High-frequency trading in a limit order book

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1 Downloaded By: [Mt Sinai School of Medicine, Levy Lirary] At: 3:19 April 00 Quantitative Finance, Vol., No. 3, April 00, 17 4 High-frequency trading in a limit order ook MARCO AVELLANEDA SASHA STOIKOV* Mathematics, New York University, 51 Mercer Street, New York, NY 1001, USA (Received 4 April 006; in final form 3 April 007) 1. Introduction The role of a dealer in securities markets is to provide liquidity on the exchange y quoting id ask prices at which he is willing to uy sell a specific quantity of assets. Traditionally, this role has een filled y marketmaker or specialist firms. In recent years, with the growth of electronic exchanges such as Nasdaq s Inet, anyone willing to sumit limit orders in the system can effectively play the role of a dealer. Indeed, the availaility of high frequency data on the limit order ook (see com) ensures a fair playing field where various agents can post limit orders at the prices they choose. In this paper, we study the optimal sumission strategies of id ask orders in such a limit order ook. The pricing strategies of dealers have een studied extensively in the microstructure literature. The two most often addressed sources of risk facing the dealer are (i) the *Corresponding author. sashastoikov@gmail.com inventory risk arising from uncertainty in the asset s value (ii) the asymmetric information risk arising from informed traders. Useful surveys of their results can e found in Biais et al. (004), Stoll (003) a ook y O Hara (1997). In this paper, we will focus on the inventory effect. In fact, our model is closely related to a paper y Ho Stoll (191), which analyses the optimal prices for a monopolistic dealer in a single stock. In their model, the authors specify a true price for the asset, derive optimal id ask quotes around this price, to account for the effect of the inventory. This inventory effect was found to e significant in an empirical study of AMEX Options y Ho Macris (194). In another paper y Ho Stoll (190), the prolem of dealers under competition is analysed the id ask prices are shown to e related to the reservation (or indifference) prices of the agents. In our framework, we will assume that our agent is ut one player in the market the true price is given y the market mid-price. Of crucial importance to us will e the arrival rate of uy sell orders that will reach our agent. In order Quantitative Finance ISSN print/issn online ß 00 Taylor & Francis DOI: /

2 1 Feature Downloaded By: [Mt Sinai School of Medicine, Levy Lirary] At: 3:19 April 00 to model these arrival rates, we will draw on recent results in econophysics. One of the important achievements of this literature has een to explain the statistical properties of the limit order ook (see Bouchaud et al. 00, Luckock 003, Potters Bouchaud 003, Smith et al. 003). The focus of these studies has een to reproduce the oserved patterns in the markets y introducing zero intelligence agents, rather than modelling optimal strategies of rational agents. One possile exception is the work of Luckock (003), who defines a notion of optimal strategies, without resorting to utility functions. Though our ojective is different to that of the econophysics literature, we will draw on their results to infer reasonale arrival rates of uy sell orders. In particular, the results that will e most useful to us are the size distriution of market orders (Maslow Mills 001, Weer Rosenow 005, Gaaix et al. 006) the temporary price impact of market orders (Bouchaud et al. 00, Weer Rosenow 005). Our approach, therefore, is to comine the utility framework of the Ho Stoll approach with the microstructure of actual limit order ooks as descried in the econophysics literature. The main result is that the optimal id ask quotes are derived in an intuitive two-step procedure. First, the dealer computes a personal indifference valuation for the stock, given his current inventory. Second, he calirates his id ask quotes to the limit order ook, y considering the proaility with which his quotes will e executed as a function of their distance from the mid-price. In the alancing act etween the dealer s personal risk considerations the market environment lies the essence of our solution. The paper is organized as follows. In section, we descrie the main uilding locks for the model: the dynamics of the mid-market price, the agent s utility ojective the arrival rate of orders as a function of the distance to the mid-price. In section 3, we solve for the optimal id ask quotes, relate them to the reservation price of the agent, given his current inventory. We then present an approximate solution, numerically simulate the performance of our agent s strategy compare its Profit Loss (P&L) profile to that of a enchmark strategy.. The model.1. The mid-price of the stock For simplicity, we assume that money market pays no interest. The mid-market price, or mid-price, of the stock evolves according to ds u ¼ dw u ð1 with initial value S t ¼ s. Here W t is a stard onedimensional Brownian motion is constant.y Underlying this continuous-time model is the implicit assumption that our agent has no opinion on the drift or any autocorrelation structure for the stock. This mid-price will e used solely to value the agent s assets at the end of the investment period. He may not trade costlessly at this price, ut this source of romness will allow us to measure the risk of his inventory in stock. In section.4 we will introduce the possiility to trade through limit orders... The optimizing agent with finite horizon The agent s ojective is to maximize the expected exponential utility of his P&L profile at a terminal time T. This choice of convex risk measure