Pietro Fodra
email: pietro.fodra@exqim.com

Mauricio Labadie
email: mauricio.labadie@exqim.com

High-frequency market-making with inventory constraints and directional bets

Keywords: Quantitative Finance, high-frequency trading, market-making, limit-order book, inventory risk, optimisation, stochastic control, Hamilton-Jacobi-Bellman, PnL distribution

In this paper we extend the market-making models with inventory constraints of Avellaneda and Stoikov (High-frequency trading in a limit-order book, Quantitative Finance Vol.8 No.3 2008) and Guéant, Lehalle and Fernandez-Tapia (Dealing with inventory risk, Preprint 2011) to the case of a rather general class of mid-price processes, under either exponential or linear PnL utility functions, and we add an inventory-risk-aversion parameter that penalises the marker-maker if she finishes her day with a non-zero inventory. This general, non-martingale framework allows a market-maker to make directional bets on market trends whilst keeping under control her inventory risk. In order to achieve this, the marker-maker places non-symmetric limit orders that favour market orders to hit her bid (resp. ask) quotes if she expects that prices will go up (resp. down).

With this inventory-risk-aversion parameter, the market-maker has not only direct control on her inventory risk but she also has indirect control on the moments of her PnL distribution. Therefore, this parameter can be seen as a fine-tuning of the marker-maker's risk-reward profile.

In the case of a mean-reverting mid-price, we show numerically that the inventory-risk-aversion parameter gives the market-maker enough room to tailor her risk-reward profile, depending on her risk budgets in inventory and PnL distribution (especially variance, skewness, kurtosis and VaR). For example, when compared to the martingale benchmark, a market can choose to either increase her average PNL by more than 15% and carry a huge risk, on inventory and PNL, or either give up 5% of her benchmark PNL to increase her control on inventory and PNL, as well as increasing her Sharpe ratio by a factor bigger than 2.

Introduction Market-makers

A market-maker is a trader who buys and sells assets in a stock exchange. The difference with any other market agent is that the market-maker is bound to make firm quotes: once she shows a buying/selling quantity at a certain price, she is engaged to trade under those conditions. As liquidity provider, a market-maker receives a compensation: she buys at a lower price (bid) and sells at a higher price (ask). This difference is called the spread.

A market maker is exposed to two main risks, adverse selection and inventory risk. Adverse selection means that if the market-maker sells (resp. buys) an asset it is not necessarily good news, it could mean that her ask (resp. bid) price is lower (resp. higher) than it should on the current market conditions. Inventory risk comes into play by inbalances in the arrival of buying and selling orders: since the marketmaker quotes both bid and ask prices, her net position depends on which quotes are executed and in which quantities. A market maker uses the spread to both control her inventory and compensate herself from adverse selection. In a nutshell, a market-maker loses money against informed traders, but she covers that loss by making noise (i.e. un-informed) traders pay the spread on each transaction.

In order to create a market-making strategy, we need to consider three factors: price, spread and inventory. The price is often the mid-price, i.e. the average between the current ask and bid prices of the market. The spread of the market-maker is her only control on her PnL and inventory throughout the trading day. It is true that the market maker can affect the mid-price by improving the current ask and bid market quotes, but since that would normally trigger a market order that consumes the offer, in a first approach we can consider that the mid price cannot be affected by the quotes of the market maker.

Main features of the present article

In this article we extend the current stochastic-control models of market-making, in particular those of Avellaneda and Stoikov [START_REF] Avellaneda | High-frequency trading in a limit-order book[END_REF] and Lehalle et al [6].

We managed to find closed-form solutions for the optimal ask/bid quotes of a market-making for midprice dynamics that are not necessarily martingale, which can be interpreted of directional bets on price trends. Although our approach is based on optimal stochastic-control and nonlinear-PDE techniques, the philosophy is very simple: given the utility function we choose a possible form of the solution of the nonlinear PDE equation (i.e. we make an ansatz ); we plug this ansatz into the equation and compute the (implicit) optimal controls; we then plug the (implicit)controls and separate the equation into several simpler ones, normally linear; we then compute explicitly the controls and the solution for the equation.

Sometimes it is impossible to find explicitly the controls for the solution, but for a sub-solution it is always possible (at least for the class of processes and utility functions we are dealing with). In that framework, the controls do not optimise the utility function but give a lower bound of the potential losses. It is worth to mention that, for practitioners, it is better to have explicit controls that minimise potential losses (a sub-solution) than highly implicit, numerically-intensive controls that optimise the PnL.

Our approach can be applied to any utility functions, not only exponential as in Avellaneda and Stoikov [START_REF] Avellaneda | High-frequency trading in a limit-order book[END_REF] and Lehalle et al [6], provided it is explicit enough to admit an ansatz. This gives a lot of flexibility to the market-maker for the choice of her risk-reward profile. Of course, selecting a mid-price dynamic determines the class of utility functions that we can choose from because the utility function has to be finite (a.s.). However, with the most recurrent mid-price models such as martingales, Brownian motion with drift, Ornstein-Uhlenbeck (resp. Black-Scholes) a linear (resp. exponential) utility function is finite.

We add a new parameter, which models the inventory-risk aversion of the market-maker. As it will be shown in the numerical simulations, this parameter guarantees that the trading algorithm will end the day with a flat inventory, which is the goal of a market-maker. Moreover, it also allows the market maker to control its directional bets via exposure to price movements intraday.

We show that the inventory-risk-aversion parameter not only exerces direct control on the inventory risk directly but it also has some indirect control on the risk in the PnL distribution of the market-maker (i.e. on the first four moments namely mean, variance, skewness and kurtosis). Moreover, this relation can be also inverted: the parameter of the exponential utility function has direct control on the PnL distribution and indirectly controls the inventory risk. This can be interpreted as a high risk -high reward scenario: big exposure to extreme events, either via fat tails or directional bets, improves the average PnL.

Organisation of the study

The goal of this study is to find the optimal ask and bid quotes for a high-frequency market-maker that, under the framework of a directional bet on the market trend, simultaneously maximise her PnL and minimise her inventory risk. The main inspiration is the paper of Avellaneda and Stoikov [START_REF] Avellaneda | High-frequency trading in a limit-order book[END_REF], who found via stochastic control the optimal bid and ask quotes for a high-frequency market-maker. The second inspiration comes from Lehalle et al [6], who formalised the findings of Avellaneda and Stoikov [START_REF] Avellaneda | High-frequency trading in a limit-order book[END_REF]. It is also worth to mention that the Hamilton-Jacobi-Bellman framework we use was originally set by Ho and Stoll [START_REF] Ho | Optimal dealer pricing under transactions and return uncertainty[END_REF], but in neither of these three articles the effect of a directional market bet on the PnL distribution was considered.

In Chapter 2 we set the framework under which we will be working, which is stochastic control and Hamilton-Jacobi-Bellman equations. In Section 3 we completely solve the optimal control problem for a linear utility function and an arbitrary Markov process by finding explicitly the unique solution. In Section 4 we perturbate the linear utility function via a quadratic inventory penalty. By using first-order approximations on the arrival of orders to the limit-order book, we find explicitly the (approximate) solutions and their corresponding optimal controls, which are perturbations of the closed-form solution and controls we found in Section 3. In Section 5 we extend the results of Avellaneda and Stoikov [START_REF] Avellaneda | High-frequency trading in a limit-order book[END_REF] and Lehalle et al [6] for exponential utility functions. We show that their approach can be used to more general process than arithmetic Brownian motion and with inventory-risk aversion. In Section 6 we perform numerical simulations to show how directional bets on the market trend affect the market-making strategies. We also assess the effect of the inventory-risk-aversion parameter on the PnL distribution and the inventory risk and show that this parameter controls directly the inventory risk and indirectly the PnL distribution.

Stochastic control framework

Setting of the problem

We suppose that the mid-price process S(t) follows an Itô diffusion, i.e.

dS(t) = b(t, S(t))dt + σ(t, S(t))dW (t) (1)
where W (t) is a standard Brownian motion in a filtered probability space (Ω, F, F = (F t ) 0≤t≤T , P).

A market-maker can control her ask and bid quotes, which we denote p + (t) and p -(t) respectively. Instead of working with the market-maker's prices we will rather work with the market-maker's spreads, i.e

δ + (t) := p + (t) -S(t), δ -(t) := S(t) -p -(t) .
The market-maker's spreads are assumed as two predictable processes. Under these variables, the marketmaker's bid-ask spread is thus δ + + δ -≥ 0 (see Figure 1).

Figure 1. Description of the intensity λ ± of limit orders (LOs) as a function of the distances (δ + , δ -).

As usual, δ ± → λ ± (δ ± ) is decreasing: the closer we are to the mid-price, the more likely our LOs are executed. Here we explicited the exponential decay we will use in our model.

Strictly speaking, we should consider that δ + ≥ 0 and δ -≥ 0. However, these constraints render the optimisation problem very hard to solve explicitly due to boundary effects at δ ± = 0. In consequence, and since our goal is to have explicitly the values of the optimal distances (δ + * , δ - * ) of the optimal market-maker quotes (p + * , p- * ), we will assume in our analysis that δ ± ∈ R. However, in our numerical simulations we interpret δ ± ≤ 0 as a market order (see Figure 2). Figure 2. Extrapolation of the intensities λ ± of limit orders (LOs) when δ ± ≤ 0 (dotted lines). For our simulations, when δ ± ≤ 0 we will assume they are market orders.

In addition to the mid-price process S(t) and the predictable processes (δ + , δ -), we will consider two other processes. On the one hand, the inventory Q(t) ∈ Z, which varies with the execution of the limit orders placed by the market-maker; on the other hand, the cash process X(t) ∈ R, which also varies as the market-maker buys or sells the asset.

We will assume that the dynamics of Q(t) and X(t) are governed by

dQ(t) = dN -(t) -dN + (t) , dX(t) = S(t) + δ + dN + (t) -S(t) -δ -dN -(t) ,
where N ± (t) are two independent Poisson processes of intensity λ ± (δ ± ) and δ ± → λ ± (δ ± ) is decreasing. In that framework, the PnL or wealth of the market-maker is

PnL(t) = X(t) + Q(t)S(t) ∈ R .
Finally, we will also assume that the market-maker has a utility function φ(s, q, x) and an associated value function u(t, s, q, x) := max

(δ + ,δ -) E t,s,q,x φ S(T ), Q(T ), X(T )
where t ∈ [0, T ] is the current time, s = S(t) is the current mid-price of the asset, x = X(t) is the current cash and q = Q(t) is the current inventory level.

It is worth to mention that all these assumptions are also present (explicitly or implicitly) in both Avellaneda and Stoikov [START_REF] Avellaneda | High-frequency trading in a limit-order book[END_REF] and Lehalle et al [6].

The Hamilton-Jacobi-Bellman equation

Let us give a heuristic interpretation of the Hamilton-Jacobi-Bellman equation in terms of the two infinitesimal operators, one on the continuous variable s and the other on the jump variables (q, x), as well as of the stochastic controls (δ + , δ -).

First, suppose q and x fixed. Since the continuous variable s follows (1), from Feynmann-Kac representation formula we have that if (t, s) → u C (t, s, q, x) satisfies

(∂ t + L) u C = 0 , L := b(t, s)∂ s + 1 2 σ 2 (t, s)∂ ss , u C (T, s, q, x) = φ(s, q, x) (2)
then (see e.g. Pham [START_REF] Pham | Continuous-time stochastic control and optimization with financial applications[END_REF])

u C (t, s, q, x) = E t,s φ S(T ), q, x) .
Second, suppose now that s is fixed. We model the arrival of orders to the limit-order Book as two independent Poisson Process, N + for the ask quotes and N -for the bid quotes , with intensity λ ± (respectively).

