On bid and ask side-specific tick sizes
Bastien Baldacci, Philippe Bergault, Joffrey Derchu, Mathieu Rosenbaum
OOn bid and ask side-specific tick sizes ∗ Bastien
Baldacci † Philippe
Bergault ‡ Joffrey
Derchu § Mathieu
Rosenbaum ¶ June 1, 2020
Abstract
The tick size, which is the smallest increment between two consecutive prices for a given asset,is a key parameter of market microstructure. In particular, the behavior of high frequency marketmakers is highly related to its value. We take the point of view of an exchange and investigatethe relevance of having different tick sizes on the bid and ask sides of the order book. Using anapproach based on the model with uncertainty zones, we show that when side-specific tick sizesare suitably chosen, it enables the exchange to improve the quality of liquidity provision.
Keywords:
High frequency trading, tick size, model with uncertainty zones, market making,stochastic control, viscosity solutions, financial regulation. ∗ This work benefits from the financial support of the Chaires Analytics and Models for Regulation, Financial Risk.Bastien Baldacci, Joffrey Derchu and Mathieu Rosenbaum gratefully acknowledge the financial support of the ERCGrant 679836 Staqamof. The authors would like to thank Shlomo Ahal for inspiring discussions during the conference"The Regulation and Operation of Modern Financial Markets 2019" in Reykjavik, Iceland. The authors are also gratefulto Mouhamad Dramé and Vincent Ragel for insightful comments. † École Polytechnique, CMAP, 91128, Palaiseau, France, [email protected]. ‡ Université Paris 1 Panthéon-Sorbonne. Centre d’Economie de la Sorbonne. 106, boulevard de l’Hôpital, 75013 Paris,France, [email protected] § École Polytechnique, CMAP, 91128, Palaiseau, France, joff[email protected] ¶ École Polytechnique, CMAP, 91128, Palaiseau, France, [email protected] a r X i v : . [ q -f i n . T R ] M a y Introduction
The tick size is the smallest increment between two consecutive prices on a trading instrument. It isfixed by the exchange or regulator and typically depends on both the price of the asset and the tradedvolume, see [16, 18]. It is a crucial parameter of market microstructure and its value is often subject ofdebates: a too small tick size leads to very frequent price changes whereas a too large tick size preventsthe price from moving freely according to the investor’s views. In this article, we focus on so-calledlarge tick assets, that is assets for which the spread is most of the time equal to one tick. Such assetsrepresent a large number of financial products, especially in Europe since MIFID II regulation, see [18].The tick size has a major influence on the ecosystem of financial markets, in particular on the activityof high frequency traders. Being usually considered as market makers, these agents are the main liq-uidity providers for most heavily traded financial assets. This means that they propose prices at whichthey are ready to buy (bid price) and sell (ask price) units of financial products. In [13], the authorsinvestigate the behavior of high frequency traders with respect to the relative tick size, which is definedas the ratio between the tick size and the price level. One of their findings is that everything else equal,stocks with a lower relative tick size attract a greater proportion of high frequency traders, see also[10, 19]. This is because they can rapidly marginally adjust their quotes to seize price priority. In thecase of a large tick asset, speed is still an important feature as market participants have to competefor queue priority in the order book, see [17, 20].Market makers (typically high frequency traders) face a complex optimization problem: making moneyout of the bid-ask spread (the difference between the bid and ask prices) while mitigating the inventoryrisk associated to price changes. This problem is usually addressed via stochastic control theory tools,see for example [2, 7, 8, 14, 15]. In classical market making models, the so-called efficient price, whichrepresents the market consensus on the value of the asset at a given time, around which the marketmaker posts his quotes, is a continuous semi-martingale. The quotes of the market maker are continu-ous in terms of price values and not necessarily multiple of the tick size. However, in actual financialmarkets, transaction prices are obviously lying on the discrete tick grid. This discreteness of prices isa key feature which cannot be neglected at the high frequency scale since it plays a fundamental rolein the design of market making strategies in practice. To get a more realistic market making model,2ne therefore needs to build a relevant continuous-time price dynamic with discrete state space to takeinto account this very important microstructural property of the asset.To this end, we borrow the framework of the model with uncertainty zones introduced in [21, 22]. Inthis model, transaction prices are discrete and the current transaction price is modified only when theunderlying continuous efficient price process crosses some predetermined zones. In our approach, wealso consider that there exists an efficient price that market participants have in mind when makingtheir trading decisions. Based on this efficient price, market participants build “fair” bid and “fair”ask prices. These two prices are lying on the tick grid and represent the views of market participantson reasonable and tradable values for buying and selling, regardless of any inventory constraint. In oursetting, depending on his views and his inventory constraint, the market maker chooses whether or notto quote a constant volume at these fair bid and ask prices. This is a stylized viewpoint as in practicethe market maker will probably quote a larger spread rather than not quoting at all. The marketmaker increases (resp. decreases) his current “fair” bid price if the efficient price becomes “sufficiently”higher (resp. lower) than his current fair bid price and similarly for the ask side. The mechanism todetermine whether the efficient price is sufficiently higher (resp. lower) than the current price is thatof the model with uncertainty zones, described in Section 2.Usual market making models include a symmetric running penalty for the inventory process, oftendefined as φ R T Q t dt where Q t is the inventory of the market maker at time t ∈ [0 , T ], φ > T is the end of the trading period. It is well-known, see for example [1], thatfor regulatory and operational reasons, market participants and especially market makers are reluctantto have a short inventory at the end of the trading day. This is mainly due to constraints imposed bythe exchange/regulator and to the overnight repo rate that they have to pay. This asymmetry betweenlong and short terminal inventory of the market maker gives the intuition of the potential relevance ofsome kind of asymmetry in the market design between buy and sell orders.If some kind of asymmetry is implemented at the microstructure level, it can have important conse-quences on the profit of exchanges, as it notably depends on the number of processed orders. Typicalways to optimize the number of orders on platforms are the choice of relevant tick sizes and suitable3ee schedules (which subsidize liquidity provision and tax liquidity consumption). In [12], the authorshighlight the importance of differentiating maker and taker fees in order to increase the trading rate. Inthe more recent studies [3, 11], optimal make-take fees schedules are designed based on contract theory.In this work, the asymmetry we consider is not between liquidity consumers and liquidity providersbut between buyers and sellers.The goal of this paper is to show the possible benefits for an exchange in terms of liquidity provisionof side-specific tick sizes. To this end, we build an agent-based model where a high frequency marketmaker acts on a large tick asset. The exchange is mitigating the activity on its platform by choosingsuitable tick sizes on the bid and ask sides. This means we have a different tick grid for buy and sellorders. For given the tick sizes chosen by the exchange, we formulate the stochastic control problemfaced by the market maker who needs to maximize his Profit and Loss (PnL for short) while controllinghis inventory risk, taking into account asymmetry between short and long inventory. We show exis-tence and uniqueness of a viscosity solution to the Hamilton-Jacobi-Bellman (HJB for short) equationassociated to this problem. Then, we derive a quasi-closed form for the optimal controls of the marketmaker (up to the value function). In particular, the role of the tick size in the decision of whetheror not to quote is explicit: essentially, a large tick size implies a large profit per trade for the marketmaker but less market orders coming from market takers, and conversely.Next, we solve the optimization problem of the exchange which can select optimal tick sizes knowingthe associated trading response of the market maker. In our model, the exchange earns a fixed fee whena transaction occurs. Therefore, its remuneration is related to the quality of the liquidity provided bythe market maker on its platform. Numerical results show that side-specific tick sizes are more suitablethan symmetric ones both for the market maker and the exchange. The former is able to trigger morealternations in the sign of market orders, which is beneficial both for spread pocketing and inventorymanagement (in contrast with the case where sequences of buy orders are followed by sequences of sellorders). The latter increases the number of transactions on its platform. We also show that a tick sizeasymmetry can offset short inventory constraints, therefore increasing the gains of both the marketmaker and the exchange. 