is particularly convenient, since it will allow us to define reservation (or indifference) prices which are independent of the agent s wealth. We first model an inactive trader who does not have any limit orders in the market simply holds an inventory of q stocks until the terminal time T. This frozen inventory strategy will later prove to e useful in the case when limit orders are allowed. The agent s value function is vðx, s, q, t ¼E t ½ expð ðx þ qs T Š, where x is the initial wealth in dollars. This value function can e written as vðx, s, q, t ¼ expð x expð qs exp q ðt t, ð3 which shows us directly its dependence on the market parameters. We may now define the reservation id ask prices for the agent. The reservation id price is the price that would make the agent indifferent etween his current portfolio his current portfolio plus one stock. The reservation ask price is defined similarly elow. We stress that this is a sujective valuation from the point of view of the agent does not reflect a price at which trading should occur. Definition 1. Let v e the value function of the agent. His reservation id price r is given implicitly y the ywe choose this model over the stard geometric Brownian motion to ensure that the utility functionals introduced in the sequel remain ounded. In practical applications, we could also use a dimensionless model such as ds u S u ¼ dw u ð with initial value S t ¼ s. To avoid mathematical infinities, exponential utility functions could e modified to a stard mean/ variance ojective with the same Taylor-series expansion. The essence of the results would remain. More details regarding the model () with mean/variance utility are given in the appendix.

3 Feature 19 Downloaded By: [Mt Sinai School of Medicine, Levy Lirary] At: 3:19 April 00 relation vðx r ðs, q, t, s, q þ 1, t ¼vðx, s, q, t: The reservation ask price r a solves vðx þ r a ðs, q, t, s, q 1, t ¼vðx, s, q, t: A simple computation involving equations (3), (4) (5) yields a closed-form expression for the two prices r a ðs, q, t ¼s þð1 q ðt t ð6 r ðs, q, t ¼s þð 1 q ðt t ð7 in the setting where no trading is allowed. We will refer to the average of these two prices as the reservation or indifference price rðs, q, t ¼s q ðt t, given that the agent is holding q stocks. This price is an adjustment to the mid-price, which accounts for the inventory held y the agent. If the agent is long stock (q40), the reservation price is elow the mid-price, indicating a desire to liquidate the inventory y selling stock. On the other h, if the agent is short stock (q50), the reservation price is aove the mid-price, since the agent is willing to uy stock at a higher price..3. The optimizing agent with infinite horizon Because of our choice of a terminal time T at which we measure the performance of our agent, the reservation price () depends on the time interval (T t). Intuitively, the closer our agent is to time T, the less risky his inventory in stock is, since it can e liquidated at the midprice S T. In order to otain a stationary version of the reservation price, we may consider an infinite horizon ojective of the form Z 1 vðx, s, q ¼E expð!t expð ðx þ qs t dt : 0 The stationary reservation prices (defined in the same way as in Definition 1) are given y r a ðs, q ¼s þ 1 ln 1 þ ð1 q! q r ðs, q ¼s þ 1 ln 1 þ ð 1 q! q, where!4ð1= q. The parameter! may therefore e interpreted as an upper ound on the inventory position our agent is allowed to take. The natural choice of! ¼ð1= ðq max þ 1 would ensure that the prices defined aove are ounded. ð4 ð5 ð.4. Limit orders We now turn to an agent who can trade in the stock through limit orders that he sets around the mid-price given y (1). The agent quotes the id price p the ask price p a, is committed to respectively uy sell one share of stock at these prices, should he e hit or lifted y a market order. These limit orders p p a can e continuously updated at no cost. The distances ¼ s p a ¼ p a s the current shape of the limit order ook determine the priority of execution when large market orders get executed. For example, when a large market order to uy Q stocks arrives, the Q limit orders with the lowest ask prices will automatically execute. This causes a temporary market impact, since transactions occur at a price that is higher than the mid-price. If p Q is the price of the highest limit order executed in this trade, we define p ¼ p Q s to e the temporary market impact of the trade of size Q. If our agent s limit order is within the range of this market order, i.e. if a 5p, his limit order will e executed. We assume that market uy orders will lift our agent s sell limit orders at Poisson rate a ( a ), a decreasing function of a. Likewise, orders to sell stock will hit the agent s uy limit order at Poisson rate ( ), a decreasing function of. Intuitively, the further away from the midprice the agent positions his quotes, the less often he will receive uy sell orders. The wealth inventory are now stochastic depend on the arrival of market sell uy orders. Indeed, the wealth in cash jumps every time there is a uy or sell order dx t ¼ p a dnt a p dnt where Nt is the amount of stocks ought y the agent Nt a is the amount of stocks sold. Nt Nt a are Poisson processes with intensities a. The numer of stocks held at time t is q t ¼ Nt Nt a : The ojective of the agent who can set limit orders is uðs, x, q, t ¼max E t ½ expð ðx T þ q T S T Š: a, Notice that, unlike the setting descried in the previous susection, the agent controls the id ask prices therefore indirectly influences the flow of orders he receives. Before turning to the solution of this prolem, we consider some realistic functional forms for the intensities a ( a ) ( ) inspired y recent results in the econophysics literature.