• For the ask quote, we assume that λ + depends on the distance to the mid-price s, i.e. λ + = λ + (δ + ).

Moreover, since buying market orders favour ask quotes with the smallest spread δ + , it is natural to assume that δ + → λ + (δ + ) is increasing, i.e. the probability of execution for the market-maker decreases as she moves her ask quote further away from the mid price.

• Analogously, for the bid quote we have λ -= λ -(δ -) and δ -→ λ -(δ -) increasing.

The jump variables (q, x) are related via the arrival of market orders that hit the quotes of the market-maker:

• Suppose a buying market order of one share of S hits the ask quote of the market-maker. It follows then that x → x + (s + δ + ) and q → q -1.

• Analogously, if a selling market order of one share hits her bid quote then x → x + (s -δ -) and q → q + 1.

Under this framework, we have that the value function of the jump (q, x) → u D (t, s, q, x) satisfies

∂ t u D + λ + (δ + ) u D (t, s, q -1, x + (s + δ + )) -u D (t, s, q, x) (3)
+λ -(δ -) u D (t, s, q + 1, x -(s -δ -)) -u D (t, s, q, x) = 0 . u(T, s, q, x) = φ(s, q, x) .

Third, let us now put together the continuous and jump dynamics (2)- [START_REF] Cartea | Buy Low Sell High: A High Frequency Trading Perspective[END_REF]. Suppose that the marketmaker's spread quotes (δ + , δ -) are known (i.e. deterministic), the value function u(t, s, q, x) satisfies has the following infinitesimal generator (see e.g. Ho and Stoll [START_REF] Ho | Optimal dealer pricing under transactions and return uncertainty[END_REF]):

(∂ t + L) u + λ + (δ + ) u(t, s, q -1, x + (s + δ + )) -u(t, s, q, x) (4)
+λ

-(δ -) u(t, s, q + 1, x -(s -δ -)) -u(t, s, q, x) = 0 .
u(T, s, q, x) = φ(s, q, x) .

Fourth, notice that (4) is valid only when the spread quotes (δ + , δ -) are known, but in the current case they are part of the set of unknowns of the problem. In consequence, from the stochastic control theory it follows that the value function u(t, s, q, x) with unknown controls (δ + , δ -) is solution of the Hamilton-Jacobi-Bellman equation

(∂ t + L) u + max δ + λ + (δ + ) u(t, s, q -1, x + (s + δ + )) -u(t, s, q, x) (5)
+ max δ -λ -(δ -) u(t, s, q + 1, x -(s -δ -)) -u(t, s, q, x) = 0 , u(T, s, q, x) = φ(s, q, x) .

In general we should use sup instead of max in ( 5), but if we assume that b(t, s) and σ(t, s) are Lipschitz and that the jump dynamic (3) is bounded then the supremum is attained and the solution u(t, s, q, x) is unique, as in the particular cases we will consider here.

Fifth, Avellaneda and Stoikov [START_REF] Avellaneda | High-frequency trading in a limit-order book[END_REF] showed that, given the empirical evidence provided by the current research in Econophysics e.g. Potters and Bouchaud [START_REF] Potters | More statistical properties of order books and price impact[END_REF], we can assume that λ ± (δ) = Ae -kδ . Under this framework, the Hamilton-Jacobi-Bellman (5) becomes

(∂ t + L) u + max δ + Ae -kδ + u(t, s, q -1, x + (s + δ + )) -u(t, s, q, x) (6)
+ max

δ - Ae -kδ - u(t, s, q + 1, x -(s -δ -)) -u(t, s, q, x) = 0 , u(T, s, q, x) = φ(s, q, x) ,
Solving the Hamilton-Jacobi-Bellman equation Equation ( 6) is the one we will consider in the rest of the present work. The steps to solve it are as follows:

1. Based on the utility function φ(s, q, x) we make an ansatz, i.e. we guess the general form of the solution of the on Hamilton-Jacobi-Bellman equation. For example, if φ(s, q, x) = x + ϕ(s, q)

we will use u(t, s, q, x) = x + v(t, s, q) .

2. We substitute the ansatz on the HJB in order to find an easier HJB equation for v. We use this new HJB equation to find the optimal controls (δ + * , δ - * ) that maximize the jump. With the ansatz (7) the optimal controls are

δ + * = 1 k -s + v(t, s, q) -v(t, s, q -1) , δ - * = 1 k + s -v(t, s, q + 1) + v(t, s, q) .
3. We substitute the optimal controls on the HJB equation: the resulting equation is called the verification equation. In our case it is

(∂ t + L) v + A k e -kδ + * + e -kδ -
where (δ + * , δ - * ) do not depend on w, the (unique) solution is (see e.g. Pham [START_REF] Pham | Continuous-time stochastic control and optimization with financial applications[END_REF]) w(t, s, q, x) = E t,s,q,x T t f (δ + * , δ - * , ξ, S(ξ), q, x)dξ + g(δ + * , δ - * , S(T ), q, x) ,

where E t,s,q,x is the conditional expectation given S(t) = s, Q(t) = q and X(t) = x 5. Alternatively, we could express the optimal quotes in terms of the market-maker's bid-ask spread

ψ * := δ + * + δ - *
and the mid-point of the spread (called the indifference price)

r * := 1 2 p + * + p - * = s + 1 2 δ + * -δ - *
.

Notice that if δ + = δ -then r * (t) = s = S(t). Therefore, r * -s measures the level of asymmetry of the quotes with respect to the mid-price s.

Linear utility function

Let us suppose that the utility function is linear, i.e.

φ(s, q, x) = x + qs .

Then the corresponding value function is u(t, s, q, x) = max

(δ + ,δ -) E t,s,q,x [X(T ) + Q(T )S(T )] , (8)
where X(t) is the cash process, Q(t) the inventory process, S(t) the price process and E t,s,q,x is the conditional expectation given S(t) = s, Q(t) = q and X(t) = x. Alternatively, this corresponds to choosing the final condition as φ(s, q, x) = x + qs. X(T ) + Q(T )S(T ) is the final value of the market-maker's portfolio, and corresponds to the final PnL of the market-maker. Indeed, X(T ) the cash she holds whilst Q(T )S(T ) the cash value of her inventory: she holds Q(T ) assets and clears them at (unitary) price S(T ) on the close auction.

Ansatz

From the final condition (i.e. the utility function) φ(s, q, x) = x + qs we will search a solution of the form

u(t, s, q, x) = x + θ 0 (t, s) + qθ 1 (t, s) . (9)
Plugging ( 9) into (6) yields

(∂ t + L) (θ 0 + qθ 1 ) + max δ + Ae -kδ + s + δ + -θ 1 + max δ -Ae -kδ - -s + δ -+ θ 1 = 0 , θ 0 (T, s) = 0 , θ 1 (T, s) = s .
Computing the optimal controls

Define f + (δ + ) := Ae -kδ + s + δ + -θ 1 .
Using Calculus we obtain that the maximum is attained at

δ + * = 1 k -s + θ 1 . Analogously, if f -(δ -) := Ae -kδ - -s + δ -+ θ 1 then δ - * = 1 k + s -θ 1 .
In consequence, the optimal quotes (δ + * , δ - * ), spread ψ * and indifference price r * are

δ ± * = 1 k ± (θ 1 -s) , ψ * = δ + * + δ - * = 2 k , r * = θ 1 .

Solving the verification equation

Since f + (δ + * ) = A ek e -k(θ1-s) , f + (δ + * ) = A ek e k(θ1-s)
it follows that the ansatz θ 0 + qθ 1 solves the verification equation

(∂ t + L) (θ 0 + qθ 1 ) + 2A ek cosh [k(θ 1 -s)] = 0 ( 10
)
θ 0 (T, s) = 0 . θ 1 (T, s) = s .
We separate [START_REF] Potters | More statistical properties of order books and price impact[END_REF] in terms of the powers of q, one equation for q 0 = 1 and another for q 1 = q. With this procedure we obtain two coupled equations,

(∂ t + L) θ 0 + 2A ek cosh [k(θ 1 -s)] = 0 (11) θ 0 (T, s) = 0 , and
(∂ t + L) θ 1 = 0 (12) θ 1 (T, s) = s .
Applying the Feynman-Kac formula to (12) and recursively to (11) yields, respectively (see e.g. Pham [START_REF] Pham | Continuous-time stochastic control and optimization with financial applications[END_REF]),

θ 1 (t, s) = E t,s [S(T )] , θ 0 (t, s) = 2A ek E t,s T t cosh k θ 1 (ξ, S(ξ)) -S(ξ) dξ .
In consequence,

δ ± * = 1 k ± (E t,s [S(T )] -s) , ψ * = 2 k , r * = E t,s [S(T )]
and

u(t, s, q, x) = x + 2A ek E t,s T t cosh k θ 1 (ξ, S(ξ)) -S(ξ) dξ + qE t,s [S(T )] .
In particular, since cosh(α) ≥ 1 and cosh(α) = 1 ⇐⇒ α = 0 we obtain that u(t, s, q, x) ≥ u(t, s, q, x)

:= x + 2A ek (T -t) + qE t,s [S(T )] and u(t, s, q, x) = u(t, s, q, x) ⇐⇒ θ 1 (ξ, S(ξ)) = S(ξ) for all ξ ∈ [t, T ]. Since t ∈ [0, T ] is arbitrary then taking ξ = t we have that u(t, s, q, x) = u(t, s, q, x) ⇐⇒ s = E t,s [S(T )], i.e. if

Results

Let us summarise all our findings.

Theorem 1 Consider the Hamilton-Jacobi-Bellman problem

(∂ t + L) u + max δ + Ae -kδ + u(t, s, q -1, x + (s + δ + )) -u(t, s, q, x) + max δ -Ae -kδ - u(t, s, q + 1, x -(s -δ -)) -u(t, s, q, x) = 0 , u(T, s, q, x) = x + qs ,
which corresponds to a linear utility function φ(s, q, x) = x + qs, value function

u(t, s, q, x) = max (δ + ,δ -) E t,s,q,x [X(T ) + Q(T )S(T )]
and stochastic controls (δ + , δ -). Then:

1. The optimal controls (δ + * , δ - * ), spread ψ * and indifference price r * (i.e. the centre of the spread) of the market-maker are

δ ± * = 1 k ± (E t,s [S(T )] -s) ψ * = δ + * + δ - * = 2 k , r * = E t,s [S(T )] .
2. The (unique) solution of the HJB problem is

u(t, s, q, x) = x + 2A ek E t,s T t cosh k θ 1 (ξ, S(ξ)) -S(ξ) dξ + qE t,s [S(T )] .
3. The solution u(t, s, q, x) is bounded from below by

u(t, s, q, x) := x + 2A ek (T -t) + qE t,s [S(T )] and u(t, s, q, x) = u(t, s, q, x) ⇐⇒ s = E t,s [S(T )] ,
i.e. if and only if S(t) is a martingale. In this case we have

r * (t) = S(t) , δ ± * = 1 k and u(t, s, q, x) = x + qs .

Remarks

• The worst price dynamic for the PnL-based utility function ( 8) is a martingale, in the sense that with any other price dynamic the PnL is greater. But observe that the optimal spread ψ * is centred around r * = E t,s [S(T )] and not around s = S(t). In consequence, if the market-maker considers that the current mid-price s has deviated from its fundamental value r * then she can make directional bets via her bid-ask quotes, which yields a higher PnL than the martingale assumption if the bet is correct.

• Applying perturbation methods on the variable q of the form

u(t, s, q, x) = x + θ 0 (t, s) + qθ 1 (t, s) + q 2 θ 2 (t, s) + • • • ,
is a very rough approximation, to say the least. Indeed, as Lehalle et al [6] pointed out, q is an integer, i.e. discrete and not small, and as such a perturbation method on q cannot be performed. However, once the ansatz is shown to solve the verification equation, then by uniqueness it coincides with the solution of the original problem. Therefore, the separation of the equation into two terms, one with q 0 and another with q 1 , is justified a posteriori via the maximum principle (i.e. existence and uniqueness) for the Hamilton-Jacobi-Bellman equation, and as such it does not rely at all on perturbation methods, as Avellaneda and Stoikov [START_REF] Avellaneda | High-frequency trading in a limit-order book[END_REF] suggested.