4he paper is organized as follows. In Section 2, we give a reminder on the model with uncertaintyzones and explain how we revisit it for market making purposes. The market maker and exchange’sproblems are described in Section 3. We also state here our results about existence and uniqueness of aviscosity solution associated to the control problem of the market maker and derive its optimal controls.Finally, Section 4 is devoted to numerical results and their interpretations. Proofs are relegated to anappendix. In this section, we provide a reminder on the model with uncertainty zones introduced in [21, 22], andwe adapt it to the framework of a market making problem with side-specific tick values. It is commonlyadmitted that low frequency financial price data behave like a continuous Brownian semi-martingale.However this is clearly not the case for high frequency data. The model with uncertainty zones repro-duces sparingly and accurately the behavior of ultra high frequency transaction data of a large tickasset. It is based on a continuous-time semi-martingale efficient price and a one dimensional parameter η ∈ [0 , ]. The key idea of the model is that when a transaction occurs at some value on the tick grid,the efficient price is close enough to this value at the transaction time. This proximity is measuredthrough the parameter η .We define the efficient price ( S t ) t ∈ [0 ,T ] on a filtered probability space (Ω , F , P ) where T is the tradinghorizon. The logarithm of the efficient price ( Y t ) t ∈ [0 ,T ] is an F t -adapted continuous Brownian semi-martingale of the form Y t = log( S t ) = log( S ) + Z t a s d s + Z t σ s − d W s , where W is an F -Brownian motion, and ( σ t ) t ∈ [0 ,T ] is an F -adapted process with càdlàg paths and( a t ) t ∈ [0 ,T ] is F -progressively measurable. Transaction prices lie on two fixed tick grids, defined by { kα a , kα b } where α a (resp. α b ) is the tick size on the ask (resp. bid) side and k ∈ N . For 0 ≤ η i ≤ and i ∈ { a, b } , we define the zone U ik = [0 , ∞ ) × ( d ik , u ik ) with d ik = ( k + 12 − η i ) α i , u ik = ( k + 12 + η i ) α i . (2.1)5herefore U ak is a band of size 2 η a α a around the ask mid-tick grid value ( k + ) α a and U bk is a bandof size 2 η b α b around the bid mid-tick grid value ( k + ) α b . We call these bands the uncertainty zones.The zones on the bid and ask sides are characterized by the parameters η b , η a which control the widthof the uncertainty zones. We will see in the next section how the fair bid and ask prices are deducedfrom the efficient price dynamics across the uncertainty zones. In particular, the larger η i , the fartherfrom the last traded price (on the bid or ask side) the efficient price has to be so that a price changeoccurs. The idea behind the model with uncertainty zones is that, in some sense, market participantsfeel more comfortable when the asset price is constant than when it is constantly moving. However,there are times when the transaction price has to change because they consider that the last tradedprice value is not reasonable anymore.For sake of simplicity, we assume that transaction prices cannot jump by more than one tick. We alsodefine the time series of bid and ask transaction times leading to a price change as ( τ bj , τ aj ) j ≥ . Thelast traded bid or ask price process is characterized by the couples of transaction times and transactionprices with price changes ( τ j ; P iτ ij ) j ≥ where P iτ ij = S ( α i ) τ ij , the superscript ( α i ) denoting the rounding tothe nearest α i .The dynamics of the ( τ ij ) will be described in Section 3. One can actually show that the efficient pricecan be retrieved from transaction data using the equation S τ ij = S ( α i ) τ ij − α i ( 12 − η i )sgn (cid:16) S ( α i ) τ ij − S ( α i ) τ ij − (cid:17) , i ∈ { a, b } , j ∈ N . This formula is particularly useful in order to derive ultra high frequency estimators of volatility andcovariation (see [22]). The parameters η i can be estimated very easily. Let N ( a ) α i ,t and N ( c ) α i ,t be respectivelythe number of alternations and continuations of one tick over the period [0 , t ]. Then, an estimator of η i over [0 , t ] is given by ˆ η α i ,t = N ( c ) α i ,t N ( a ) α i ,t . We refer to [21, 22] for further details on these estimation procedures. In this paper, we use the model An alternation/continuation corresponds to two consecutive price changes in the opposite/same direction.
We consider a high frequency marker maker acting on an asset whose efficient price S t has the dynamicsd S t = σ d W t , where σ > η a , η b , andthe tick sizes α a , α b . If η a is small (resp. large), the market maker changes more (resp. less) frequentlyhis ask price, and similarly for the bid price with η b . This leads to the following definition of fair bidand ask prices of the market maker S a , S b : S at = S at − + α a { S t − S at − > ( + η a ) α a } − α a { S t − S at − < − ( + η a ) α a } ,S bt = S bt − + α b { S t − S bt − > ( + η b ) α b } − α b { S t − S bt − < − ( + η b ) α b } . Thus the fair bid (resp. ask) is modified when the efficient price is close enough to a new tradable priceon the tick grid with mesh α b (resp. α a ). Remark 3.1.
Note that in the case α a = α b , η a = η b , the fair best bid is equal to the fair best ask. Thismeans that at a given time, a buy or sell order would be at the same price. In this situation, in ourstylized view, the market maker would probably quote only on one side (bid or ask). It is consistent withthe standard form of the model with uncertainty zones, where, at a given time, transactions can onlyhappen only on one side of the market, depending on the location of the efficient price. Still, the market Note that we can have situations where the bid price is above the ask price. However, recall that S a and S b are onlyviews about the fair bid and ask prices under the constraint that they have to lie on the tick grids. aker collects the spread from transactions occurring at different times as it is the case in practice. We assume a constant volume of transaction equal to one. The market maker can choose to be presentor not for a transaction at the bid (with a price S b ) or at the ask (with a price S a ). The correspondingcash process at terminal time T is given by X T = Z T (cid:18) S at d N at − S bt d N bt (cid:19) , where the N it represent the number of transactions on the bid or ask side between 0 and t . In thisframework, the inventory of the market maker is given by Q t = N bt − N at ∈ Q = [ − ˜ q, ˜ q ] where ˜ q isthe risk limit of the market maker. For i ∈ { a, b } , the dynamics of N it is that of a point process withintensity λ ( ‘ it , Q t ) := λ‘ it κα i ) { φ ( i ) Q t > − q } , φ ( i ) = { i = a } − { i = b } . The process ‘ it ∈ { , } is the market maker’s control which lies in the set of F − predictable processeswith values in { , } denoted by L . The parameter κ > α i , and λ > S b (resp. S a ) he sets ‘ b = 0 (resp. ‘ a = 0) and conversely. In our large tickasset setting, the situation where the market maker is not present is a simplified way to model the casewhere the market maker’s quote is higher than the best possible limit. At a given time t ∈ [0 , T ], when ‘ bt = 0 (resp ‘ at = 0), the intensity of the point process N bt (resp. N at ) is equal to zero so that there areno incoming transactions. In addition to this, market takers are more confident to send market orderswhen the tick size is small, as the market maker has more flexibility to adjust his bid and ask prices. This explains the decreasing shape of the intensities of market order arrivals from market takers withrespect to the tick size. The chosen parametric form for the intensities ensures no degenerate behaviorwhen the tick size gets close to zero.The marked-to-market value of the market maker’s portfolio at time t is defined as Q t S t . His optimiza- When the tick size is smaller, the market takers are more willing to trade. This does not necessarily lead to a highernumber of orders as it depends on the market maker’s presence. ‘ ∈L E (cid:20) X T + Q T ( S T − AQ T ) − φ Z T Q s ds − φ − Z T | Q s | Q s < ds (cid:21) , (3.1)where φ > φ − > , T ] and AQ T , with A >