4 0 Feature Downloaded By: [Mt Sinai School of Medicine, Levy Lirary] At: 3:19 April The trading intensity One of the main ojectives of the econophysics community has een to descrie the laws governing the microstructure of financial markets. Here, we will e focusing on the results which address the Poisson intensity with which a limit order will e executed as a function of its distance to the mid-price. In order to quantify this, we need to know statistics on (i) the overall frequency of market orders, (ii) the distriution of their size (iii) the temporary impact of a large market order. Aggregating these results suggests that should decay as an exponential or a power law function. For simplicity, we assume a constant frequency of market uy or sell orders. This could e estimated y dividing the total volume traded over a day y the average size of market orders on that day. The distriution of the size of market orders has een found y several studies to oey a power law. In other words, the density of market order size is f Q ðx /x 1 for large x, with ¼ 1.53 in Gopikrishnan et al. (000) for US stocks, ¼ 1.4 in Maslow Mills (001) for shares on the NASDAQ ¼ 1.5 in Gaaix et al. (006) for the Paris Bourse. There is less consensus on the statistics of the market impact in the econophysics literature. This is due to a general disagreement over how to define it how to measure it. Some authors find that the change in price p following a market order of size Q is given y p / Q, ð9 ð10 where ¼ 0.5 in Gaaix et al. (006) ¼ 0.76 in Weer Rosenow (005). Potters Bouchaud (003) find a etter fit to the function p / lnðq: ð11 Aggregating this information, we may derive the Poisson intensity at which our agent s orders are executed. This intensity will depend only on the distance of his quotes to the mid-price, i.e. ( ) for the arrival of sell orders a ( a ) for the arrival of uy orders. For instance, using (9) (11), we derive ð ¼Pðp4 ¼ Pð lnðq4k ¼ PðQ4 expðk ¼ Z 1 expðk ¼ A expð k x 1 dx ð1 where A ¼ / k ¼ K. In the case of a power price impact (10), we otain an intensity of the form ð ¼B = : Alternatively, since we are interested in short term liquidity, the market impact function could e derived directly y integrating the density of the limit order ook. This procedure is descried in Smith et al. (003) Weer Rosenow (005) yields what is sometimes called the virtual price impact. 3. The solution 3.1. Optimal id ask quotes Recall that our agent s ojective is given y the value function uðs, x, q, t ¼max E t ½ expð ðx T þ q T S T Š ð13 a, where the optimal feedack controls a will turn out to e time state dependent. This type of optimal dealer prolem was first studied y Ho Stoll (191). One of the key steps in their analysis is to use the dynamic programming principle to show that the function u solves the following Hamilton Jacoi Bellman equation u t þ 1 u ss þ max ð uðs, x s þ, q þ 1, t >< uðs, x, q, t þ max a ð a uðs, x þ s þ a, q 1, t a uðs, x, q, t ¼ 0, >: uðs, x, q, T ¼ expð ðx þ qs: The solution to this nonlinear PDE is continuous in the variales s, x t depends on the discrete values of the inventory q. Due to our choice of exponential utility, we are ale to simplify the prolem with the ansatz uðs, x, q, t ¼ expð x expð ðs, q, t: ð14 Direct sustitution yields the following equation for : t þð1= ss ð1= s ð þ max ½1 e ðs r >< Š ð15 a ð a þ max >: ðs, q, T ¼qs: ½1 e ðsþa r a Š ¼ 0; Applying the definition of reservation id ask prices (given in section.) to the ansatz (14), we find that r r a depend directly on this function. Indeed, r ðs; q; t ¼ðs; q þ 1; t ðs; q; t ð16 is the reservation id price of the stock, when the inventory is q r a ðs, q, t ¼ðs, q, t ðs, q 1, t ð17 is the reservation ask price, when the inventory is q. From the first-order optimality condition in (15),