• In Theorem 1 we have implicitly assumed that the value function u(t, s, q, x) is finite when we applied the Fenmann-Kac formula. However, this is valid if and only if

E t,s T t cosh k θ 1 (ξ, S(ξ)) -S(ξ) dξ < ∞ , θ 1 (t, s) = E t,s [S(T )] . (13)
If S(t) is a martingale then (13) holds trivially. For a non-martingale mid-price process S(t), two sufficient conditions for (13) to hold are (i) the conditional expectation E t,s [S(T )] is affine on s and (ii) the moment-generating function

M Z (λ) = E [exp{λS(t)}]
is finite for all λ ∈ R. This is the case for any Gaussian Markov process, e.g. an arithmetic Brownian motion with drift and the Ornstein-Uhlenbeck process. However, (13) does not hold for the geometric Brownian motion with drift.

Linear utility function with inventory penalty

With the linear utility function there is no penalty if at the end of the trading day the market-maker carries a huge inventory. In order to force a liquidation of the inventory before the end of the day, we propose the following utility function,

φ(s, q, x) = x + qs -ηq 2 , η ≥ 0 ,
which is the PnL with a quadratic penalty on the inventory. The associated value function is u(t, s, q, x) = max

(δ + ,δ -) E t,s,q,x [X(T ) + Q(T )S(T ) -ηQ 2 (T )] , (14)
A quadratic penalty function for a market-maker is already known in the literature (see e.g. Stoll [START_REF] Stoll | The supply of dealer services in securities markets[END_REF]).

Ansatz

Given the form of the utility function, we will search a solution of the form

u(t, s, q, x) = x + θ 0 (t, s) + qθ 1 (t, s) -ηq 2 θ 2 (t, s) . (15)
Plugging ( 15) into ( 6) yields

(∂ t + L) (θ 0 + qθ 1 -ηq 2 θ 2 ) + max δ + Ae -kδ + s + δ + -θ 1 -η(1 -2q)θ 2 + max δ - Ae -kδ - -s + δ -+ θ 1 -η(1 + 2q)θ 2 = 0 , θ 0 (T, s) = 0 , θ 1 (T, s) = s . θ 2 (T, s) = 1 .

Computing the optimal controls

As in the previous section, if

f + (δ + ) := Ae -kδ + s + δ + -θ 1 -η(1 -2q)θ 2 then δ + * = 1 k -s + θ 1 + η(1 -2q)θ 2 .
On the other hand, if

f -(δ -) := Ae -kδ - -s + δ -+ θ 1 -η(1 + 2q)θ 2 then δ - * = 1 k + s -θ 1 + η(1 + 2q)θ 2 .
In consequence, the optimal quotes (δ + * , δ - * ), spread ψ * and indifference price r * are

δ ± * = 1 k + ηθ 2 ± (θ 1 -s -2qηθ 2 ) , ψ * = δ + * + δ - * = 2 k + 2ηθ 2 , r * = θ 1 -2ηqθ 2 .

Solving the equation with linear jumps

We fix q ∈ Z and define the jump functional

J q (δ + * , δ - * ) := A ek e 1-kδ + * + e 1-kδ - *
, whose first-order Taylor expansion (i.e. its Fréchet derivative) is

J q (δ + * , δ - * ) = A ek 4 -k(δ + * + δ - * ) + O |1 -kδ + * | 2 + |1 -kδ - * | 2 = A ek (2 -kηθ 2 ) + O |1 -kδ + * | 2 + |1 -kδ - * | 2 .
Since at first order the jumps are independent of q, it follows that

θ(t, s, q) = θ 0 (t) + qθ 1 (t, s) -ηq 2 θ 2 (t) solves (∂ t + L) θ + A ek (2 -kηθ 2 ) = 0 θ(T, s, q) = qs -ηq 2 .
We separate (16) in terms of the powers of q, one equation for q 0 = 1 and another for q 1 = q. With this procedure we obtain three coupled equations,

∂ t θ 0 + A ek (2 -kηθ 2 ) = 0 (16) θ 0 (T ) = 0 , (∂ t + L) θ 1 = 0 (17) θ 1 (T, s) = s , and
∂ t θ 2 = 0 (18) θ 2 (T ) = 1 .
In consequence,

θ 2 = 1 , θ 1 (t, s) = E t,s [S(T )] , θ 0 (t) = A ek (2 -kη) (T -t) .
Finding a sub-solution

Since A ek e 1-kδ + * + e 1-kδ - * ≥ A ek 4 -k(δ + * + δ - * )
then the solution of the equation with linear jump (i.e. first-order Taylor) is a sub-solution of the original problem with exponential jump. In other words, if we define

u(t, s, q, x) := x + θ 0 (t) + qθ 1 (s, t) -ηq 2 θ 2 (t) ,
where θ 0 (t), θ 1 (s, t) and θ 2 (t) are defined as above then u(t, s, q, x) ≤ u(t, s, q, x) ,

i.e. it is a sub-solution of the HJB equation.

Results

Let us summarise all our findings.

Theorem 2 Consider the Hamilton-Jacobi-Bellman problem

∂ t u + Lu + max δ + Ae -kδ + u(t, s, q -1, x + (s + δ + )) -u(t, s, q, x) + max δ -Ae -kδ - u(t, s, q + 1, x -(s -δ -)) -u(t, s, q, x) = 0 , u(T, s, q, x) = x + qs -ηq 2 ,
which corresponds to a linear utility function with quadratic inventory penalty φ(s, q, x) = x + qs -ηq 2 , value function u(t, s, q, x) = max

(δ + ,δ -) E t,s,q,x X(T ) + Q(T )S(T ) -ηQ 2 (T )
and stochastic controls (δ + , δ -). Then:

1. Let u(t, s, q, x) be the (unique) solution of the HJB equation. Then

u(t, s, q, x) := x + A ek (2 -kη) (T -t) + qE t,s [S(T )] -ηq 2
is a sub-solution of the HJB equation and u(t, s, q, x) ≥ u(t, s, q, x).

2. With the linear aproximation of the jumps, or equivalently using the HJB of the sub-solution u, the optimal controls (δ + , δ -), spread ψ * and indifference price r * are

δ ± * = 1 k + η ± (E t,s [S(T )] -s -2qη) , ψ * = δ + * + δ - * = 2 k + 2η , r * = E t,s [S(T )] -2ηq .

General inventory penalties

Suppose that the utility function is now

φ(s, q, x) = x + qs -ηq 2 π(s) , η ≥ 0 ,
where s → π(s) is continuous and for s ≥ 0 it is non-decreasing and non-negative. For example, if π ≡ 1 we recover the previous case whilst if π(s) = s 2 we recover the classical mean-variance PnL criterion.

The associated value function is u(t, s, q, x) = max

(δ + ,δ -) E t,s,q,x [X(T ) + Q(T )S(T ) -ηQ 2 (T )π(S(T ))] . (19)
With the ansatz u(t, s, q, x) = x + θ 0 (t, s)

+ qθ 1 (t, s) -ηq 2 θ 2 (t, s) (20)
it can be shown that the functions θ 0 -θ 2 solve the equations

(∂ t + L) θ 0 + A ek (2 -kηθ 2 ) = 0 (21) θ 0 (T ) = 0 , (∂ t + L) θ 1 = 0 (22) θ 1 (T, s) = s , and
(∂ t + L) θ 2 = 0 (23) θ 2 (T ) = π(s) ,
whose explicit solutions are

θ 2 (t, s) = E t,s [π(S(T ))] , θ 1 (t, s) = E t,s [S(T )] , θ 0 (t, s) = 2A ek (T -t) - ηA e E t,s T t θ 2 (ξ, S(ξ))dξ .
In the light of these results, we have the following extension of Theorem 2 to a general penalty function π.

Theorem 3 Consider the Hamilton-Jacobi-Bellman problem

∂ t u + Lu + max δ + Ae -kδ + u(t, s, q -1, x + (s + δ + )) -u(t, s, q, x) + max δ - Ae -kδ - u(t, s, q + 1, x -(s -δ -)) -u(t, s, q, x) = 0 , u(T, s, q, x) = x + qs -ηq 2 π(s) ,
which corresponds to a linear utility function with quadratic inventory penalty φ(s, q, x) = x+qs-ηq 2 π(s), value function u(t, s, q, x) = max

(δ + ,δ -) E t,s,q,x X(T ) + Q(T )S(T ) -ηQ 2 (T )π(S(T ))
and stochastic controls (δ + , δ -). Then:

1. If u(t, s, q, x) is the (unique) solution of the HJB equation then

u(t, s, q, x) := x + 2A ek (T -t) - ηA e E t,s T t θ 2 (ξ, S(ξ))dξ + qE t,s [S(T )] -ηq 2 θ 2 (t, s) ,
where

θ 2 (t, s) = E t,s [π(S(T ))]
, is a sub-solution of the HJB equation and u(t, s, q, x) ≤ u(t, s, q, x).

2. With the linear aproximation of the jumps, or equivalently using the HJB of the sub-solution u, the optimal controls (δ + , δ -), spread ψ * and indifference price r * are

δ ± * = 1 k + η ± (E t,s [S(T )] -s -2qηE t,s [π(S(T ))]) , ψ * = δ + * + δ - * = 2 k + 2η , r * = E t,s [S(T )] -2ηqE t,s [π(S(T ))] .

Remarks

• The optimal controls depend on the ansatz we make on the utility function u(t, s, q, x), i.e. on the functions θ 1 (t, s) and θ 2 (t). However, without the linear aproximation of the jumps our ansatz cannot give the solution: in fact, there is no solution with the chosen ansatz. That said, given that the sub-solution u is explicit, and by definition it provides a lower bound on the real solution u, we can consider that the quotes we have found are optimal for the sub-solution, which is a lower bound on the (penalised) PnL. Under that spirit, the optimal quotes can be interpreted as conservative estimates of the real optimal quotes, since they minimise the potential drops on the PnL.

• If η = 0 we recover the optimal controls and the sub-solution of the linear case without inventory penalty, i.e. Theorem 1. Therefore, our linear approximation of the jumps is consistent, in the sense that it provides a perturbation of the optimal quotes in terms of the "inventory-risk" or "risk-aversion" parameter η.

• When η > 0 the spread ψ * widens and the indifference price r * shifts downwards (resp. upwards) if the inventory is positive (resp. negative), which is in line with the intuition on the inventory risk. Indeed, if the net position of the market-maker is long (resp. short) then she will improve the current ask (resp. bid) quote to lure buyers (resp. sellers), and simultaneously she will try to hide her bid (resp. ask) quote deep into the limit-order Book to deter sellers (resp. buyers). By doing so, she favours the probability of being executed in the direction that makes her to go back to zero.

• Notice that since the linear approximation of the jump functional (i.e. its Fréchet derivative) does not depend on q, the solution u of the approximate verification equation does not rely on perturbation methods and asymptotic expansions on q. Of course, the solution u to the verification equation with exponential jumps will indeed depend on q, and as such an approach similar to Lehalle et al [6] is needed in order to deal with the discrete variable q.

• For Theorems 2 and 3 we are assuming that the corresponding value function u(t, s, q, x) is finite in order to apply the Feynmann-Kac formula. In the current framework, given a mid-price dynamic we choose the penalty function π such that u(t, s, q, x) is finite. In that spirit, if S(t) is Gaussiann (e.g. arithmetic Brownian motion or Ornstein-Ulenbeck) or a martingale (even with jumps) then π ≡ 1 suffices, whilst if S(t) is a geometric Brownian motion then π(s) = s 2 is a viable candidate.