0, is a penalty termfor the terminal inventory position regardless of its sign. In this setting, the market maker wishes tohold a terminal inventory close to zero because of the quadratic penalty AQ T . The term φ R T Q s ds penalizes long or short positions over the trading period. Problem (3.1) can of course be rewritten assup ‘ ∈L E (cid:20) Q T ( S T − AQ T ) + Z T n S as λ ( ‘ as ) − S bs λ ( ‘ bs ) − φQ s − φ − Q s Q s < o ds (cid:21) . We define the corresponding value function h defined on the open set D = (cid:26) ( S a , S b , S ) ∈ α a Z × α b Z × R such that − (cid:18)
12 + η a (cid:19) α a < S − S a < (cid:18)
12 + η a (cid:19) α a (cid:19) . and − (cid:18)
12 + η b (cid:19) α b < S − S b < (cid:18)
12 + η b (cid:19) α b (cid:27) by h ( t, S a , S b , S, q ) = sup ‘ ∈L t E t,S a ,S b ,S,q (cid:20) Q T ( S T − AQ T ) + Z Tt (cid:26) S as λ ( ‘ as ) − S bs λ ( ‘ bs ) − φQ s − φ − Q s Q s < (cid:27) d s (cid:21) , (3.2)where L t denotes the restriction of admissible controls to [ t, T ]. We define the boundary ∂ D of D as ∂ D = (cid:26) ( S a , S b , S ) ∈ α a Z × α b Z × R such that S − S a = ± (cid:18)
12 + η a (cid:19) α a and/or S − S b = ± (cid:18)
12 + η b (cid:19) α b . (cid:27) , and write ¯ D = D ∪ ∂ D . For given ( S a , S b ), if ( S a , S b , S ) ∈ ∂ D , it means that S corresponds to anefficient price value that triggers a modification of the fair bid or ask price.9he Hamilton-Jacobi-Bellman equation associated to this stochastic control problem is given by0 = ∂ t h ( t, S a , S b , S, q ) − φq − φ − q q< + 12 σ ∂ SS h ( t, S a , S b , S, q )+ λ κα a ) max ‘ a ∈{ , } ‘ a (cid:18) S a + h ( t, S a , S b , S, q − ‘ a ) − h ( t, S a , S b , S, q ) (cid:19) + λ κα b ) max ‘ b ∈{ , } ‘ b (cid:18) ( − S b ) + h ( t, S a , S b , S, q + ‘ b ) − h ( t, S a , S b , S, q ) (cid:19) , (3.3)for ( t, S a , S b , S, q ) ∈ [0 , T ) × D × Q , with terminal condition h ( T, S a , S b , S, q ) = q ( S − Aq ) . (3.4)Let us consider the function h defined in 3.2. For ( t, S a , S b , S, q ) ∈ [0 , T ) × ∂D ×Q , and ( t n , S an , S bn , S n , q n ) n ∈ N a sequence in [0 , T ) × D × Q which converges to ( t, S a , S b , S, q ), we will show that h ( t n , S an , S bn , S n , q n )converges independently of the sequence and we denote by h ( t, S a , S b , S, q ) its limit. On [0 , T ) × ∂ D×Q ,we will show the following boundary conditions (which we will naturally impose for the solution of 3.3):0 = { S − S a =( + η a ) α a , S − S b < ( + η b ) α b } (cid:16) h ( t, S a + α a , S b , S, q ) − h ( t, S a , S b , S, q ) (cid:17) + { S − S a < ( + η a ) α a , S − S b =( + η b ) α b } (cid:16) h ( t, S a , S b + α b , S, q ) − h ( t, S a , S b , S, q ) (cid:17) + { S − S a =( + η a ) α a , S − S b =( + η b ) α b } (cid:16) h ( t, S a + α a , S b + α b , S, q ) − h ( t, S a , S b , S, q ) (cid:17) + { S − S a = − ( + η a ) α a , S − S b > − ( + η b ) α b } (cid:16) h ( t, S a − α a , S b , S, q ) − h ( t, S a , S b , S, q ) (cid:17) + { S − S a > − ( + η a ) α a , S − S b = − ( + η b ) α b } (cid:16) h ( t, S a , S b − α b , S, q ) − h ( t, S a , S b , S, q ) (cid:17) + { S − S a = − ( + η a ) α a , S − S b = − ( + η b ) α b } (cid:16) h ( t, S a − α a , S b − α b , S, q ) − h ( t, S a , S b , S, q ) (cid:17) . (3.5)In other words, the value function varies continuously when the efficient price leaves an uncertaintyzone and the prices S a and S b are modified. In the following, we say that a function defined on[0 , T ) × D × Q satisfies the continuity conditions if it satisfies (3.5).The following proposition is of particular importance for the existence and uniqueness of a viscositysolution associated to the control problem of the market maker. Note that, as the terminal condition does not depend on S a and S b , it also satisfies this boundary condition on ∂ D . roposition 3.2. The function h defined in Equation (3.2) is continuous on D and satisfies thecontinuity conditions (3.5) . The proof is given in Appendix A.1 and relies on the specific structure of our model based on hittingtimes of a Brownian motion. We now state the main theorem of this article, whose proof is relegatedto Appendix A.2.
Theorem 1.
The value function h is the unique continuous viscosity solution to Equation (3.3) on [0 , T ) × D × Q with terminal condition (3.4) and satisfying the continuity conditions. The value function depends on five variables. However, as ( S a , S b ) takes value in α a N × α b N , it canessentially be reduced to three variables as we now explain. For any ( i, j ) ∈ N , we introduce thefunction h i,j defined on[0 , T ] × (cid:18) α a i − ( 12 + η a ) α a , α a i + ( 12 + η a ) α a (cid:19) ∩ (cid:18) α b j − ( 12 + η b ) α b , α b j + ( 12 + η b ) α b (cid:19)| {z } = D i,j ×Q by h i,j ( t, S, q ) = h ( t, α a i, α b j, S, q ). Then h i,j is the solution of the following HJB equation:0 = ∂ t h i,j ( t, S, q ) − φq − φ − ( q ) − q< + 12 σ ∂ SS h i,j ( t, S, q )+ λ κα a ) max ‘ a ∈{ , } ‘ a (cid:18) α a i + h i,j ( t, S, q − ‘ a ) − h i,j ( t, S, q ) (cid:19) + λ κα b ) max ‘ b ∈{ , } ‘ b (cid:18) − α b j + h i,j ( t, S, q + ‘ b ) − h i,j ( t, S, q ) (cid:19) , with terminal condition h i,j ( T, S, q ) = q ( S − AQ ) and natural Dirichlet boundary conditions for S ∈ D i,j : h i,j ( t, S, q ) = h i +1 ,j ( t, S, q ) { S − α a i =( + η a ) α a , S − α b j< ( + η b ) α b } + h i,j +1 ( t, S, q ) { S − α a i< ( + η a ) α a , S − α b j =( + η b ) α b } + h i +1 ,,j +1 ( t, S, q ) { S − α a i =( + η a ) α a , S − α b j =( + η b ) α b } + h i − ,j ( t, S, q ) { S − α a i = − ( + η a ) α a , S − α b j> − ( + η b ) α b } + h i,j − ( t, S, q ) { S − α a i> − ( + η a ) α a , S − α b j = − ( + η b ) α b } + h i − ,j − ( t, S, q ) { S − α a i = − ( + η a ) α a , S − α b j = − ( + η b ) α b } . From this, we derive the optimal controls of the market maker as ‘ ?a ( t, i, j, S, q ) = { α a i + h i,j ( t,S,q − − h i,j ( t,S,q ) > } ,‘ ?b ( t, i, j, S, q ) = {− α b j + h i,j ( t,S,q +1) − h i,j ( t,S,q ) > } . The practical interest of Theorem 1 is that it allows us to compute the value function and optimalcontrols based on a finite difference scheme. Examples of computations of the value function are givenin Section 4 and Appendix A.3. Having described the problem of the market maker, we now turn tothe optimization problem of the platform.
The market maker acts on a platform whose goal is to maximize the number of market orders on [0 , T ].The intensities of arrival of market orders are functions of ‘ a , ‘ b , which are themselves functions of α a , α b . We assume that the platform is risk-neutral and earns a fixed taker cost c > Therefore its optimization problem is defined assup ( α a ,α b ) ∈ R E l ?a ,l ?b h X pT i , More complex fee schedules can be handled in this framework. We can for example add a component which isproportional to the amount of cash traded. l ?a , l ?b ) of the market maker and X pt = c ( N at + N bt ).It is easy to observe that this problem boils down to maximizing the function v defined below over R : v ( α a , α b ) := E "Z T cλ ( ‘ ?a ( t, S at , S bt , S t , q t )1 + ( κα a ) + ‘ ?b ( t, S at , S bt , S t , q t )1 + ( κα b ) ) d t . Here we clearly see the tradeoff of the platform. A small tick size α a increase the term (1 + ( κα a ) ) − .This is because it attracts more buy market orders. However, the optimal control ‘ ?,a is more oftenequal to zero: the gain of the market maker may be too small if he quotes at the price S a , thereforehe regularly sets ‘ ?,a = 0. The problem is similar on the bid side. On the other hand, a large tick sizeincreases the gain of the market maker if a transaction occurs, but decreases the number of marketorders sent by market takers, hence decreasing the trading volume.We study numerically this problem in the next section by computing the value of v on a two dimensionalgrid and finding its maximum. In this section, we show from numerical experiments the benefits of side-specific tick values in termsof increase of their value function for both the market maker and the platform. Also, we fix referencevalues η and α . From them, to choose the parameter η i associated to a given tick size α i we use aresult from [10] which gives the new value of the parameter η i in case of a change of tick size from α to α i . This formula writes η i = η r α α i . (4.1)In the following, we only consider values of α a and α b such that the underlying remains a large tickasset both on the bid and ask sides, that is η a ≤ , η b ≤ .For the first experiments, we set T = 40 s , q = 5, σ = 0 . s − , A = 0 . , κ = 10 , φ = 0 . , λ = 4, η = 0 . α = 0 .
