ) and f -(δ - * ,t ) are computed separately, the term J

(δ - * ,t , δ + * ,t ) = f + (δ + * ,t ) + f -(δ - * ,t ) is deduced: J(δ - * ,t , δ + * ,t ) = f + 0 (δ + L, * ,t ) + M 1 λ + (δ + L, * ,t ) + f - 0 (δ - L, * ,t ) -M 2 λ -(δ - L, * ,t ) + O 2 , = J 0 (δ - L, * ,t , δ + L, * ,t ) + M 1 (t, s, y, q 1 )λ + (δ + L, * ,t ) -M 2 (t, s, y, q 1 )λ -(δ - L, * ,t ) + O 2 ,
The terms of J(δ - * ,t , δ + * ,t ) are classified according to their power in :

J(δ - * ,t , δ + * ,t ) = J 0 (t, s, y) + J 1 (t, s, y, q 1 ) + O 2 ,
where J 1 (t, s, y, q 1 ) = J 1,0 (t, s, y) + q 1 J 1,1 (t, s, y) and:

J 1,0 (t, s, y) = λ + (δ + L, * ,t )(-θ 3 + θ 4 ) -λ -(δ - L, * ,t )(-θ 3 -θ 4 ), J 1,1 (t, s, y) = -2θ 4 λ + (δ + L, * ,t ) -λ -(δ - L, * ,t ) ,
The HJB equation can be separated into several terms according to the order of the parameter . By cancelling the term of order 0 in , it can be obtained that:

(∂ t + L 1 + L 2 )(x + θ 0 + q 1 θ 1 ) + J 0 = 0, with the final conditions:

θ 0 (T, s, y) = 0, θ 1 (T, s, y) = h(s).

The functions θ 0 and θ 1 are equivalent to those found in the case of a linear utility function without inventory constraints, thus:

Verification theorem in the case of linear utility

It is sufficient to prove that u 0 , the solution to the HJB equation (4.2), is finite, smooth and has a polynomial growth. By using the concavity of the square root function, we obtain:

|J 0 (t, s, y)| ≤ 2A B γ (2γ -1) |M 0 | + B(2γ -1) 2γ -1 .
Under the simplifying assumption that the traded option is a call or a put, there exists C 1 > 0 such that:

t ,

where N -and N + are Poisson processes with intensities λ -and λ + respectively, where ∀τ ≥ 0:

λ + τ = λ + (δ + τ ), λ - τ = λ -(δ - τ ). Let t N = T ∧{τ > t, |S τ -s| ≥ n}∧{|y τ -y| ≥ n}∧ N + τ -N + t ≥ n ∧ N - τ -N - t
≥ n . Since u 0 is smooth, we have: u 0 (t n , S tn , y tn , q 1,tn , X tn ) = u 0 (t, s, y, q 1 , x) + tn τ =t (∂ t + L 1 + L 2 ) u 0 (τ, s τ , y τ , q 1,τ , x τ )dτ + tn τ =t ∂u ∂s S τ σ(y τ )dW (1) τ + ∂u 0 ∂y b R (y τ )dW (2) τ -∂u 0 ∂x q 1,τ ∆ τ S τ σ(y τ )dW (1) τ + tn τ =t λ + (δ + ) u 0 (τ, s τ , y τ , q 1,τ --1, x τ -+ C Q + δ + ) -u 0 (τ, s τ , y τ , q 1,τ -, x τ -) dτ

+ tn τ =t
λ -(δ -) u 0 (τ, s τ , y τ , q 1,τ -+ 1, x τ --(C Q -δ -)) -u 0 (τ, s τ , y τ , q 1,τ -, x τ -) dτ