• As we mentioned above, it is easy to find sufficient conditions to ensure boundedness of the value function u η for η > 0. If u 0 is the explicit solution without inventory constraints of theorem 1 then the maximum principle shows that u η ≤ u η ≤ u 0 . Therefore, if u 0 is finite then u η is bounded. This is true for processes S(t) such that E t,s [S(T )] is affine in s, i.e.

E t,s [S(T )] = α(t) + β(t)s , α, β ∈ C 0 [0, T ] .
In particular, u η is bounded if S(t) is an arithmetic Brownian motion or an Ornstein-Uhlenbeck process.

Exponential utility function

This case has been entirely solved by Avellaneda and Stoikov [START_REF] Avellaneda | High-frequency trading in a limit-order book[END_REF] and Lehalle et al [6] when the mid-price is a Brownian motion. In this section we show that their approach can be easily extended to several other mid-price dynamics, e.g. Ornstein-Uhlenbeck.

Let us suppose that the utility function is exponential φ(s, q, x) = -exp{-γ(x + qs)}, whose coresponding value function is

u(t, s, q, x) = max (δ + ,δ -) E t,s,q,x -exp -γ X(T ) + Q(T )S(T ) , (24)

Ansatz

From the form of the utility function we will search a solution of the form u(t, s, q, x) = -exp{-γ(x + θ(t, s, q)} , θ(t, s, q) = θ 0 (t) + qθ 1 (t, s) + q 2 θ 2 (t) .

Plugging (25) into ( 6) yields the Hamilton-Jacobi-Bellman for θ(t, s, q), i.e.

(∂ t + L) θ - 1 2 σ 2 γ (∂ s θ) 2 + A γ max δ + e -kδ + 1 -exp{-γ(s + δ + -θ 1 + (1 -2q)θ 2 )} + A γ max δ - Ae -kδ - 1 -exp{-γ(-s + δ -+ θ 1 + (1 + 2q)θ 2 )} ] = 0 , (26)
θ(T, s) = x + qs ,

Computing the optimal controls

For the function

f + (δ + ) := A γ e -kδ + 1 -exp{-γ(s + δ + -θ 1 + (1 -2q)θ 2 )}
its maximum is attained at

δ + * = 1 γ log 1 + γ k -s + θ 1 -(1 -2q)θ 2 .
Analogously, if

f -(δ -) := A γ max δ -Ae -kδ - 1 -exp{-γ(-s + δ -+ θ 1 + (1 + 2q)θ 2 )} then δ - * = 1 γ log 1 + γ k + s -θ 1 -(1 + 2q)θ 2 .
In consequence, the optimal quotes (δ + * , δ - * ), spread ψ * and indifference price r * are

δ ± * = 1 γ log 1 + γ k -θ 2 ± (θ 1 -s + 2qθ 2 ) , ψ * = 2 γ log 1 + γ k -2θ 2 , r * = θ 1 + 2qθ 2 . ( 27
)

Solving the equation with linear jumps

For q ∈ Z fixed we define the jump functional as J q : R 2 → R as

J q (δ + * , δ - * ) := A k + γ e -kδ + * + e -kδ - * . ( 28
)
The first-order Taylor expansion of J q (i.e. its Fréchet derivative) is

J q (δ + * , δ - * ) = A k + γ 2 -k(δ + * + δ - * ) + O |1 -kδ + * | 2 + |1 -kδ - * | 2 = 2A k + γ 1 - k γ log 1 + γ k + kθ 2 + O |1 -kδ + * | 2 + |1 -kδ - * | 2 .
Therefore, at first order we have that J q = J 0 for all q ∈ Z, i.e. the jumps are independent of q. In consequence, θ(t, s, q) = θ 0 (t) + qθ 1 (t, s) + q 2 θ 2 (t) solves

(∂ t + L) (θ 0 + qθ 1 + q 2 θ 2 ) - 1 2 q 2 σ 2 γ (∂ s θ 1 ) 2 + 2A k + γ 1 - k γ log 1 + γ k + kθ 2 = 0 θ(T, s, q) = qs .
We separate (29) in terms of the powers of q, one equation for q 0 = 1 and another for q 1 = q. With this procedure we obtain three coupled equations,

∂ t θ 0 + 2A k + γ 1 - k γ log 1 + γ k + kθ 2 = 0 (29) θ 0 (T ) = 0 , (∂ t + L) θ 1 = 0 (30) θ 1 (T, s) = s , and
∂ t θ 2 - 1 2 σ 2 γ (∂ s θ 1 ) 2 = 0 (31) θ 2 (T ) = 0 .
Applying the Feynman-Kac formula to (30) we find

θ 1 (t, s) = E t,s [S(T )] .
Integrating (31) we obtain

-θ 2 (t) = 1 2 γ T t σ 2 (ξ, S(ξ)) ∂ s θ 1 (ξ, S(ξ)) 2 dξ .
However, the ansatz we have made implies that θ 2 is independent of s. Therefore, in order to solve (31) we need to assume the following conditions on the price process S(t):

σ = σ(t) , E t,s [S(T )] = α(t, T ) + sβ(t, T ) .
In consequence,

θ 2 (t) = - 1 2 γ T t σ 2 (ξ)β 2 (ξ, T )dξ .
Finally, integrating (30) yields

θ 0 (t) = 2A k + γ 1 - k γ log 1 + γ k (T -t) - kγA k + γ T t T ζ σ 2 (ξ)β 2 (ξ, T )dξ dζ .

Finding a sub-solution

Since A k + γ e -kδ + * + e -kδ - * ≥ A k + γ 2 -k(δ + * + δ - * )
then the solution of the equation with linear jump (i.e. first-order Taylor expansion)is a sub-solution of the original problem with exponential jump. In other words, if we define u(t, s, q, x) := -exp -γ x + θ 0 (t) + qθ 1 (s, t) + q 2 θ 2 (t) ,

where θ 0 (t), θ 1 (s, t) and θ 2 (t) are defined as above then u(t, s, q, x) ≤ u(t, s, q, x) .

Adding a quadratic inventory penalty

We modify the exponential utility function (24) by adding a quadratic inventory penalty:

u(t, s, q, x) = max (δ + ,δ -) E t,s,q,x -exp -γ X(T ) + Q(T )S(T ) -ηQ 2 (T ) . ( 32
)
Under this new penalty framework, the computations are exactly the same as before. The only thing that changes is the equation solved by θ 2 (t), i.e.

∂ t θ 2 - 1 2 σ 2 γ (∂ s θ 1 ) 2 = 0 (33) θ 2 (T ) = -η ,
whose solution is

θ 2 (t) = -η - 1 2 γ T t σ 2 (ξ)β 2 (ξ, T )dξ .
This has an impact on θ 0 (t), which has the new form

θ 0 (t) = 2A k + γ 1 - k γ log 1 + γ k -kη (T -t) - kγA k + γ T t T ζ σ 2 (ξ)β 2 (ξ, T )dξ dζ ,
as well as on the optimal quotes, spread and indifference price (27).

Results

Let us summarise all our findings.

Theorem 4 Consider the Hamilton-Jacobi-Bellman problem

∂ t u + Lu + max δ + Ae -kδ + u(t, s, q -1, x + (s + δ + )) -u(t, s, q, x) + max δ - Ae -kδ - u(t, s, q + 1, x -(s -δ -)) -u(t, s, q, x) = 0 , u(T, s, q, x) = -exp{-γ(x + qs -ηq 2 )} ,
which corresponds to an exponential utility function φ(s, q, x) = -exp{-γ(x + qs -ηq 2 )}, value function u(t, s, q, x) = max

(δ + ,δ -) E t,s,q,x -exp -γ X(T ) + Q(T )S(T ) -ηQ 2 (T ) ,
and stochastic controls (δ + , δ -). Assume further that the mid-price process

dS(t) = b(t, S(t))dt + σ(t)dW (t) satisfies E t,s [S(T )] = α(t, T ) + sβ(t, T ) , ∀(t, s) .
Then:

1. With the linear approximation of the jumps, or equivalently using the HJB of the sub-solution u, the optimal controls (δ + , δ -), spread ψ * and indifference price r * are

δ ± * = 1 γ log 1 + γ k -θ 2 ± (θ 1 -s + 2qθ 2 ) , ψ * = 2 γ log 1 + γ k -2θ 2 , r * = θ 1 + 2qθ 2 ,
where

θ 0 (t) = 2A k + γ 1 - k γ log 1 + γ k -kη (T -t) - kγA k + γ T t T ζ σ 2 (ξ)β 2 (ξ, T )dζ dξ , θ 1 (t, s) = E t,s [S(T )] , θ 2 (t) = -η - 1 2 γ T t σ 2 (ξ)β 2 (ξ, T )dξ .
2. Let u(t, s, q, x) be the (unique) solution of the HJB equation. Then u(t, s, q, x) := -exp -γ x + θ(t, s, q) , θ(t, s, q) = θ 0 (t) + qθ 1 (t, s) + q 2 θ 2 (t) is a sub-solution of the HJB equation and u(t, s, q, x) ≥ u(t, s, q, x).

Remarks

• The linear approximation of the jumps turns out to be independent of q, although for higher orders this is no longer true. This means that we are not performing perturbation methods and asymptotic expansions on the (discrete) variable q when we solved the approximate verification equation. Of course, if we want to solve the real HJB problem then we need to take into account the discrete variable q, which leads to an infinite system of equations as in Lehalle et al [6].

• In the case of η = 0 the inventory penalty tends to zero as t → T . This implies that the penalisation is not stong enough to force the market-maker to finish her day with a flat inventory, as it will be shown in the numerical simulations. Therefore, it was necesary to add an "inventory-risk" parameter η > 0 in order to ensure a flat inventory at the end of the day.

Examples

• If S(t) is an arithmetic Brownian motion with drift, i.e.

dS(t) = bdt + σdW (t) then E t,s [S(T )] = s + b(T -t) ,
which implies that

θ 1 = s + b(T -t) , θ 2 = -η - 1 2 γσ 2 (T -t) .
In consequence, the optimal controls are

δ ± * = 1 γ log 1 + γ k + η + 1 2 γσ 2 (T -t) ± b(T -t) -q[2η + γσ 2 (T -t)] , ψ * = 2 γ log 1 + γ k + 2η + γσ 2 (T -t) , r * = s + b(T -t) -q 2η + γσ 2 (T -t) .
In particular, if b = η = 0 we recover the results of Avellaneda and Stoikov [START_REF] Avellaneda | High-frequency trading in a limit-order book[END_REF].

• If S(t) is an Ornstein-Uhlenbeck process, i.e.

dS(t) = a(µ -S(t))dt + σdW (t) then E t,s [S(T )] = se -a(T -t) + µ 1 -e -a(T -t) ,
which implies that

θ 1 = se -a(T -t) + µ 1 -e -a(T -t) , θ 2 = -η - γσ 2 4a 1 -e -2a(T -t) .
In consequence, the optimal controls are

δ ± * = 1 γ log 1 + γ k + η + γσ 2 4a 1 -e -2a(T -t) ± (µ -s) 1 -e -a(T -t) -q 2η + γσ 2 2a 1 -e -2a(T -t) , ψ * = 2 γ log 1 + γ k + 2η + γσ 2 2a 1 -e -2a(T -t) , r * = se -a(T -t) + µ 1 -e -a(T -t) -q 2η + γσ 2 2a 1 -e -2a(T -t) .

Solving the nonlinear equation: Lehalle's approach

Assuming that S(t) is a Brownian motion, Lehalle et al [6] found that the solution u(t, s, q, x) of the nonlinear problem can be explicitly found, but that study can be easily extended to arithmetic Brownian motions with time-dependent volatility and drift.