01 which correspond to reasonable values to model a liquid asset. To remain in13he large tick regime, we investigate values of α i satisfying 0 . ≤ α i ≤ .
05 for i = a, b . In this section we investigate the case where α a = α b . We plot in Figure 1 the value functions of themarket maker and the exchange, respectively h and v , for various values of α = α a = α b . We fix theefficient price S = 10 .
5, the inventory q = 0 and we only consider values of α so that 0 . /α ∈ N .Figure 1: Value function h (on the left) and v (on the right) for φ − = 0 in blue, φ − = 0 . φ − = 0 .
005 in green, as a function of α = α a = α b .When φ − = 0, the value of the exchange reaches its maximum at α ’ . φ − leadsto a reduction of the number of transactions. However the optimal tick value for the exchange is notsignificantly modified.The optimal tick value for the market maker is larger than that of the exchange. This is because theexchange is only interested in attracting orders while the market maker’s gain per trade (not takinginto account the inventory risk) is linear with respect to the tick value. The trade-off of the exchange isthe following: on the one hand, he would like to implement a quite small tick value (to attract marketorders) but on the other hand, he must ensure a reasonable presence of market maker.When φ − increases, the value function of the market maker decreases, for all tick values. This is nosurprise since φ − corresponds to an inventory penalization, hence reducing the market maker’s PnL.In Figure 2, we substract the value function when φ − = 0 to the other value functions displayed inFigure 1. We remark that for the market maker, the larger the tick the more significant the penalization14f short inventory in terms of value function. We observe the opposite phenomenon for the exchange:the difference is essentially slightly increasing with respect to α . In particular, we see a quite strongimpact of the penalization on the value function of the exchange when the tick size is small.Figure 2: Variation of the functions h and v (difference between φ − = 0 in blue, φ − = 0 . φ − = 0 .
005 in green, and φ − = 0 as a function of α = α a = α b .We now study the case of side-specific tick values. We set α b = 0 . α a vary. We plot thevalue functions of the market maker and the exchange in Figure 3. Again we observe that both valueFigure 3: Value function h (on the left) and v (on the right) as functions of α a , for α b = 0 . φ − = 0 in blue, φ − = 0 . φ − = 0 .
005 in green.functions are decreasing with respect to φ − . From the point of view of the market maker, having nonside-specific tick values is sub-optimal, even in the case φ − = 0. This is because when the two tick valuesare different, it is possible for S a to be greater than S b and orders to arrive with the same intensities15n both sides: the market maker can collect the spread. It is not possible in the non side-specific case,where the market maker can only pocket the spread from buy and sell orders at two different times.Side-specific tick values are also clearly beneficial for the exchange. The transaction flow increases for α a > α b because of the good liquidity provided by the market maker, and for α a < α b because of thehigh number of incoming market orders. Remark 4.1.
Remark that with shifted grids (same tick values on both sides but with a grid shiftedcompared to the other), those additional opportunities for the market maker would remain. In section4.3, we will see however, that, from the point of view of the exchange, side-specific tick values are muchmore interesting. φ − We plot the two-dimensional value functions of the market maker and the exchange for side-specifictick values.First we take φ − = 0 in Figure 4. We note that the opportunity for the market maker mentioned aboveremains present for all tick values and that the value functions are symmetric around the axis α b = α a (side-specific tick values are preferred). Furthermore, we see that the exchange prefers smaller tickvalues than the market maker. The optimal values for the exchange lie on an anti-diagonal which goesfrom ( α a = 0 . , α b = 0 . α a = 0 . , α b = 0 . α a , α b ) mentioned above.If the tick values are too large the intensities of the market orders become too small and the numberof transactions diminishes. If both ticks are too small, the market maker does not trade much becausethe gain per trade becomes too little compared to the inventory cost (recall that the intensity of marketorders is upper bounded). However, the case where one tick is quite small and the other is large issuitable for the market maker: for example, if α a < α b his strategy is to be long and liquidate hislong position fast if needed thanks to the small value of α a which ensures a large number of incomingmarket orders. This explains why the optimal tick values given by the exchange are side-specific andsymmetric with respect to the axis α a = α b . More precisely, the choice of ticks ( α a = 0 . , α b = 0 . α a = 0 . , α b = 0 . h (on the left) and v (on the right) as functions of α a and α b , for φ − = 0.We now plot in Figure 5 the value function for φ − = 0 . α b = α a .Figure 5: Value function h (on the left) and v (on the right) as functions of α a and α b , for φ − = 0 . φ − = 0 .
005 and when φ − = 0 asa function of α a , α b . We see that the added component is not symmetric regarding to the axis α b = α a and both the market maker and the exchange tend to prefer the case α b > α a . It is particularly clear forthe market maker’s problem where the difference between the values at ( α a = 0 . , α b = 0 . h (on the left) and v (on the right) as functions of α a and α b , between the case φ − = 0 .
005 and the case φ − = 0.( α a = 0 . , α b = 0 . .
03 which is roughly 10% of the value function. Indeed, asexplained above, having α b quite large and α a rather small essentially ensures that the market makercan maintain a positive inventory all along the trading trajectory: attractive PnL for incoming buyorders and possibility to quickly reduce a positive inventory.The exchange is also more satisfied by the choice α b > α a . To see that more clearly, we fix α a = 0 . h and v , as functions of α b , for different values of φ .Figure 7: Value functions h and v for α a = 0 . α b , for different values of φ .The value function of the market maker is increasing in α b . This is the same phenomenon as alreadyobserved in Figure 5. The value of the exchange has a maximum which is increasing in φ − : as thepenalization gets more side-specific, the optimal tick values displayed by the exchange become moreasymmetric. Indeed, for φ − = 0, the optimum is reached for α b ’ . φ − = 0 . α b ’ . φ − . By choosing a new tick size optimally when going from φ − = 0 to φ − = 0 . α b = 0 .
024 would lead to a lossof 15%. Note that the compensation can be total for the market maker (and even exceeds the loss)but is only partial for the exchange.