+ tn τ =t
λ + (δ + ) u 0 (τ, s τ , y τ , q 1,τ --1, x τ -+ C Q + δ + ) -u 0 (τ, s τ , y τ , q 1,τ -, x τ -) dM + τ + tn τ =t λ -(δ -) u(τ, s τ , y τ , q 1,τ -+ 1, x τ --(C Q -δ -)) -u(τ, s τ , y τ , q 1,τ -, x τ -) dM - τ where M + and M -are the compensated processes associated to N + and N -respectively. Using the polynomial growth of u and the fact that the functions λ + and λ -are bounded, one can argue as in [START_REF] Pham | Optimisation et contrôle stochastique appliqués à la finance[END_REF] that the local martingales in the previous equation are martingales. Thus, by taking expectations on both sides of the last equation, we obtain: E (u 0 (t n , S tn , y tn , q 1,tn , X tn )) = u 0 (t, s, y, q 1 , x) + E tn τ =t (∂ t + L 1 + L 2 ) u 0 (τ, s τ , y τ , q 1,τ , x τ )dτ + E tn τ =t λ + (δ + ) u 0 (τ, s τ , y τ , q 1,τ --1, x τ -+ C Q + δ + ) -u 0 (τ, s τ , y τ , q 1,τ -, x τ -) dτ

+ E tn τ =t
λ -(δ -) u 0 (τ, s τ , y τ , q 1,τ -+ 1, x τ --(C Q -δ -)) -u 0 (τ, s τ , y τ , q 1,τ -, x τ -) dτ

Using again the polynomial growth of u 0 , we can deduce that u 0 is integrable on [0, t n ], and the dominated convergence theorem yields that lim n→+∞ E (u 0 (t n , S tn , y tn , q 1,tn , X tn )) = E (u 0 (T, S T , y T , q 1,T , X T )) ,

and the equation becomes:

E (u 0 (T, S T , y T , q 1,T , X T )) = u 0 (t, s, y, q 1 , x) + E T τ =t (∂ τ + L 1 + L 2 ) u 0 (τ, s τ , y τ , q 1,τ , x τ )dτ + E T τ =t λ + (δ + ) u 0 (τ, s τ , y τ , q 1,τ --1, x τ -+ C Q + δ + ) -u 0 (τ, s τ , y τ , q 1,τ -, x τ -) dτ

+ E T τ =t
λ -(δ -) u 0 (τ, s τ , y τ , q 1,τ -+ 1, x τ --(C Q -δ -)) -u 0 (τ, s τ , y τ , q 1,τ -, x τ -) dτ .

Recalling that u 0 is the solution of the HJB equation (4.2), we have for (δ - t , δ + t ) ∈ A:

(∂ t + L 1 + L 2 ) u 0 + λ + (δ + ) u 0 (τ, s τ , y τ , q 1,τ --1, x τ -+ C Q + δ + t ) -u 0 (τ, s τ , y τ , q 1,τ -, x τ -) + λ -(δ -) u 0 (τ, s τ , y τ , q 1,τ -+ 1, x τ --(C Q -δ - t )) -u 0 (τ, s τ , y τ , q 1,τ -, x τ -) ≤ 0, which means that E (U (T, S T , y T , q 1,T , X T )) ≤ u 0 (t, s, y, q 1 , x) and then u(t, s, y, q 1 , x) = Sup (δ -,δ + )∈A E (U (T, S T , y T , q 1,T , X T )) ≤ u 0 (t, s, y, q 1 , x).

In addition, since u 0 solves (4.2), then for (δ ) = u 0 (t, s, y, q 1 , x), and then u 0 (t, s, y, q 1 , x) ≤ Sup (δ ) = u(t, s, y, q 1 , x).

Finally, we conclude that u 0 (t, s, y, q 1 , x) = u(t, s, y, q 1 , x).

Figure 1 : 5 Final

 15 Figure 1: Statistics for β = 0.5

Figure 2 :

 2 Figure 2: Statistics for β = 1

Figure 3 : 5 Final

 35 Figure 3: Statistics for β = 0.5

Figure 4 :

 4 Figure 4: Statistics for β = 1

Figure 5 : 5 Final

 55 Figure 5: Statistics for β = 0.5

Figure 6 :

 6 Figure 6: Statistics for β = 1

Figure 7 : 5 Final

 75 Figure 7: Statistics for β = 0.5

Figure 8 :

 8 Figure 8: Statistics for β = 1

θ 1

 1 (t, s, y) = C P (t, s, y) -µE P t,s,y T t ∆(u, S u , y u )S u du , θ 0 (t, s, y) = E t,s,y T t J 0 (u, S u , y u )du .

  max

  4.2.1 Analytic solution Proposition 4.2. Let S = B

γ , and M 0 (t, s, y) = C Q (t, s, y)-C P (t, s, y)+µE P t,s,y T t ∆ u S u du .