Suppose that the mid-price process is of the form

dS(t) = b(t)dt + σ(t)dW (t) ,
where b(t) and σ(t) are uniformly bounded in [0, T ]. Let us make the ansatz

u(t, s, q, x) = -exp{-γ(x + qs)}v q (t) -γ/k ; v q ∈ C 1 (0, T ) , q ∈ Z . ( 34
)
Following Lehalle et al [6] it can be shown that the optimal quotes are

δ + * = 1 γ log 1 + γ k + 1 k log v q (t) v q-1 (t) , δ - * = 1 γ log 1 + γ k - 1 k log v q+1 (t) v q (t) ,
where (v q (t)) q∈Z solves the (infinite) ODE system

v q (t) = kγq 2 2 σ 2 (t) -γqb(t) v q (t) -A 1 + k γ -1-k/γ v q+1 (t) + v q-1 (t) , (35)
v q (T ) = 1 .

Lehalle et al [6] used a constructive proof to show that (35) has a unique, strictly positive solution in C ∞ [0, T )); 2 (Z) . However, this result can be easily proven in a non-constructive fashion.

Let E be a Banach space and consider a system on E of the form

v q (t) = F q (t, v(t)) , (36)
v q (T ) = v q,T > 0 ,
where

F (t, v) = (F q (t, v)) q∈Z : [0, T ] × E → E is Lipshitz in v uniformly in t ∈ [0, T ].
Applying the Cauchy-Picard Theorem and the maximum principle for ODEs on the Banach space C 1 [0, T ); E yield existence, uniqueness and positivity of the solution v of (35) (see e.g. Brézis [START_REF] Brézis | Analyse Fonctionnelle: théorie et applications[END_REF]). Unfortunately, since the linear function F q (t, v) corresponding to (35) is proportional to q 2 , we cannot apply the Cauchy-Picard Theorem directly. However, if we define

E = ∞ (Z) , F := v ∈ E : sup q∈Z q 2 v q < +∞ , (37)
then for t fixed we have that F (t, •) : F → E is linear and bounded, and the bound is uniform in t.

Therefore, F (t, v) is Lipschitz in v, uniformly in t. In consequence, we can now apply the Cauchy-Picard Theorem to ensure that there exists a unique positive solution of (35).

Remarks on Lehalle's approach

• If instead of the Banach spaces in (37) we use the Hilbert spaces

E = 2 (Z) , F :=    v ∈ E : q∈Z q 2 v q < +∞    , (38)
we recover the framework of Lehalle et al [6].

• Our proof is non-constructive, which implies that we cannot provide explicit asymptotic estimates of the solution v = (v q ) q∈Z . Lehalle et al [6], on the contrary, constructed the operator explicitly, and thus they were able to show the asymptotic behaviour of v based on the spectrum of the linear operator.

• From the ansatz (34) we see that the coefficients in the ODE system (35) cannot depend on s. This rules out mid-price processes whose drift and volatility depend on s, e.g. Ornstein-Uhlenbeck and geometric Brownian Motion. However, arithmetic Brownian motions with time-dependent drift and volatility can be used.

Numerical Simulations and sample paths

We performed several simulations of the optimal market-making strategy, i.e. the spread ψ * and the indifference price r * , under an Ornstein-Uhlenbeck mid-price process, i.e. a mean-reverting price dynamic of the form dS(t) = a(µ -S(t))dt + σdW (t) .

We considered four strategies: linear/exponential utility and mean-reverting /martingale market-making assumption. This allows us to assess the effect of the directional bet µ on the PnL of the market-making strategy. The parameters we used are k = 100, A = 1500, T = 1 (which corresponds to one trading day), n = 1000 (which corresponds to 1000 bid/ask limit orders sent per day, approx. once every 30 seconds), S(0) = 1, σ = 0.05 (daily volatility of 5%), a = 1 and η = 0.0001 (which is a very small inventory-risk aversion but enough to force the inventory to end the day flat in average). Under these parameters we have a constant linear spread of ψ * = 0.0202 whilst the exponential spread is time-dependent.

We chose three different values of µ: 0.98, 1.00 and 1.02. This corresponds, respectively, to a bet that the price will go down by 2%, will oscillate around its open price or go up by 2%. We used the linear utility function with inventory penalty and two strategies, a mean-reverting strategy (Ornstein-Uhlenbeck) with the correct directional bet, and the martingale strategy (arithmetic Brownian motion) with no directional bet. The martingale strategy performs a pure market-making strategy under inventory constraints. On the other hand, the mean-reverting strategy performs the same market-market strategy than the martingale but it also places directional bets, which can be seen not only in the agressiveness of the ask and bid quotes but also in the fact that the algorithm can place market orders. Figure 3. Simulation of the market-making strategy under a mean-reverting mid-price dynamic with (asymptotic) mean µ = 0.98. Upper-Left: mid-price (black), optimal ask quote for the market-maker (dark blue), optimal bid quote (light green), µ (light blue). Lower-Left: Inventory for the mean-reverting process (black) vs the inventory for the martingale (blue). Upper-Right: cash. Lower-Right: PnL of the mean reverting process (black) compared with the benchmark, i.e. the PnL of the martingale strategy (blue).

In Figure 3 we have plotted a realisation of the market-making strategy for µ = 0.98, i.e. assuming that the price will go down by 2% at the end of the day. If the market mid-price -black line-is above (resp. below) the optimal ask quote -dark blue-(resp. the optimal bid quote -light green-) then the market-maker sells (resp. buys) at market price, which we assume to coincide with the mid-price.

• The mid-price starts at s = 1.00, it goes up to s = 1.03 at t = 0.1 and stays above s = 1.01 up to t = 0.2. Since the bet is that the price will converge down to µ = 0.98, the market-maker sells the asset at market price. At t = 0.2 the market-maker has an inventory of q = -170 and a PnL below the martingale benchmark: as she sold her assets at the market price (mid-price), she paid the spread to mount her directional bet.

• During the time interval (0.20, 0.50) the market-maker buys back her position via limit-orders. In order to favour the arrival of selling orders and deter buying orders she plays very aggressive bid quotes and very conservative ask quotes, hence the mid-price is closer to her bid quote than her ask quotes. The strategy paid well because at t = 0.5 the mid-price converged to µ = 0.98, her inventory went back to zero and her PnL outperformed the martingale benchmark.

• On (0.50, 1.00) the market-maker does not make any directional bet, she only plays the bid-ask spread because the mid-price oscillates around µ = 0.98. As it can be seen, her bid and ask quotes are rather symmetric with respect to the mid-price, i.e. during all this non-directional period the strategy makes the same PnL than the martingale benchmark because both lines are almost parallel. In Figure 4 the market-maker assumes that the price will oscillate around µ = 1.00. Therefore, her quotes are symmetric near this threshold and are tilted when the prices wander far from it (i.e. she makes mean-reverting bets). Therefore, her inventory oscillates from positive on (0.00, 0.2) to negative on (0.25, 0.45), then back to positive and negative again. On (0.85, 1.00) there is a huge drop in the mid-price, which in absence of inventory risk would imply a consequent positive inventory due to the mean-reverting dynamic. However, the inventory-risk-aversion forces her to avoid a directional bet at the end of the day, and as such her strategy is similar to the martingale case. It is worth to mention that the mid-price finished the day below the target of 1.00. In consequence, the market-making strategy is insensitive to peaks at the end of the day. In Figure 5 the market-maker assumes that the price will converge to µ = 1.02. On (0.00, 0.10) she mounts an inventory of q = 50 because she bets the mid-price will hit µ = 1.02. She eliminates her inventory during (0.20, 0.40) using limit-orders with generous ask prices and not-very competitive bid prices, which translates into a greater flow of buying orders than selling orders. At t = 0.20 we can see that her directional bet has beaten the martingale benchmark. On (0.40, 0.70) the mid-price rises from µ = 1.02 and comes back, and since the market-maker is betting for a mean-reverting dynamic she builds up an inventory of q = -90 at t = 0.55. On (0.85, 1.00) the mid-price falls, but instead of making a U-turn in her inventory and turn it positive, as a mean-reverting dynamic suggests, she rather eliminates slowly her negative inventory by tilting her quotes towards the buying side. By doing so the market-maker avoids directional bets and minimises her market impact at the end of the trading day.

Statistics of the PnL distributions

Comparing linear and exponential utility strategies

We performed 100,000 simulations with η ∈ {0, 0.0001, 0.001}, γ = 1 and µ ∈ {0.98, 1.00, 1.02}. The rest of the parameters have the same values as for the previous simulations.

Q(T) 90% [-3,3] [-3,3] [-3,3] [-3,3]
Table 3C. Statistics of the PnL distribution. µ = 1.02, η = 0.001.

In Tables 1A-3C we can observe the effect of η on the distribution of PNL and inventory:

• The "Sharpe" ratio (that is mean over standard deviation) increased dramatically as η increases for the linear utility models: the linear ABM has its "Sharpe" increased by a factor of 3 whilst the linear MR has a factor of 26. The same effect is seen for exponential MR with a factor of 1.5. However, for the exponential ABM there is not a significant change.

• The inventory risk has an impressive reduction as η increases: by a factor of 18 for linear ABM, 230 for linear MR, 4 for exponential ABM and 8 for exponential MR.

• The PNL mean decreases as η increases: -5% for linear ABM, -25% for linear MR, -6% for exponential ABM and -7% for exponential MR.

• None of the the PnL distributions are not normal, but in all cases the inventory distributions can be considered as Gaussian because of their low Jarque-Bera scores.

In consequence, the total effect of η in the linear models is remarkable: it not only reduces the inventory risk, as expected, but as a welcoming side effect it also reduces the risks on the PNL distribution. Of course, nothing is free and this control on the risks comes with a reduction of the PNL mean. However, with the current framework a market-maker has enough room and tools to improve her PNL, given her risk budgets on inventory and PNL distribution.

The effect of η on the linear-utility strategy

As we have already seen, η has some indirect control on the the PnL distribution, but we wanted to have a more detailed view of this fact. We performed 20,000 Monte-Carlo simulations for µ = 1.00. The rest of the parameters are as before. As we can see in Figure 7 and Table 6, between the linear martingale strategy and the exponential MR strategy with γ = 1 we have a better Sharpe of a factor of 2, reduce our VaR(5%) by 8%, our VaR(1%) by 14% and our inventory risk by 50%. In consequence, γ controls directly the PnL distribution and also has indirect control on the inventory, but the effect of η seems to be stronger in both risk factors.

Observe that the indirect control of γ on the inventory is less impressive than the indirect control of η on the PNL distribution. However, with γ the market-maker is not sacrificing any PNL at alln unlike she does with η.

Comparing η = 0.0001 and γ = 0.3

From Figures 67we have that the linear MR with η = 0.0001 and the exponential MR with γ = 0.3 seem to have the same (statistical) mode than the bechmark linear martingale. Here we compare these two distributions. We performed 20,000 Monte-Carlo simulations for µ = 1.00 and kept the other parameters unchanged. As we can see in Table 7, seems that the exponential MR has better control on the PnL distribution than the linear MR: higher mean, "Sharpe" and quantiles at 1% and 5% (VaR); smaller standard deviation, (absolute) skewness and kurtosis. However, the inventory of exponential MR is much higher than the inventory of linear MR (5 times bigger).

Conclusions

On the market-making model

• In Theorem 4 we generalised the Avellaneda and Stoikov [START_REF] Avellaneda | High-frequency trading in a limit-order book[END_REF] approach for an exponential utility to any Markov process, provided its conditional expectation E t,s [S(T )] is affine in s, which includes processes like arithmetic Brownian motion with drift or Ornstein-Uhlenbeck process. This allowed us to assess the effect of directional bets on the market mid-price on the PnL distribution of a high-frequency market-maker. Moreover, we also showed that the results of Lehalle et al [6] can be generalised to arithmetic Brownian motions with time-dependent drift and volatility, and that their approach cannot be extended further with the ansatz they chose.