A suitable choice of tick values by the exchange is a subtle equilibrium. If the platform imposes thesame tick value on the bid and ask sides, it has to be sufficiently large to ensure significant PnL pertrade for the market maker and therefore good liquidity provision, and sufficiently small to attractmarket orders from market takers. When allowing for side-specific tick values with no constraint onshort inventory, the optimal tick values for the exchange are of the form ( α ? , α ? ) or symmetrically( α ? , α ? ) with α ? < α ? . In this case, the market maker can take advantage from additional tradingopportunities and increase his activity. The exchange benefits from this situation because of the highernumber of trades on his platform. Moreover, when there is a penalty for short inventory positionsof the market maker, there is only one optimal couple of tick values. In this case, the market makerand subsequently the exchange prefer α b > α a and the difference between α a and α b at the optimumbecomes larger. Finally, note that side-specific tick values could have subtle consequences in a multi-platform setting. This issue is left for further study, as well as the situation where market takers aremore strategic in their execution. A Appendix
A.1 Proof of Proposition 3.2
First we prove the continuity of h on D × [0 , T ).Let q ∈ Q , t ∈ [0 , T ), ( s a , s b , s ) ∈ D . Note that { s ∈ R , ( s a , s b , s ) ∈ D} is an open interval containing s , which we denote by ( s ← , s → ). If the process S t starts from a point s ∈ ( s ← , s → ) with S at = s a and S bt = s b , S at and S bt will not jump as long as S t stays in ( s ← , s → ). We will prove that the function19 t, s ) ∈ [0 , T ) × ( s ← , s → ) h ( t, s a , s b , s, q ) is continuous at ( t , s ).We fix η >
0. There is a ball with positive diameter B in [0 , T ) × ( s ← , s → ) centered on ( t , s ) andsome (cid:15) > t , s ) ∈ B , then E [ τ − t | S t = s ] < η, E [ τ − t | S t = s ] < η, (A.1) P [ τ < T | S t = s ] > − η, P [ τ < T | S t = s ] > − η, (A.2)and inf ‘ ∈L P [ inf t ≤ s ≤ τ Q s = sup t ≤ s ≤ τ Q s = q | S t = s , Q t = q ] > − η, inf ‘ ∈L P [ inf t ≤ s ≤ τ Q s = sup t ≤ s ≤ τ Q s = q | S t = s , Q t = q ] > − η, (A.3)where we write τ = T ∧ inf { t ≥ t , S t ,s t = ( s ∨ s ) + (cid:15) or S t ,s t = ( s ∧ s ) − (cid:15) } ,τ = T ∧ inf { t ≥ t , S t ,s t = ( s ∨ s ) + (cid:15) or S t ,s t = ( s ∧ s ) − (cid:15) } . The quantities τ and τ are stopping times such that t ≤ τ ≤ T a.s. and t ≤ τ ≤ T a.s. Weimpose s ← < ( s ∧ s ) − (cid:15) < s < ( s ∨ s ) + (cid:15) < s → , s ← < ( s ∧ s ) − (cid:15) < s < ( s ∨ s ) + (cid:15) < s → for any ( t , s ) ∈ B by taking a smaller ball B and a smaller (cid:15) if necessary. In particular, this tells usthat if ( S t , S at , S bt ) = ( s , s a , s b ), S at does not jump between t and τ . Similarly, if ( S t , S at , S bt ) =( s , s a , s b ), S bt does not jump between between t and τ .Let some arbitrary ( t , s ) ∈ B and τ and τ the associated stopping times. Using the dynamic These inequalities can be attained independently of the control ‘ as S is independent from Q . h ( t , s a , s b , s , q ) = sup ‘ ∈L E " h ( τ ,S aτ ,S bτ ,S τ ,Q τ )+ Z τ t n − φQ t − φ − Q t Q t < o dt (cid:12)(cid:12)(cid:12)(cid:12) S t = s , S at = s a , S bt = s b , Q t = q . This can be rewritten as h ( t , s a , s b , s , q ) = sup ‘ ∈L E " X ¯ q ∈Q (cid:16) h ( τ , s a , s b , S τ , ¯ q ) { Q τ =¯ q } + Z τ t n − φQ t − φ − Q t Q t < o { Q t =¯ q } (cid:17) dt (cid:12)(cid:12)(cid:12) S t = s , Q t = q . Recalling that h is bounded, we deduce by A.3 that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ( t , s a , s b , s , q ) − sup ‘ ∈L E (cid:20) h ( τ , s a , s b , S τ , q ) + Z τ t n − φq − φ − ( q ) − o dt (cid:12)(cid:12)(cid:12) S t = s , Q t = q (cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cη for a constant C , independent from ( t , t , s , s ), and ( q ) − = q q< . The expectation above does notdepend on the control ‘ , hence we drop the supremum and fix an arbitrary control ‘ = 0. We denoteby E the expectation under the probability measure given by this control. The expectation neitherdepends on the process Q t , so we drop the conditioning with respect to Q t .This leads to (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ( t , s a , s b , s , q ) − E (cid:20) h ( τ , s a , s b , S τ , q ) + Z τ t n − φq − φ − ( q ) − o dt (cid:12)(cid:12)(cid:12) S t = s (cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cη.
Similarly, starting from ( t , s a , s b , s , q ) with ( t , s ) ∈ B , we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ( t , s a , s b , s , q ) − E (cid:20) h ( τ , s a , s b , S τ , q ) + Z τ t n − φq − φ − ( q ) − o dt (cid:12)(cid:12)(cid:12) S t = s (cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cη, | h ( t , s a , s b , s , q ) − h ( t , s a , s b , s , q ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E (cid:20) h ( τ , s a , s b , S τ , q ) + Z τ t n − φq − φ − ( q ) − o dt | S t = s (cid:21) − E (cid:20) h ( τ , s a , s b , S τ , q ) + Z τ t n − φq − φ − ( q ) − o dt | S t = s (cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 2 Cη ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E (cid:20) h ( τ , s a , s b , S τ , q ) | S t = s (cid:21) − E (cid:20) h ( τ , s a , s b , S τ , q ) | S t = s (cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) φq − φ − ( q ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) E h τ − t | S t = s i + E h τ − t | S t = s i(cid:19) + 2 Cη. (A.4)Using (A.1), we get (cid:12)(cid:12)(cid:12)(cid:12) E h τ − t | S t = s i + E h τ − t | S t = s i(cid:12)(cid:12)(cid:12)(cid:12) < η. (A.5)Also, the conditional laws (cid:16) τ | S t = s , S τ = ( s ∨ s ) + (cid:15), τ < T (cid:17) , (cid:16) τ | S t = s , S τ = ( s ∧ s ) − (cid:15), τ < T (cid:17) , (cid:16) τ | S t = s , S τ = ( s ∨ s ) + (cid:15), τ < T (cid:17) , (cid:16) τ | S t = s , S τ = ( s ∧ s ) − (cid:15), τ < T (cid:17) , have bounded continuous densities, which we denote by f , + , f , − , f , + and f , − respectively (see forexample [5], Formula 3 . . S τ and S τ , we can write (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E (cid:20) h ( τ , s a , s b , S τ , q ) | S t = s (cid:21) − E (cid:20) h ( τ , s a , s b , S τ , q ) | S t = s (cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) X j ∈{ + , −} Z T h ( t, s a , s b , s j , q )( f ,j ( t ) P [ S τ = s j , τ < T | S t = s ] − f ,j ( t ) P [ S τ = s j , τ < T | S t = s ]) dt (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) E h h ( τ , s a , s b , S τ , q ) { S τ = s + ,S τ = s − }∪{ τ = T } | S t = s i(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) E h h ( τ , s a , s b , S τ , q ) { S τ = s + ,S τ = s − }∪{ τ = T } | S t = s i(cid:12)(cid:12)(cid:12)(cid:12) (A.6)where s + = s ∨ s + (cid:15) and s − = s ∧ s − (cid:15) . Remark that the event S τ = s + , S τ = s − , S t = s happens only if τ = T so that P [ { S τ = s + , S τ = s − } ∪ { τ = T }| S t = s ] < η by (A.2). Similarly22 [ { S τ = s + , S τ = s − } ∪ { τ = T }| S t = s ] < η by (A.2). As a consequence, using again (A.4), (A.5)and the fact that h is bounded, we get (cid:12)(cid:12)(cid:12) h ( t , s a , s b , s , q ) − h ( t , s a , s b , s , q ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) X j ∈{ + , −} Z h ( t, s a , s b , s j , q )( f ,j ( t ) P [ S τ = s j , τ < T | S t = s ] − f ,j ( t ) P [ S τ = s j , τ < T | S t = s ]) dt (cid:12)(cid:12)(cid:12)(cid:12) + (2 | φq − φ − ( q ) − | + 4 C ) η. Recall that the f , + , f , − , f , + and f , − depend on s and t . We have | P [ S τ = s + , τ < T | S t = s ] f , + − P [ S τ = s + , τ < T | S t = s ] f , + | → ( t ,s ) → ( t ,s ) , | P [ S τ = s − , τ < T | S t = s ] f , − − P [ S τ = s − , τ < T | S t = s ] f , − | → ( t ,s ) → ( t ,s ) , T ] directly by [5] Formula 3.0.6 and Appendix 11. Having fixed (cid:15) and using again [5]Formula 3.0.6 and Appendix 11, we see that the above functions are uniformly bounded with respect to( s , t ) ∈ B . So, using that h is bounded, we can apply the dominated convergence theorem to deducethat (cid:12)(cid:12)(cid:12)(cid:12) X j ∈{ + , −} Z T h ( t, s a , s b , s j , q )( f ,j ( t ) P [ S τ = s j , τ < T | S t = s ] − f ,j ( t ) P [ S τ = s j , τ < T | S t = s ]) dt (cid:12)(cid:12)(cid:12)(cid:12) → ( t ,s ) → ( t ,s ) . Thus we have shown that h is continuous at the point ( t , S a , S b , s , q ). The case t = T is treated thesame way.The continuity conditions can be proved using the same lines: fixing q ∈ Q , t ∈ [0 , T ) and ( S a , S b , s ) ∈ ∂ D , choosing ( t , s ) close enough to ( t , s ) and applying the dynamic programming principle between t and τ , and t and τ , for τ and τ two well-chosen stopping times (for example τ = T ∧ inf n t > t , S t = s + (cid:15) or S t = s ← ∧ s o , τ = T ∧ inf n t > t , S t = s + (cid:15) or S t = s ← ∧ s o . with (cid:15) > .2 Proof of Theorem 1 We first prove that the value function of the market maker’s problem is indeed a viscosity solution of(3.3).
Proposition A.1.