Table 1 :

 1 Statistics on final option inventory q 1,T

	Cases	Median Mean	Std	Skewness Kurtosis
	β = 1 2 and with optimal quotes β = 1 2 and with zero-intelligence	23.00 1.00	38.42 58.66 0.80 19.05	0.98 -0.14	0.63 -0.02
	β = 1 and with optimal quotes	29.00	72.94 114.14	1.52	1.94
	β = 1 and with zero-intelligence	1.00	1.13 19.90	-0.07	0.00

Table 2 :

 2 Statistics on final wealth W T

Table 3 :

 3 Statistics on final option inventory q 1,T

	Cases	Median Mean	Std	Skewness Kurtosis
	β = 1 2 and ρ R = ρ I β = 1 2 and ρ R = ρ I	23.00 42.50	38.41 57.30 165.14 58.66	0.97 0.19	0.62 -1.16
	β = 1 and ρ R = ρ I	29.00	72.942 114.13	1.51	1.93
	β = 1 and ρ R = ρ I	105.50 112.38 699.43	-0.049	-1.48

Table 4 :

 4 Statistics on final wealth W T .

		Median Mean	Std	Skewness Kurtosis
	β = 1 2 and ρ R = ρ I β = 1 2 and ρ R = ρ I	356.89 379.63 176.13 331.58 584.06 772.01	0.72 1.29	9.38 2.01
	β = 1 and ρ R = ρ I	144.54 154.90 187.97	0.74	12.24
	β = 1 and ρ R = ρ I	102.22 1485.31 3780.57	1.20	1.26

Table 5 :

 5 Statistics on final option inventory q 1,T . Since θ R < θ I , we have C P (t, S t , y t ) < C Q (t, S t , y t ). The market maker posts aggressive ask quotes and conservative bid quotes. Therefore, she should be more likely to finish with a short option position, a fact that is confirmed by the simulations.

	Cases	Median Mean	Std	Skewness Kurtosis
	β = 1 2 and θ R = θ I β = 1 2 and θ R = θ I	23.00 -90.50	38.42 -73.81	58.66 68.76	0.98 1.22	0.63 1.45
	β = 1 and θ R = θ I	29.00	72.94 114.14	1.52	1.94
	β = 1 and θ R = θ I -693.00 -636.17 204.57	1.51	2.52
	Cases	Median Mean	Std	Skewness Kurtosis
	β = 1 2 and θ R = θ I β = 1 2 and θ R = θ I	356.89 379.64 176.13 856.03 972.96 504.59	0.72 1.26	9.38 1.91
	β = 1 and θ R = θ I	144.54 154.90 187.97	0.75	12.24
	β = 1 and θ R = θ I 3029.61 3737.93 2834.59	1.18	1.26

Table 6 :

 6 Statistics on final wealth W T

Table 7 :

 7 Statistics on final option inventory q 1,T .

		Median Mean	Std	Skewness Kurtosis
	β = 1 2 and η R = η I β = 1 2 and η R = η I	23.00 48.00	38.42 58.66 57.00 58.72	0.98 0.34	0.63 -0.83
	β = 1 and η R = η I	29.00	72.94 114.14	1.52	1.94
	β = 1 and η R = η I	38.50	86.20 105.06	1.13	0.19
	Cases	Median Mean	Std	Skewness Kurtosis
	β = 1 2 and η R = η I 356.89 379.64 176.13 β = 1 2 and η R = η I 251.15 250.49 227.15	0.72 0.22	9.38 2.98
	β = 1 and η R = η I	144.54 154.90 187.97	0.75	12.24
	β = 1 and η R = η I	34.57	81.90 269.94	0.44	2.27