• If instead of an exponential utility function we choose a linear utility function with inventory constraints, in Theorems 1-3 we showed that using the same stochastic-control approach we can find the optimal controls (i.e. the market-maker bid and ask quotes) for very general Markov processes, even with jumps (e.g. Lévy processes), provided the inventory-risk penalty π(s) is chosen appropriately to ensure the boundedness of the value function u(t, s, q, x). Moreover, the optimal controls in the linear case are independent of the volatility of the asset, which is very hard to estimate for high-frequency data, and as such the linear case is easier to calibrate with real data than the exponential case.

• Our approach, although based on optimal-control and nonlinear-PDE techniques, is very intuitive: choose the right ansatz for the solution, compute the (implicit) controls, plug them into the equation, separate the equation into smaller and easier parts and solve them all to have the explicit form of the solution and the controls. Moreover, if the full equation is not explicitly solvable, approximate the jump part due to inventory: the resulting control is thus optimal for a sub-solution, and as such we are controlling the PnL utility function from below, which translates into optimal quotes to reduce losses.

On the role of the parameters η and γ

• If η increases then the inventory risk decreases and eventually becomes negligeable. This was expected because the parameter η was added as an inventory penalty. However, as a nice side effect we have that η also reduces the risk on the PnL, in the sense that the variance and the kurtosis of the PnL distribution decrease as η increases. In other words, η controls perfectly the inventory risk, and by doing so it indirectly controls the PnL distribution.

• For γ we have the a similar effect, but the other way around. As γ increases the risk on the PnL distribution decrease, in particular the first four moments and the VaR. Moreover, as a side effect the inventory risk decreases as well. This implies that γ directly controls the PnL distribution and indirectly controls the inventory risk.

• This mirror-like, intertwined role of η and γ highlights a intimate relation between inventory risk and the risk on the PnL distribution: if one decreases then necessarily the other has to do the same. This implies that a market-maker who chooses to reduce the risk on her PnL distribution necessarily reduces her inventory risk; conversely, a market-maker who reduces her inventory risk also reduces the risk on her PnL distribution. However, this risk reduction also implies a reduction on the average PnL.

• It seems that the control provided by η on the linear MR is better than the control provided by γ on the exponential MR. However, this claim has to be taken with a grain of salt because, for η = 0.0001 and γ = 0.3 (where qualitatively both distributions are comparable to the martingale benchmark) the linear MR is less risky in terms of inventory whilst the exponential MR is less risky in terms of PnL distribution. That said, our numerical simulations show that an increase on η (from 0.0001 to 0.001) renders a more drastic reduction on inventory risk and stronger structural changes on PnL distribution than an increase on γ of the same order of magnitude (from 0.1 to 1).

On the directional bets and the risk profiles

As we have shown in our simulations, there is a clear (and expected) relation between risk and reward.

• When the market-maker makes a directional bet she improves her PnL up to 25% with respect to the martingale benchmark. However, by doing so she has to accept more risk, either on her PnL distribution (measured in terms of variance, skewness and kurtosis) or on her inventory.

• After a directional bet, the market maker can trade some of her excess PnL (over the martingale benchmark) for a direct control on her risks. If she chooses to gain direct control on her PnL distribution directly (resp. inventory risk) then she gains some reduction on her inventory risk (resp. PnL distribution), but the latter cannot be controlled directly. In that spirit, the marketmaker can choose to use an exponential MR strategy if she prices more dearly her risk on the PnL distribution and a linear MR strategy if her biggest concern is the inventory risk.

• In summary, directional bets enhance the PnL of the market-maker but add extra risk on her inventory and PnL distribution; both risks are positively correlated, so if she reduces one directly the other is indirectly reduced as well. Therefore, a market-maker in practice has to assess three factors, i.e. PnL increase, inventory risk and PnL distribution, in order to choose her optimal trading strategy because, in the current set-up at least, there is no strategy that is optimal in all three factors.

Further developments

• In our model there are no market orders, just limit orders. For the simulations we assumed that if δ + * ≤ 0 (resp. δ - * ≤ 0) then the market-maker sends a selling (resp. buying) market order that is executed at the mid-price, i.e. we assumed that the market spread is zero and their impact (or cost) on the PnL is |δ ± | ≥ 0.

• For a detailed market-impact analysis, we should add the market spread as another state variable and consider that the market-maker's limit orders affect both the market spread and the mid-price. For example, if she improves the best ask (resp. bid) then she reduces the market spread by one tick and pushes down (resp. up) the mid-price by half a tick.

• Another possibility is to incorporate market orders directly into the model and making a clear distinction between market and limit orders (see e.g. Guilbaud and Pham [START_REF] Guilbaud | Optimal high frequency trading with limit and market orders[END_REF]). However, the current framework does not seem to be easily extended for that purpose, which suggests that a new framework is needed. We are currently working on that direction.

Figure 4 .

 4 Figure 4. Simulation for µ = 1.00.

Figure 5 .

 5 Figure 5. Simulation for µ = 1.02.

Figure 7 .

 7 Figure 7. Histogram of the PnL as a function of γ for an exponential utility function (µ = 1.00). Blue: Linear martingale. Red: Exponential MR. Upper-Left: γ = 0.1, Lower-Left: γ = 0.3, Upper-right: γ = 0.7, Lower-Right: γ = 1.0.

Figure 8 .

 8 Figure 8. Comparing the PnL histograms for linear MR (η = 0.0001) and exponential MR (γ = 0.3).Blue: Linear MR. Red: Exponential MR.

Table 1A .

 1A Statistics of the PnL distribution. µ = 0.98, η = 0. ABM = arithmetic Brownian Motion without drift, MR = mean-reverting. The "Sharpe" value is the normalised return per risk unit, i.e. mean / std dev.

			linear ABM linear MR exp ABM exp MR
	PnL	mean	11.039	14.290	10.668	11.084
		std dev	1.013	13.678	0.356	0.520
		"Sharpe"	10.90	1.04	29.97	21.32
		skewness	0.075	-0.550	0.026	-1.008
		kurtosis	5.721	4.692	3.043	6.176
		Jarque Bera	30947.9	16970.7	19.1	58949.6
		VaR 5%	9.430	-10.909	10.086	10.177
		VaR 1%	8.346	-25.726	9.842	9.465
	Inv	mean	0.087	-333.297	0.004	-3.311
		std dev	33.258	418.200	7.672	15.227
		skewness	-0.005	0.622	0.003	0.007
		kurtosis	2.981	2.346	3.001	3.010
		Jarque Bera	1.8	8232.0	0.2	1.3
		Q(T) 90%	[-55,55]	[-847,463]	[-13,13]	[-28,22]
			linear ABM linear MR exp ABM exp MR
	PnL	mean	10.982	11.576	10.607	10.945
		std dev	0.412	1.541	0.347	0.444
		"Sharpe"	26.66	7.51	30.57	24.65
		skewness	0.023	-2.164	0.038	-0.605
		kurtosis	3.016	11.418	3.007	4.460
		Jarque Bera	10.2	373263.9	24.7	14982.6
		VaR 5%	10.308	8.640	10.041	10.183
		VaR 1%	10.026	5.794	9.809	9.691
	Inv	mean	-0.020	-1.739	0.008	-0.785
		std dev	5.025	8.428	4.574	5.696
		skewness	0.006	0.004	-0.001	-0.003
		kurtosis	3.024	2.995	2.965	3.000
		Jarque Bera	3.0	0.4	5.2	0.2
		Q(T) 90%	[-8,8]	[-16,12]	[-8,8]	[-10,9]

Table 1B .

 1B Statistics of the PnL distribution. µ = 0.98, η = 0.0001.

			linear ABM linear MR exp ABM exp MR
	PnL	mean	10.435	10.494	10.000	10.234
		std dev	0.336	0.364	0.324	0.342
		"Sharpe"	31.06	28.83	27.47	29.92
		skewness	0.023	-0.158	0.037	-0.034
		kurtosis	3.013	3.306	3.021	3.073
		Jarque Bera	9.5	806.5	24.3	41.2
		VaR 5%	9.886	9.889	9.470	9.672
		VaR 1%	9.659	9.594	9.252	9.426
	Inv	mean	-0.001	-0.018	0.005	-0.020
		std dev	1.667	1.680	1.674	1.673
		skewness	-0.011	-0.014	-0.010	-0.001
		kurtosis	3.004	3.030	3.022	3.029
		Jarque Bera	2.2	7.1	3.6	3.5
		Q(T) 90%	[-3,3]	[-3,3]	[-3,3]	[-3,3]

Table 1C .

 1C Statistics of the PnL distribution. µ = 0.98, η = 0.001.

			linear ABM linear MR exp ABM exp MR
	PnL	mean	11.030	13.457	10.670	11.039
		std dev	0.992	11.708	0.356	0.501
		"Sharpe"	11.12	1.15	29.97	22.03
		skewness	0.014	-0.676	0.022	-0.864
		kurtosis	5.666	5.176	2.996	5.632
		Jarque Bera	29607.8	27355.7	8.4	41315.1
		VaR 5%	9.460	-8.180	10.086	10.167
		VaR 1%	8.368	-22.205	9.850	9.524
	Inv	mean	-0.021	0.055	-0.013	0.031
		std dev	33.126	468.213	7.682	15.245
		skewness	-0.009	0.000	0.007	0.007
		kurtosis	2.990	1.849	3.024	2.988
		Jarque Bera	1.8	5523.2	3.2	1.5
		Q(T) 90%	[-55,55]	[-732,731]	[-13,13]	[-25,25]

Table 2A .

 2A Statistics of the PnL distribution. µ = 1.00, η = 0.

			linear ABM linear MR exp ABM exp MR
	PnL	mean	10.982	11.453	10.606	10.913
		std dev	0.411	1.317	0.345	0.433
		"Sharpe"	26.72	8.70	30.74	25.20
		skewness	0.029	-2.070	0.036	-0.554
		kurtosis	3.031	10.934	2.992	4.443
		Jarque Bera	17.7	333649.4	22.1	13802.7
		VaR 5%	10.312	8.956	10.043	10.178
		VaR 1%	10.037	6.601	9.813	9.699
	Inv	mean	-0.006	-0.046	0.016	-0.018
		std dev	5.034	8.378	4.570	5.664
		skewness	-0.003	0.006	0.008	-0.001
		kurtosis	2.992	2.974	3.008	2.969
		Jarque Bera	0.5	3.4	1.5	4.0
		Q(T) 90%	[-8,8]	[-14,14]	[-7,8]	[-9,9]

Table 2B .

 2B Statistics of the PnL distribution. µ = 1.00, η = 0.0001.

			linear ABM linear MR exp ABM exp MR
	PnL	mean	10.436	10.483	10.000	10.226
		std dev	0.335	0.356	0.325	0.338
		"Sharpe"	31.15	29.45	30.77	30.25
		skewness	0.031	-0.097	0.031	-0.023
		kurtosis	3.001	3.162	2.988	3.069
		Jarque Bera	15.5	266.4	16.5	28.6
		VaR 5%	9.888	9.889	9.469	9.668
		VaR 1%	9.658	9.623	9.246	9.431
	Inv	mean	-0.011	-0.009	0.005	0.006
		std dev	1.673	1.684	1.672	1.676
		skewness	0.006	-0.004	-0.001	0.002
		kurtosis	3.026	2.992	3.025	3.026
		Jarque Bera	3.6	0.5	2.7	3.0
		Q(T) 90%	[-3,3]	[-3,3]	[-3,3]	[-3,3]

Table 2C .