The value function h is a continuous viscosity solution on [0 , T ) × D × Q of (3.3) .Furthermore, h ( T, S a , S b , S, q ) = q ( S − Aq ) for all ( S a , S b , S, q ) ∈ D × Q , and h ( t, S a , S b , S, q ) = { S − S a =( + η a ) α a , S − S b < ( + η b ) α b } h ( t, S a + α a , S b , S, q )+ { S − S a < ( + η a ) α a , S − S b =( + η b ) α b } h ( t, S a , S b + α b , S, q )+ { S − S a =( + η a ) α a , S − S b =( + η b ) α b } h ( t, S a + α a , S b + α b , S, q )+ { S − S a = − ( + η a ) α a , S − S b > − ( + η b ) α b } h ( t, S a − α a , S b , S, q )+ { S − S a > − ( + η a ) α a , S − S b = − ( + η b ) α b } h ( t, S a , S b − α b , S, q )+ { S − S a = − ( + η a ) α a , S − S b = − ( + η b ) α b } h ( t, S a − α a , S b − α b , S, q ) , for all ( t, S a , S b , S, q ) ∈ [0 , T ) × ∂ D × Q . Proof.
Let ( ¯ S a , ¯ S b , ¯ q ) ∈ α a N × α b N × Q , and ( t n , S n ) n ∈ N ∈ [0 , T ] × R be a sequence such that t n → n → + ∞ ˆ t ∈ [0 , T ) ,S n → n → + ∞ ˆ S ∈ R ,h ( t n , ¯ S a , ¯ S b , S n , ¯ q ) → n → + ∞ h (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) , with ( ¯ S a , ¯ S b , ˆ S ) ∈ D . Without loss of generality we can assume that ( ¯ S a , ¯ S b , S n ) ∈ D for all n ∈ N . Let us first consider the case ˆ t = T . Let us take two arbitrary controls ‘ as = ‘ bs = 0, for all s ∈ [0 , T ) , then for all n ∈ N we have h ( t n , ¯ S a , ¯ S b , S n , ¯ q ) ≥ E t n , ¯ S a , ¯ S b ,S n , ¯ q Q T ( S T − AQ T ) − φ Z Tt n Q s d s − φ − Z Tt n Q s Q s < d s , h ( T, ¯ S a , ¯ S b , ˆ S, ¯ q ) ≥ ¯ q ( ˆ S − A ¯ q ) . Now let us consider the case ˆ t < T.
Let ϕ : [0 , T ) × D × Q → R be a continuous function, C in t , C in S and such that 0 = min [0 ,T ) ×D ( h − ϕ ) = ( h − ϕ )(ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) . We also assume that h = ϕ only at thepoint (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ). Let us assume that there exists η > η ≤ ∂ t ϕ (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) − φ ¯ q − φ − ¯ q ¯ q< + 12 σ ∂ SS ϕ (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q )+ λ κα a ) max ‘ a ∈{ , } ‘ a (cid:18) ¯ S a + ϕ (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q − ‘ a ) − ϕ (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) (cid:19) + λ κα b ) max ‘ b ∈{ , } ‘ b (cid:18) − ¯ S b + ϕ (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q + ‘ b ) − ϕ (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) (cid:19) . Then we must have0 ≤ ∂ t ϕ ( t, ¯ S a , ¯ S b , S, ¯ q ) − φ ¯ q − φ − ¯ q ¯ q< + 12 σ ∂ SS ϕ ( t, ¯ S a , ¯ S b , S, ¯ q )+ λ κα a ) max ‘ a ∈{ , } ‘ a (cid:18) ¯ S a + ϕ ( t, ¯ S a , ¯ S b , S, ¯ q − ‘ a ) − ϕ ( t, ¯ S a , ¯ S b , S, ¯ q ) (cid:19) + λ κα b ) max ‘ b ∈{ , } ‘ b (cid:18) − ¯ S b + ϕ ( t, ¯ S a , ¯ S b , S, ¯ q + ‘ b ) − ϕ ( t, ¯ S a , ¯ S b , S, ¯ q ) (cid:19) , for all ( t, S ) ∈ B = (cid:16) (ˆ t − r, ˆ t + r ) ∩ [0 , T ) (cid:17) × (cid:16) ˆ S − r, ˆ S + r (cid:17) for some r >
0. Without loss of generality,we can assume that B contains the sequence ( t n , S n ) n and that for all ( t, S ) ∈ B, we have ( ¯ S a , ¯ S b , S ) ∈D . We can choose the value of η such that ϕ ( t, ¯ S a , ¯ S b , S, ¯ q ) + η ≤ h ( t, ¯ S a , ¯ S b , S, ¯ q )on ∂ p B := (cid:16) (ˆ t − r, ˆ t + r ) ∩ [0 , T ) (cid:17) × (cid:16)n ˆ S − r o ∪ n ˆ S + r o(cid:17) ! ∪ { ˆ t + r } × h ˆ S − r, ˆ S + r i ! . We can alsoassume that ϕ ( t, S a , S b , S, q ) + η ≤ h ( t, S a , S b , S, q ) , t, S a , S b , S, q ) ∈ ˜ B with˜ B = ( ( t, ¯ S a , ¯ S b , S, q ) (cid:12)(cid:12)(cid:12) ( t, S ) ∈ B, q ∈ { ¯ q − , ¯ q + 1 } ∩ Q ) . We introduce the set B D = n ( t, ¯ S a , ¯ S b , S, ¯ q ) (cid:12)(cid:12)(cid:12) ( t, S ) ∈ B o and set π n = inf { t ≥ t n | ( t, S at , S bt , S t , q t ) / ∈ B D } with S it n = ¯ S i , q t n = ¯ q, S t n = S n , where the processesare controlled by ‘ at = { S at + ϕ ( t,S at ,S bt ,S t ,q t − − − ϕ ( t,S at ,S bt ,S t ,q t − ) > } ,‘ bt = {− S bt + ϕ ( t,S at ,S bt ,S t ,q t − +1) − ϕ ( t,S at ,S bt ,S t ,q t − ) > } . Using Itô’s formula and noting that S at , S bt do not jump between t n and π n , we derive ϕ ( π n , S aπ n , S bπ n , S π n ,q π n ) = ϕ ( t n , ¯ S a , ¯ S b , S n , ¯ q ) + Z π n t n (cid:26) ∂ t ϕ ( t, S at , S bt , S t , q t ) + 12 σ ∂ SS ϕ ( t, S at , S bt , S t , q t ) (cid:27) dt + Z π n t n λ ( ‘ at ) n ϕ ( t, S at , S bt , S t , q t − − ‘ at ) − ϕ ( t, S at , S bt , S t , q t − ) o dt + Z π n t n λ ( ‘ bt ) n ϕ ( t, S at , S bt , S t , q t − + ‘ bt ) − ϕ ( t, S at , S bt , S t , q t − ) o dt + Z π n t n σ∂ S ϕ ( t, S at , S bt , S t , q t ) dW t + Z π n t n n ϕ ( t, S at , S bt , S t , q t − − ‘ at ) − ϕ ( t, S at , S bt , S t , q t − ) o d ˜ N at + Z π n t n n ϕ ( t, S at , S bt , S t , q t − + ‘ bt ) − ϕ ( t, S at , S bt , S t , q t − ) o d ˜ N bt ≥ ϕ ( t n , ¯ S a , ¯ S b , S n , ¯ q ) − Z π n t n n S at λ ( ‘ at ) − S bt λ ( ‘ bt ) − φq t − φ − q t q t < o dt + Z π n t n σ∂ S ϕ ( t, S at , S bt , S t , q t ) dW t + Z π n t n n ϕ ( t, S at , S bt , S t , q t − − ‘ at ) − ϕ ( t, S at , S bt , S t , q t − ) o d ˜ N at + Z π n t n n ϕ ( t, S at , S bt , S t , q t − + ‘ bt ) − ϕ ( t, S at , S bt , S t , q t − ) o d ˜ N bt . ϕ ( t n , ¯ S a , ¯ S b , S n , ¯ q ) ≤ E " ϕ ( π n , S aπ n , S bπ n , S π n , q π n ) + Z π n t n n S at λ ( ‘ at ) − S bt λ ( ‘ bt ) − φq t − φ − q t q t < o dt . Thus ϕ ( t n , ¯ S a , ¯ S b , S n , ¯ q ) ≤ − η + E " h ( π n , S aπ n , S bπ n , S π n , q π n ) + Z π n t n n S at λ ( ‘ at ) − S bt λ ( ‘ bt ) − φq t − φ − q t q t < o dt . As ϕ ( t n , ¯ S a , ¯ S b , S n , ¯ q ) → n → + ∞ ϕ (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) = h (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) ,h ( t n , ¯ S a , ¯ S b , S n , ¯ q ) → n → + ∞ h (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) , there exists n ∈ N such that for all n ≥ n , h ( t n , ¯ S a , ¯ S b , S n , ¯ q ) − η ≤ ϕ ( t n , ¯ S a , ¯ S b , S n , ¯ q ) and we deduce h ( t n , ¯ S a , ¯ S b , S n , ¯ q ) ≤ − η E " h ( π n , S aπ n , S bπ n , S π n , q π n ) + Z π n t n n S at λ ( ‘ at ) − S bt λ ( ‘ bt ) − φq t − φ − q t q t < o dt , which contradicts the dynamic programming principle. Therefore,0 ≥ ∂ t ϕ (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) − φ ¯ q − φ − ¯ q ¯ q< + 12 σ ∂ SS ϕ (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q )+ λ κα a ) max ‘ a ∈{ , } ‘ a (cid:18) ¯ S a + ϕ (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q − ‘ a ) − ϕ (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) (cid:19) + λ κα b ) max ‘ b ∈{ , } ‘ b (cid:18) − ¯ S b + ϕ (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q + ‘ b ) − ϕ (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) (cid:19) , and h is a viscosity supersolution of the HJB equation on [0 , T ) × D × Q . The proof for the subsolution part is identical.For a given ρ >
0, we introduce the function ˜ h such that˜ h ( t, S a , S b , S, q ) = e ρt h ( t, S a , S b , S, q ) ∀ ( t, S a , S b , S, q ) ∈ [0 , T ] × D × Q . h is a viscosity solution of the following HJB equation:0 = − ρ ˜ h ( t, S a , S b , S, q ) + ∂ t ˜ h ( t, S a , S b , S, q ) − φq − φ − q q< + 12 σ ∂ SS ˜ h + λ κα a ) max ‘ a ∈{ , } ‘ a (cid:18) e ρt S a + ˜ h ( t, S a , S b , S, q − ‘ a ) − ˜ h ( t, S a , S b , S, q ) (cid:19) + λ κα b ) max ‘ b ∈{ , } ‘ b (cid:18) e ρt ( − S b ) + ˜ h ( t, S a , S b , S, q + ‘ b ) − ˜ h ( t, S a , S b , S, q ) (cid:19) , (A.7)and we see that proving a maximum principle for (A.7) is equivalent to proving one for (3.3). Definition A.2.