Table 8 :

 8 Statistics on final wealth W T .

  2 > 0 with E (|M 0 (t, s, y)|) ≤ C 2 (1 + s). It follows that there exists C 2 > 0, such that |θ 0 (t, s, y)| ≤ C 2 (1 + s). Using Young's inequality ∀a, b ∈ R, ab ≤ a 2 +b 22 , we can find C 2 > 0 such that: |u 0 (t, s, y, q 1

  (C P (t, s, y), C Q (t, s, y)) ≤ C 1 (1 + S t ) and we also have|∆(t, s, y)| ≤ 1. It follows that ∃M 1 > 0, ∀u < T , |M 0 (u, S u , y u )| ≤ M 1 (1 + S u ). Using this result, we can state that: -t) + M 1 S t e µ(T -t) -1 µ . (t, s, y, q 1 , x)| ≤ |x| + |θ 0 (t, s, y)| + |q 1 | C P (t, s, y) + S t e µ(T -t) , then ∃C 2 > 0 such that |u 0 (t, s, y, q 1 , x)| ≤ C 2 1 + x 2 + s 2 + y 2 + q 2 1 ,which proves that u 0 is finite and has a quadratic growth. Moreover, u 0 is smooth since it is a combination of smooth functions.Let us now prove that u 0 coincides with the value function. Let then δ = (δ -, δ + ) be an admissible control process, and consider the following processes:dq 1,t = dN - t -dN + t , dX t = (C Q (t, S t , y t ) + δ + t )dN + t -(C Q (t, S t , Y t ) -δ - t )dN - t + q 2,t dS t , dS t S t = µdt + σ(y t )dW

	and then:
	|θ 0 (t, s, y)| ≤ (T Since obviously 2A B γ M 1 + B 2γ -1
	|u 0 (1)

|E t,s,y (J 0 (u, S u , y u ))| ≤ E t,s,y |J 0 (u, S u , y u )| , ≤ 2A B γ (2γ -1) (2γ -1) M 1 + M 1 S t e µ(u-t) + B(2γ -1) , t , dy t = a R (y t )dt + b R (y t )dW

http://www.mesocentre.ecp.fr

Therefore, the intensity of order arrivals can be written as λ(δ) = As regards the parameter γ, several empirical studies show that γ > 1, see e.g. [START_REF] Gopikrishnan | Statistical properties of share volume traded in financial markets[END_REF] (γ = 1.53), [START_REF] Maslov | Price fluctuations from the order book perspective -empirical facts and a simple model[END_REF] (γ = 1.4) or [START_REF] Gabaix | Institutional investors and stock market volatility[END_REF] (γ = 1.5).

Solution of the HJB equation with linear impact and linear utility

The utility function U given in (4.1) is linear. Therefore, we make the following Ansatz for u:

Let f + 0 = J + , this function represents the jump part due to the execution of the ask quote. Using (8.3), the function f + 0 can be written explicitly as:

In order to determine δ + L, * ,t = ArgM ax {x≥0} f + 0 (x), the derivative of the function f + 0 is computed:

If M 0 (t, s, y) ≥ B γ , then ∀ δ + ≥ 0, (f + 0 ) (δ + ) ≤ 0, and then δ + L, * ,t = 0. If M 0 (t, s, y) ≤ B γ , the function (f + 0 ) changes sign on [0, +∞[ and gets null at x + = B-γM 0 (t,s,y) γ-1

. Using the sign of (f + 0 ) on [0, x + ] and [x + , +∞[, it can be deduced that δ + L, * ,t = x + . In conclusion, δ + L, * ,t can be determined as follows:

, and:

The same approach can be applied to the function f - 0 = J -. Indeed, using the form of the value function suggested in (8.3), the function f - 0 can be written as follows: f - 0 (δ -) = λ -(δ -)(δ --M 0 (t, s, y)). The derivative of f - 0 writes:

λ -(δ -) B + δ -((1 -γ)δ -+ (B + γM 0 (t, s, y))).