 2C Statistics of the PnL distribution. µ = 1.00, η = 0.001.

			linear ABM linear MR exp ABM exp MR
	PnL	mean	11.038	14.230	10.670	11.081
		std dev	1.011	13.653	0.356	0.522
		"Sharpe"	10.92	1.04	29.97	21.23
		skewness	0.015	-0.554	0.021	-0.987
		kurtosis	5.638	4.702	2.986	5.941
		Jarque Bera	29009.6	17181.5	8.0	52288.0
		VaR 5%	9.435	-10.866	10.084	10.168
		VaR 1%	8.332	-26.196	9.849	9.451
	Inv	mean	0.045	335.118	-0.013	3.297
		std dev	33.253	417.302	7.641	15.295
		skewness	0.008	-0.626	-0.003	0.006
		kurtosis	3.004	2.358	3.024	2.992
		Jarque Bera	1.2	8241.5	2.5	0.9
		Q(T) 90%	[-55,55]	[-459,847]	[-13,13]	[-22,28]

Table 3A .

 3A Statistics of the PnL distribution. µ = 1.02, η = 0.

			linear ABM linear MR exp ABM exp MR
	PnL	mean	10.982	11.581	10.605	10.945
		std dev	0.412	1.517	0.345	0.446
		"Sharpe"	26.66	7.63	30.74	24.54
		skewness	0.031	-2.074	0.028	-0.647
		kurtosis	3.052	10.611	3.011	4.642
		Jarque Bera	27.2	313076.3	13.9	18207.8
		VaR 5%	10.308	8.659	10.039	10.182
		VaR 1%	10.026	5.977	9.807	9.662
	Inv	mean	-0.003	1.712	0.005	0.795
		std dev	5.024	8.420	4.574	5.681
		skewness	0.001	-0.005	0.010	0.008
		kurtosis	3.006	2.976	2.993	2.980
		Jarque Bera	0.2	2.8	1.9	2.8
		Q(T) 90%	[-8,8]	[-12,16]	[-7,8]	[-9,10]

Table 3B .

 3B Statistics of the PnL distribution. µ = 1.02, η = 0.0001.

			linear ABM linear MR exp ABM exp MR
	PnL	mean	10.436	10.495	10.000	10.234
		std dev	0.335	0.365	0.326	0.342
		"Sharpe"	31.15	28.75	30.67	29.92
		skewness	0.038	-0.154	0.033	-0.061
		kurtosis	3.004	3.348	2.990	3.103
		Jarque Bera	24.0	901.2	18.5	106.9
		VaR 5%	9.890	9.889	9.468	9.669
		VaR 1%	9.666	9.598	9.252	9.414
	Inv	mean	0.006	0.022	-0.006	0.018
		std dev	1.678	1.677	1.672	1.671
		skewness	0.002	-0.001	-0.018	-0.001
		kurtosis	3.001	3.030	3.027	3.023
		Jarque Bera	0.0	3.8	8.5	2.2

Table 6 .

 6 Statistics of the PnL distribution for exponential MR as a function of γ (η = 0).

	7	γ = 1

Table 7 .

 7 Statistical comparison between linear MR η = 0.001 and exponential γ = 0.3. The % change is the variation of the linear MR with respect to the exponential MR. Skewness change is in absolute value.

	% change

Pietro FODRA • Mauricio LABADIE

Acknowledgements

The authors would like to thank Prof. Huyên Pham (University Paris-Diderot, France) and Prof. Mathieu Rosenbaum (University Pierre et Marie Curie, France) for their suggestions and kind advice.

• We have assumed that the mid-price is continuous, which can be interpreted in the discretisation for our simulations as assuming that the bid-ask spread of the market-maker is large with respect to small changes on the mid-price. This is true for assets whose spread is large with respect to the tick size (e.g. equities), but not for futures for which the spread is in average 1-2 ticks, so the smallest price move can make the market-maker's quotes cross the spread. Therefore, another framework is necessary to deal with assets whose spread and tick size are comparable. We will address this problem in a future work.

• Our mid-price S(t) is a Markov processes, just like in Avellaneda and Stoikov [START_REF] Avellaneda | High-frequency trading in a limit-order book[END_REF] and Lehalle et al. This means that the infinitesimal generator is local, S(t) has zero auto-correlation and the Hamilton-Jacobi-Bellman equation is a nonlinear PDE. Now, if we consider a mid-price process with non-zero autocorrelation (e.g. fractional Brownian Motion, multifractal processes or Hawkes process) then its infinitesimal operator is non-local and its HJB equation is an integro-differential equation. However, in this case we cannot invoke the Feynman-Kac formula, which means that our approach is no longer valid and another model is necessary (see e.g. Cartea and Jaimungai [START_REF] Cartea | Buy Low Sell High: A High Frequency Trading Perspective[END_REF] and [START_REF] Cartea | Risk Measures and Fine Tuning of High Frequency Trading Strategies[END_REF]).


Viet Tran Khac
email: tran-khv@leesu.enpc.fr

Brigitte Vinçon-Leite

Gilles Varrault

Bruno Jacques Lemaire

Bruno Tassin

Nilo Nascimento

Urban lakes: interaction between phytoplankton dynamics and trace metal speciation

Keywords: Urban lakes, trace metal, phytoplankton, transparent exopolymer particles, speciation

Urban lakes have a particular influence on the water cycle in urban catchments. Thermal stratification and a longer residence time of the water within the lake can boost the phytoplankton production. On the other hand, trace metals are naturally found in environment in trace amounts due to erosion and physic-chemical or biological alteration processes. Trace metals are essential to growth and reproduction of organisms. However, they are also well known for their toxic effects on animals and humans. Their ecotoxicity depends on metal properties and chemical speciation (particulate, dissolved: labile or bioavailable and inert fractions). The interaction between phytoplankton production and trace metal speciation is not well understood until present time. The aim of this research is twofold: (1) to analyze the driving processes of the phytoplankton production in an urban lake through physical-chemical and biological field data collection and (2) simultaneously to survey the chemical speciation of trace metals in this lake. To address these issues, a field survey and laboratory analysis, including physical-chemical, biological variables, organic matter and trace metal concentrations, have been designed and implemented in an urban study site: Lake Créteil in France

INTRODUCTION

Urban lakes are different from rural and natural lakes: they are specific elements of storm water drainage networks, highly artificial, small and shallow water bodies. These water bodies, such as ponds and grave-pits lakes, are common in urban and peri-urban landscape. Urban lakes play a key role in recreation and storing rainwater (Stephen Birch, 1999). According to Downing et al. (2006), 99,2% of water bodies in the world is 10ha or less. These aquatic ecosystems are highly impacted by watershed urbanization and by urban activities such as industry, traffic waste and drainage systems (Friese et al., 2010). Moreover, they are also very sensitive to a large variety of planetary environmental changes (PECs) such as increased CO 2 , increased air temperature and atmospheric pollution (Poff et al., 2002). Despite the abundance and the usefulness of these water bodies for societies, we notice a lack of research on these urban lakes in comparison with large and deep lakes (Jeppesen 1998, Moss 2010, Davies et al. 2008).

Because urban lakes are often small and shallow, they should respond rapidly and strongly to climatic fluctuations (Whitehead et al. 2007). Climate change could increase wet periods during winter and spring and increase drought during summer time. Consequently, water residence time would be modified (Bates et al. 2008). During extreme climatic events, rain runoff effluents lead to a decline of bacteriological and chemical water quality and thus impacting the sanitary state and the functioning of aquatic ecosystems (Characklis et al. 2005, Patz et al. 2005, Allan & Castilla 2007). Heat wave events can enhance phytoplankton production rates (Nõges 2009, Moijj et al. 2005). Urban lakes are generally polymictic lakes and they may be particularly vulnerable to changes in the water temperature and the mixing patterns caused by climatic extremes (Wagner & Adrian 2009).

All these environmental and anthropic disturbances can modify both hydrodynamic and physical-chemical conditions of urban lakes, which can favor phytoplankton development or even reinforce eutrophication due to higher nutriment internal cycling and better light climate (Nõges 2009, Bertolo et al. 1999 ).

Scientific context of the research

Two sources of trace metals exist in aquatic environment: natural source and anthropic source. Trace metals are naturally present in environment due to erosion and physical-chemical or biological alteration processes. Human activities represent an anthropic source in aquatic ecosystem. Trace metals enter urban water bodies via several pathways such as atmospheric deposition, industrial and domestic wastewater discharge, agricultural activities or storm water runoff (Thévenot et al, 2009).

Trace metals are vital components for growth, development and reproduction of living organisms (Buffle, 1988;Tessier et Turner, 1995). A lack of metals can be a limited factor for development of organism, for example a lack of iron can cause a diminution of phytoplankton development (Martin et Fitzwater, 1988;Martin et al., 1991). However, a high concentration of metals can be extremely toxic. Indeed, trace metal toxicity for aquatic organisms have been known for a long time and are considered as a serious threat for human and animal health (Bryan and Langston, 1992).

Ecotoxicity and bioavailability of trace metal are not directly related to total metal concentration but depends on the form and the nature of chemical species (Florence, 1986, Scoullos and Pavlidou, 1997, Lores and Pennock, 1998). In aquatic systems, trace metals can exist in many different chemical forms, which determine bioavailability and toxicity to aquatic organisms (Hirose, 2006, Morel et al., 2003). Trace metals can be particulate or dissolved (free or complexed to organic and inorganic ligands). Under different meteorological and physicalchemical conditions (pH, salinity, hardness of water, organic carbon content…), trace metal form can change (Forstner and Wittmann 1979). In dissolved metal fraction, metal can be labile (free or weakly linked to organic or inorganic ligand) or inert (no interaction with surrounding environment). Labile metal is defined as free metal or weakly linked to mineral ions (hydroxyl, carbonate) or to organic molecules (Thevenot, Meybeck et al. 2002). Labile metal is also defined as "easily modifiable, exchangeable by easy and rapid interaction with surrounding solution" (Tusseau-Vuillemin 2005). In other word, labile metal is constituted of cationic free form or weakly associated to mineral ions (Thevenot, Meybeck et al. 2002). Because of their mobility in aquatic environment, this fraction is directly related to bioavailability of metal. These labile metals, which can be easily linked to aquatic organisms, are a good indicator of toxicity. Therefore, it is important to study the speciation of trace metal in aquatic systems.

So far, we know that dissolved organic matter (DOM) plays a key role on trace metal speciation in aquatic systems. DOM can be divided into 2 categories: the humic substances and the nonhumic substances. According to Thurman (1985), humic substances are majority, up to 70% of DOM in comparison with non humic substances, estimated at 25%. Humic and fulvic acids are the most studied because they play a key role on complexing free metal ion in most of the aquatic systems. However, in urban areas, the non-humic, mainly hydrophilic DOM, can constitute a significant fraction as a result of treated or untreated wastewater discharge (Pernet-Coudrier et al., 2011). The binding ability of wastewater organic matter has been highlighted during an acute toxicity test of copper with Daphnia Magna (Pernet-Coudrier et al., 2008). In aquatic ecosystems, non-humic DOM can also be produced by microbiological activity, mainly phytoplankton and bacteria. The complexing ability of trace-metal of this natural aquagenic DOM, called extracellular polymeric substances (EPS) is still unclear.

Our research will focus on one type of EPS, called Transparent Exopolymer Particles (TEP). TEP are defined as particles formed from acid polysaccharides. During phytoplankton blooms in freshwater and marine systems, phytoplankton and bacteria abundantly exude TEPs. Diatoms are especially known for producing TEPs during their growth (Passow, U. 2002). TEPs can play a role on trace metal speciation because of the large affinity of trace metal to surface-active exopolymers (Tye, Jespen, & Lick, 1996). Between 40-90% of trace elements can be adsorbed by the TEPs in marine system (Passow, U. 2002). However, until now, the TEPs have been more widely studied in marine systems than in freshwaters. Moreover, very few results exist about the TEP characteristics in urban lakes.