Let U : [0 , T ) × D × Q → R be continuous with respect to ( t, S ) . For (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) ∈ [0 , T ) × D × Q , we say that ( y, p, A ) ∈ R is in the subjet P − U (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) (resp. the superjet P + U (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) ) if U ( t, ¯ S a , ¯ S b , S, ¯ q ) ≥ U (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) + y ( t − ˆ t ) + p ( S − ˆ S ) + 12 A ( S − ˆ S ) + o (cid:16) | t − ˆ t | + | S − ˆ S | (cid:17) , (cid:18) resp. U ( t, ¯ S a , ¯ S b , S, ¯ q ) ≤ U (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) + y ( t − ˆ t ) + p ( S − ˆ S ) + 12 A ( S − ˆ S ) + o (cid:16) | t − ˆ t | + | S − ˆ S | (cid:17) (cid:19) , for all ( t, S ) such that ( t, ¯ S a , ¯ S b , S, ¯ q ) ∈ [0 , T ) × D × Q . We also define ¯ P − U (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) as the set of points ( y, p, A ) ∈ R such that there exists a sequence ( t n , ¯ S a , ¯ S b , S n , ¯ q, y n , p n , A n ) ∈ [0 , T ) × D × Q × P − U ( t n , ¯ S a , ¯ S b , S n , ¯ q ) satisfying ( t n , ¯ S a , ¯ S b , S n , ¯ q, y n , p n , A n ) → n → + ∞ (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q, y, p, A ) . The set ¯ P + U (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) is defined similarly. Let us recall one of the definitions of viscosity solutions which we are going to use for the proof of theuniqueness.
Lemma A.3.
A continuous function ˜ U is a viscosity supersolution (resp. subsolution) to (A.7) on [0 , T ) ×D×Q if and only if for all (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) ∈ [0 , T ) ×D×Q and all (ˆ y, ˆ p, ˆ A ) ∈ ¯ P − U (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q )28 resp. ¯ P + U (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) ), we have − ρ ˜ U (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) + ˆ y − φ ¯ q − φ − ¯ q q< + 12 σ A + λ κα a ) max ‘ a ∈{ , } ‘ a (cid:18) e ρt ¯ S a + U (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q − ‘ a ) − U (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) (cid:19) + λ κα b ) max ‘ b ∈{ , } ‘ b (cid:18) e ρt ( − ¯ S b ) + U (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q + ‘ b ) − U (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) (cid:19) ≤ (resp. ≥ ). We refer to [6] for a proof of this result. We can now state a maximum principle from which theuniqueness can be easily deduced:
Proposition A.4.
Let U (resp. V ) be a continuous viscosity supersolution (resp. subsolution) of (3.3) with polynomial growth on [0 , T ) × D × Q and satisfying the continuity conditions (3.5) . If U ≥ V on { T } × D × Q , then U ≥ V on [0 , T ) × D × Q . Proof.
As before, we introduce the functions ˜ U and ˜ V such that˜ U ( t, S a , S b , S, q ) = e ρt U ( t, S a , S b , S, q ) and ˜ V ( t, S a , S b , S, q ) = e ρt V ( t, S a , S b , S, q ) . Then ˜ U and ˜ V are respectively viscosity supersolution and subsolution of Equation (A.7) on [0 , T ) ×D × Q with ˜ U ≥ ˜ V on { T } × D × Q . To prove the proposition, it is enough to prove that ˜ U ≥ ˜ V on[0 , T ) × D × Q . We proceed by contradiction. Let us assume that sup [0 ,T ) ×D×Q ˜ V − ˜ U > . Let p ∈ N ∗ suchthat lim k S k → + ∞ sup t ∈ [0 ,T ] ,q ∈Q ( S,S a ,S b ) ∈D | ˜ U ( t, S a , S b , S, q ) | + | ˜ V ( t, S a , S b , S, q ) | k S k p = 0 , where k · k is the Euclidian norm. Then there exists (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) ∈ [0 , T ] × D × Q such that0 < ˜ V (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) − ˜ U (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) − φ (ˆ t, ˆ S, ˆ S, ¯ q )= sup ( t,S a ,S b ,S,q ) ˜ V ( t, S a , S b , S, q ) − ˜ U ( t, S a , S b , S, q ) − φ ( t, S, S, q ) , where φ ( t, S, R, q ) := εe − µt (1 + k S k p + k R k p ) , ε > , µ >
0. The choice of the function φ allows us to look for a supremum in a bounded setwith respect to ( S, S a , S b ). Then the supremum is either reached for a point in [0 , T ] × D × Q or on[0 , T ] × ∂ D × Q (recall that D is open). But the continuity conditions tell us that if the supremum isreached on [0 , T ] × ∂ D × Q , it is also reached in [0 , T ] × D × Q . Since ˜ U ≥ ˜ V on { T } × D × Q , it isclear that ˆ t < T .Then, for all n ∈ N ∗ , we can find ( t n , S n , R n ) ∈ [0 , T ] × R such that ( ¯ S a , ¯ S b , S n ) , ( ¯ S a , ¯ S b , R n ) ∈ D and0 < ˜ V ( t n , ¯ S a , ¯ S n , S n , ¯ q ) − ˜ U ( t n , ¯ S a , ¯ S b , R n , ¯ q ) − φ ( t n , S n , R n , ¯ q ) − n | S n − R n | − (cid:16) | t n − ˆ t | + | S n − ˆ S | (cid:17) = sup ( t,S,R ) ˜ V ( t, ¯ S a , ¯ S b , S, ¯ q ) − ˜ U ( t, ¯ S a , ¯ S b , R, ¯ q ) − φ ( t, S, R, ¯ q ) − n | S − R | − (cid:16) | t − ˆ t | + | S − ˆ S | (cid:17) . Then, we have ( t n , S n , R n ) → n → + ∞ (ˆ t, ˆ S, ˆ S ) , and ˜ V ( t n , ¯ S a , ¯ S b , S n , ¯ q ) − ˜ U ( t n , ¯ S a , ¯ S b , R n , ¯ q ) − φ ( t n , S n , R n ) − n | S n − R n | − (cid:16) | t n − ˆ t | + | S n − ˆ S | (cid:17) → n → + ∞ ˜ V (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) − ˜ U (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) − φ (ˆ t, ˆ S, ˆ S ) . For n ∈ N ∗ , let us write for ( t, S, R ) ∈ [0 , T ] × R ϕ n ( t, S, R ) := φ ( t, S, R ) + n | S − R | + | t − ˆ t | + | S − ˆ S | . Then Ishii’s Lemma (see [4, 9]) guarantees that for any η > , we can find( y n , p n , A n ) ∈ ¯ P + ˜ V ( t n , ¯ S a , ¯ S b , S n , ¯ q ) and ( y n , p n , A n ) ∈ ¯ P − ˜ U ( t n , ¯ S a , ¯ S b , R n , ¯ q ) such that y n − y n = ∂ t ϕ n ( t n , S n , R n ) , ( p n , p n ) = ( ∂ S ϕ n , − ∂ R ϕ n ) ( t n , S n , R n )30nd A n − A n ≤ H SR ϕ n ( t n , S n , R n ) + η ( H SR ϕ n ( t n , S n , R n )) , where H SR ϕ n ( t n , ., . ) denotes the Hessian of ϕ n ( t n , ., . ) . Applying Lemma A.3, we get ρ (cid:16) ˜ V ( t n , ¯ S a , ¯ S b , S n , ¯ q ) − ˜ U ( t n , ¯ S a , ¯ S b , R n , ¯ q ) (cid:17) ≤ y n − y n + 12 σ ( A n − A n )+ λ κα a ) max ‘ a ∈{ , } ‘ a (cid:18) e ρt n ¯ S a + ˜ V ( t n , ¯ S a , ¯ S b , S n , ¯ q − ‘ a ) − ˜ V ( t n , ¯ S a , ¯ S b , S n , ¯ q ) (cid:19) + λ κα b ) max ‘ b ∈{ , } ‘ b (cid:18) e ρt n ( − ¯ S b ) + ˜ V ( t n , ¯ S a , ¯ S b , S n , ¯ q + ‘ b ) − ˜ V ( t n , ¯ S a , ¯ S b , S n , ¯ q ) (cid:19) − λ κα a ) max ‘ a ∈{ , } ‘ a (cid:18) e ρt n ¯ S a + ˜ U ( t n , ¯ S a , ¯ S b , R n , ¯ q − ‘ a ) − ˜ U ( t n , ¯ S a , ¯ S b , R n , ¯ q ) (cid:19) − λ κα b ) max ‘ b ∈{ , } ‘ b (cid:18) e ρt n ( − ¯ S b ) + ˜ U ( t n , ¯ S a , ¯ S b , R n , ¯ q + ‘ b ) − ˜ U ( t n , ¯ S a , ¯ S b , R n , ¯ q ) (cid:19) . Moreover, we have H SR ϕ n ( t n , S n , R n ) = ∂ SS φ ( t n , S n , R n ) + 2 n + 12( S n − ˆ S ) ∂ SR φ ( t n , S n , R n ) − n∂ SR φ ( t n , S n , R n ) − n ∂ SR φ ( t n , S n , R n ) + 2 n . It follows that ρ (cid:16) ˜ V ( t n , ¯ S a , ¯ S b , S n , ¯ q ) − ˜ U ( t n , ¯ S a , ¯ S b , R n , ¯ q ) (cid:17) ≤ ∂ t φ ( t n , S n , R n ) + 2( t n − ˆ t )+ 12 σ (cid:16) ∂ SS φ ( t n , S n , R n ) + ∂ RR φ ( t n , S n , R n ) + 2 ∂ SR φ ( t n , S n , R n ) + 12( S n − ˆ S ) (cid:17) + ηC n + λ κα a ) max ‘ a ∈{ , } ‘ a (cid:18) e ρt n ¯ S a + ˜ V ( t n , ¯ S a , ¯ S b , S n , ¯ q − ‘ a ) − ˜ V ( t n , ¯ S a , ¯ S b , S n , ¯ q ) (cid:19) + λ κα b ) max ‘ b ∈{ , } ‘ b (cid:18) e ρt n ( − ¯ S b ) + ˜ V ( t n , ¯ S a , ¯ S b , S n , ¯ q + ‘ b ) − ˜ V ( t n , ¯ S a , ¯ S b , S n , ¯ q ) (cid:19) − λ κα a ) max ‘ a ∈{ , } ‘ a (cid:18) e ρt n ¯ S a + ˜ U ( t n , ¯ S a , ¯ S b , R n , ¯ q − ‘ a ) − ˜ U ( t n , ¯ S a , ¯ S b , R n , ¯ q ) (cid:19) − λ κα b ) max ‘ b ∈{ , } ‘ b (cid:18) e ρt n ( − ¯ S b ) + ˜ U ( t n , ¯ S a , ¯ S b , R n , ¯ q + ‘ b ) − ˜ U ( t n , ¯ S a , ¯ S b , R n , ¯ q ) (cid:19) , C n does not depend on η. Therefore, as the maximums on the right-hand side are always positive,we deduce that for all n ∈ N ∗ , ρ (cid:16) ˜ V ( t n , ¯ S a , ¯ S b , S n , ¯ q ) − ˜ U ( t n , ¯ S a , ¯ S b , R n , ¯ q ) (cid:17) ≤ ∂ t φ ( t n , S n , R n ) + 2( t n − ˆ t )+ 12 σ (cid:16) ∂ SS φ ( t n , S n , R n ) + ∂ RR φ ( t n , S n , R n ) + 2 ∂ SR φ ( t n , S n , R n ) + 12( S n − ˆ S ) (cid:17) + λ κα a ) max ‘ a ∈{ , } ‘ a (cid:18) e ρt n ¯ S a + ˜ V ( t n , ¯ S a , ¯ S b , S n , ¯ q − ‘ a ) − ˜ V ( t n , ¯ S a , ¯ S b , S n , ¯ q ) (cid:19) + λ κα b ) max ‘ b ∈{ , } ‘ b (cid:18) e ρt n ( − ¯ S b ) + ˜ V ( t n , ¯ S a , ¯ S b , S n , ¯ q + ‘ b ) − ˜ V ( t n , ¯ S a , ¯ S b , S n , ¯ q ) (cid:19) . As ˜ V is continuous and ( t n , S n ) n converges to (ˆ t, ˆ S ), the last two terms are bounded from above bysome constant M. Then by sending n to infinity, we get ρ (cid:16) ˜ V (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) − ˜ U (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) (cid:17) ≤ ∂ t φ (ˆ t, ˆ S, ˆ S )+ 12 σ (cid:16) ∂ SS φ (ˆ t, ˆ S, ˆ S ) + ∂ RR φ (ˆ t, ˆ S, ˆ S ) + 2 ∂ SR φ (ˆ t, ˆ S, ˆ S ) (cid:17) + M. For µ > ρ > V (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) − ˜ U (ˆ t, ¯ S a , ¯ S b , ˆ S, ¯ q ) < , hence the contradiction. A.3 Effects of the uncertainty zones on h We keep the same parameters as in Section 4 and take α a = 0 .
01 and α b = 0 . h ) on some small range of values of S . Note that S = 10 . • S a = α a b S/α a c and S b = α b j S/α b k (green dots), • S a = α a b S/α a c and S b = α b l S/α b m (red dash-dots), • S a = α a d S/α a e and S b = α b j S/α b k (orange dash), • S a = α a d S/α a e and S b = α b l S/α b m (blue solid).32igure 8: Value function h of the market maker for q = 0, as a function of S .Note that depending on the value of S , some of those cases can be excluded. The solid vertical redand black lines represent respectively the values on the ask ( α a N ) and the bid grid ( α b N ). The dottedvertical lines represent the limits of the uncertainty zones on each side.In the uncertainty zones, the value function h depends non-trivially on S a and S b . Thanks to thecontinuity conditions at the boundaries of the uncertainty zones, we get a smooth behavior of h when S exits a zone. Remark that when S ∈ [10 ± (( − η a ) α a ) ∧ (( − η b ) α b )], necessarily S a = S b = 10.In our example, α a > α b and ( − η a ) α a > ( − η b ) α b . So, if S is in (10 + ( − η b ) α b , ( − η a ) α a ),necessarily S a = 10, but S b can take either the value 10 or 10 + α b depending on whether S comes fromhigher prices or lower prices. This is why there are two curves in the interval (10+( − η b ) α b , ( − η a ) α a ).At ( − η a ) α a , two additional curves appear as S a can also be two different values. References [1] T. Adrian, A. Capponi, E. Vogt, and H. Zhang. Intraday market making with overnight inventorycosts.
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