In this context, the main question of our research is: How are the TEPs generated by phytoplankton and bacteria during blooms events influencing the trace-metal speciation in an urban lake?

To achieve this issue, monthly field campaigns aimed at measuring physical-chemical and biological variables and trace metal concentrations have been implemented in Lake Créteil (France). Physical parameters (temperature, pH, conductivity, oxygen, photosynthesis active radiations (PAR)) are monitored in situ with probes. Laboratory analysis (total phosphorus, phosphates, major cations and anions, TEP…) are performed on water samples. In this paper, the organization of the field campaigns and the laboratory analysis will be presented. The first results will be exposed and discussed.

A second study site (not be presented in this paper) with different climate and socio-economic context has been selected, Lake Pampulha (Brazil). Lake Créteil survey takes part of the research project PULSE (Peri-Urban Lakes, Society and Environment) supported by ANR (National Research Agency). Lake Pampulha survey takes part of the research project MAPLU (Storm water management project, funded by the Brazilian Agency for Funding Study and Projects -FINEP.

MATERIAL & METHODS

In order to investigate the interactions between phytoplankton and trace metals, we have designed a field survey aimed at monitoring the main environmental parameters driving the phytoplankton dynamics, the transparent exopolymer particles production which could consequently influence the metal speciation.

The field survey includes high-frequency measurements of water temperature aimed at providing continuous time series at 3 depths, giving access to the stratification and mixing patterns of the water column. Water temperature is an essential parameter, which directly, through algal growth rate, plays a key role in phytoplankton production. Moreover, mixing or stratification of the water column will control in a different way the growth of the algal species and will lead to different algal succession. These high-frequency measurements will allow us to understand at small time scale the physical control of phytoplankton dynamics.

The field campaigns also include, at a regular frequency, (1) vertical profiles of physical parameters aimed at describing the phytoplankton growth conditions: temperature, pH, conductivity, photosynthesis active radiations (PAR); (2) vertical profiles of parameters associated to the phytoplankton biomass: Chlorophyll a produced by 4 main phytoplankton groups and oxygen concentrations; (3) water sampling for further analysis of nutrients and TEPs concentrations and (4) specific water sampling and field filtration for trace metal speciation analysis .

More details on the field and the analysis protocols are given in a following paragraph.

Description of study site

Lake Créteil is located in Ile-de-France region (48°46'N 02°27'E), in an urbanized zone at 15km from Paris. The lake originates from an excavation of alluvial sediments near the confluence of the Rivers Seine and Marne. Its surface is 40.8 ha, the volume is 1.5 106 m3, the maximum depth 6m and the mean depth 4.5 m. The lake is mainly supplied by groundwater but also supplied by a stormwater outlet of the urbanized Mont-Mesly area. This lake is used for recreational activities (sailing, windsurfing, fishing...) and plays an important role in providing quality of life for the inhabitants of this highly urbanized area. Lake Créteil is polymictic with a density stratification lasting from days to weeks during the summer, which impacts the primary production. At the central point C (see Figure 2), three temperature sensors installed at three fixed depths (0.55m, 0.94 m and 3.62m) will measure continuously the water temperature. The measurement time step is 30mn. This sensor chain has been working since August 2011.

The monthly field campaigns

The monitoring campaigns, conducted on Lake Créteil, include three sampling points located on an horizontal transect, from the stormwater outlet to the lake outlet : the « S » point is close to the stormwater outlet, the « R » point is located next to a rich organic reedbed area and the central point « C », is located between S and R (Figure 2). For each sampling point, three depths are defined: surface (0.5 meters), middle (1.5 meters) and bottom (3.5 meters). For each campaign, a "lake average sample", named "M" is mixed from 9 samples collected at three depths at the three points (S1, S2, S3, C1, C2, C3, R1, R2, R3). Water samples are collected using a NISKIN bottle. For each campaign, 6 samples are collected for trace metal and TEP analysis: S2 (middle of the water column at point S), C1, C2, C3 (3 depths at central point), R2 (middle of the water column at point R) and M (lake average point).

Water sampling

Trace-metals

Trace metal analysis suffers of very high contamination risks. In order to avoid these risks, all materials used are in high-density polyethylene and have been prepared in a clean room. These materials have been washed according to a specific procedure: first washing cycle in detergent solution EXTRA at 5% prepared with reverse osmosis water during 24h and second washing cycle in nitric acid solution at 5% prepared with ultrapure water.

For each field campaign, 3 metal fractions are analyzed: (1) total metals, (2) total dissolved metals and (3) inert metals. Total metal, analyzed in the raw water sample (50mL), corresponds to particulate and dissolved metal fractions. Total dissolved metal is operationally defined as the filtrate of 0.45-micron filtration. The sample volume is 20mL. Inert metals are determined using chelating disks where a chelating resin retains labile metal. The chelating disk (EMPORE TM ) contains a chelating resin "Chelex 100", constituted of styrene-divinylbenzene co-polymer containing imino-diacetic acid groups. Imino-diacetic acid groups have a strong affinity toward transition metal that allows fixing them by a coordinate bond. Inert metals go through the chelating disk. The sample volume for inert metals is 20mL. The labile metal fraction is quantified by difference between total dissolved metal and inert metal.

The concentrations of the 3 metal fractions mentioned above are determined using inductively coupled plasma mass spectrometry (ICP-MS). Trace metals determined are: Al, Cr, Mn, Fe, Ni, Cu, Zn, Cd, Pb, Ti, V, Co, Ag.

Transparent exopolymer particles (TEP)

Transparent exopolymer particles (TEP) are operationally defined as particles retained on polycarbonate filters (0.45µm), which stains with the cationic dye alcian blue. The dying blue alcian solution 8GX, which can stain both sulfated and carboxylated polysaccharides, is prepared at concentration 0.02% m/v with 0.06% acetic acid to maintain pH at 2.5 (Passow & Alldredge, 1995b).

TEP concentration is measured using Passow and Alldredge's method (1995). 100 mL of raw water are filtered through 0.45µm pore size polycarbonate filters and stained with alcian blue.

Stained filters are rinsed with ultrapure water to remove excess blue alcian and keep in frozen until analysis. TEP stained by blue alcian are re-dissolved in 6mL of 80% sulphuric acid for 3h. Absorption at 787nm was measured spectrophotometrically. Measures are conducted in triplicate. Average concentration will be expressed as a standardized Xanthan equivalent in µg.l -1 (µgXeq.l -1 ).

Laboratory analysis

Water samples are analyzed in laboratory to determine nutrient (total phosphorus, phosphate, nitrate) and major anion and cation concentration. Total Phosphorus and phosphate analysis are measured according to AFNOR norm. Total Phosphorus is mineralized with persulfate and phosphate is colored by ascorbic acid. Absorption at 880nm is measured with a spectrophotometer. Nitrate concentration is determined by ionic chromatography.

PRELIMINARY RESULTS

Our preliminary results allow us to present the seasonal physical-chemical and biological patterns of Lake Creteil recorded from August 2011 to March 2012.

First results of metal concentration by ICP-MS were obtained very recently (30 th March 2012) and the data treatment is under progress. We are still working on the validation of these results.

TEP analysis planned for monthly field campaigns will start in May 2012. Presently, we are conducting laboratory tests in order to validate our final protocol.

Conductivity is very high in Lake Creteil; it was nearly constant at about 1600 µS/cm between October 2011 and March 2012. This value corresponds to historical data of the lake. Lake Créteil is rich on chloride and sulfate, which contribute to a high conductivity (Bonneau et al.

Sulfate and chloride

Ionic chromatography analysis confirmed that sulfate and chloride are the major anions in the lake. The concentration of these anions in the lake is largely higher in comparison to the others. Sulfate and chloride concentrations are respectively about 450 mg/L and 150 mg/L. We have to notice that in February, air temperature was negative during 2 weeks. The lake was frozen during the field campaign. It was not possible to reach the regular sampling points and we had to conduct measurements close to point position. It can explain why there was an extremely high phosphorus total concentration at the reedbed point R2. Phosphate concentration seems to be more homogenous for March than February.

Phosphorus total and phosphate

Chlorophyll a

Between October 2011 and March 2012 chlorophyll a concentration measured by Fluoroprobe BBE varies from 5 -20 µg/L.

CONCLUSION AND PERSPECTIVES

In order to investigate interactions between trace metal and phytoplankton, we suppose that organic matter at the lake central point C is essentially produced by phytoplankton. On the contrary, organic matter from storm water effluent will be mostly of urban origin and the organic matter in the reedbed area will have a more humic structure. Comparing and analyzing the data set of these 3 points will allow us to disentangle the influence of phytoplankton on trace metal speciation.

Monitoring of physical parameters and laboratory analysis will be useful to understand hydrodynamical and environmental conditions of the lake. In a second step, we intend to synthesize the knowledge of the lake functioning in an hydrodynamical and biological model, which can simulate and explain phytoplankton development in the lake. Understanding of phytoplankton dynamic helps us to understand better TEP production, which can explain trace metal speciation and bioavailability.

• Thurman. 1985. Organic geochemistry of natural waters. Dordrecht: Kluwer Academic Publishers Group. • Tushara Chaminda, G. G., F. Nakajima, et al. (2009). "Zn and Cu complexation with DOM in wastewater treatment plant effluent." 19ème congrès international de la SETAC. • Tye, R., Jepsen, R., & Lick, W. (1996). Effects of colloids, flocculation, particle size, and organic matter on the adsorption of hexachlorobenzene to sediments. Environmental Toxicology and Chemistry, 15, 643-651. 13-WWW- YES-2012-Tran-Khac-Paper-2012-06-15.doc

Figure 1 :

 1 Figure 1: Lake Créteil location (Créteil, France) (www.geoportail.fr)

Figure 2 :

 2 Figure 2: Location of the three sampling points on the lake Créteil (Left) and sampling depth (black rectangle) for laboratory analysis Probe measurements Vertical profiles of physical (salinity, temperature, oxygen, conductivity, photosynthetically active radiation (PAR), water transparency) and biological (chlorophyll a) parameters are also monitored in situ using probes: CTD Seabird43, LI-COR193 (underwater PAR sensor), Fluoroprobe BBE.

  2005) pH is about 7.8 between October 2011 and March 2012. Between August 2011 and March 2012, the surface temperature at the central point C was maximal at 23°C in 05-August 2011. Temperature profile was homogenous in the water column from December 2011 to March 2012. The temperature was minimal (1.6°C) in February 2012, when the lake was frozen. The temperature gradients recorded in the lake during the summer 2011 are low, around 0.4°/m.

•

  Wagner, C. & R. Adrian. 2009. Cyanobacteria dominance: Quantifying the effects of climate change. Limnology and Oceanography, 54: 2460-2468. • Whitehead, P.G., R.L. Wilby, R.W. Battarbee, M. Kernan & A.J.Wade. 2009. A review of the potential impacts of climate change on surface water quality. Hydrological Sciences, 54: 101-123.

Table 1 Total Phosphorus and phosphate concentrations for February and March 2012

 1

			February		March
	Points Phosphate (µg/L)	Total Phosphorus (µg/L)	Phosphate (µg/L)	Total Phosphorus (µg/L)
	M	9.1	36.5	2.5	137.4
	S2	21.2	52.6	2.8	32.3
	R2	8.2	305.8	2.5	56.7
	C1	29.9	65.6	5.5	34.1
	C2	9.0	35.2	1.7	33.3
	C3	8.2	36.6	5.6	35.2

12thWWW YES, Arcueil, France -20-26 June 2012

• Allan D.J., and Castillo M.M., 2007. Stream ecology: structure and function of running waters.

Springer, Dordrecht, the Netherlands, pp. 436.