5 Feature 1 Downloaded By: [Mt Sinai School of Medicine, Levy Lirary] At: 3:19 April 00 we otain the optimal distances a. They are given y the implicit relations s r ðs; q; t ¼ 1 ð ln 1 ð@ =@ð ð1 r a ðs, q, t s ¼ a 1 a ð a ln 1 ð@ a =@ð a : ð19 In summary, the optimal id ask quotes are otained through an intuitive, two-step procedure. First, we solve the PDE (15) in order to otain the reservation id ask prices r (s, q, t) r a (s, q, t). Second, we solve the implicit equations (1) (19) otain the optimal distances (s, q, t) a (s, q, t) etween the mid-price optimal id ask quotes. This second step can e interpreted as a caliration of our indifference prices to the current market supply dem a. 3.. Asymptotic expansion in q The main computational difficulty lies in solving equation (15). The order arrival terms (i.e. the terms to e maximized in the expression) are highly nonlinear may depend on the inventory. We therefore suggest an asymptotic expansion of in the inventory variale q, a linear approximation of the order arrival terms. In the case of symmetric, exponential arrival rates a ð ¼ ð ¼Ae k ; ð0 the indifference prices r a (s, q, t) r (s, q, t) coincide with their frozen inventory values, as descried in section.. Sustituting the optimal values given y equations (1) (19) into (15) using the exponential arrival rates, we otain >< t þ 1 ss 1 s þ A k þ ðe ka þ e k ¼0, >: ðs, q, T ¼qs: ð1 Consider an asymptotic expansion in the inventory variale ðq, s, t ¼ 0 ðs, tþq 1 ðs, tþ 1 q ðs, tþ: ð The exact relations for the indifference id ask prices, (16) (17), yield r ðs, q, t ¼ 1 ðs, tþð1 þ q ðs, tþ r a ðs, q, t ¼ 1 ðs, tþð 1 þ q ðs, tþ: ð3 ð4 Using equations (4) (3), along with the optimality conditions (1) (19), we find that the optimal pricing strategy amounts to quoting a spread of a þ ¼ ðs; tþ ln 1 þ ð5 k around the reservation price given y rðs, q, t ¼ r a þ r ¼ 1 ðs, tþq ðs, t: The term 1 can e interpreted as the reservation price, when the inventory is zero. The term may e interpreted as the sensitivity of the market maker s quotes to changes in inventory. For instance, since will turn out to e negative, accumulating a long position q40 will result in aggressively low quotes. The id ask spread in (5) is independent of the inventory. This follows from our assumption of exponential arrival rates. The spread consists of two components, one that depends on the sensitivity to changes in inventory one that depends on the intensity of arrival of orders, through the parameter k. Taking a first-order approximation of the order arrival term A k þ ðe ka þ e k ¼ A k þ kða þ þ, ð6 we notice that the linear term does not depend on the inventory q. Therefore, if we sustitute () (6) into (1) group terms of order q, we otain < t 1 þ 1 ss 1 ¼ 0, ð7 : 1 ðs, T ¼s; whose solution is 1 (s, t) ¼ s. Grouping terms of order q yields < t þ 1 ss 1 ðs 1 ¼ 0 ð : ðs, T ¼0: whose solution is ¼ ð1= ðt t. Thus, for this linear approximation of the order arrival term, we otain the same indifference price rðs; t ¼s q ðt t ð9 as for the frozen inventory prolem from section.. We then set a id/ask spread given y a þ ¼ ðt tþ ln 1 þ ð30 k around this indifference or reservation price. Note that if we had taken a quadratic approximation of the order arrival term, we would still otain 1 ¼ s, ut the sensitivity term (s, t) would solve a nonlinear PDE. Equations (9) (30) thus provide us with simple expressions for the id ask prices in terms of our model parameters. This approximate solution

6 Feature Downloaded By: [Mt Sinai School of Medicine, Levy Lirary] At: 3:19 April 00 also simplifies the simulations we perform in the next section Numerical simulations We now test the performance of our strategy, focusing primarily on the shape of the P&L profile the final inventory q T. We will refer to our strategy as the inventory strategy, compare it to a enchmark strategy that is symmetric around the mid-price, regardless of the inventory. This strategy, which we refer to as the symmetric strategy, uses the average spread of the inventory strategy, ut centres it around the mid-price, rather than the reservation price. In practice, the choice of time step dt is a sutle one. On the one h, dt must e small enough so that the proaility of multiple orders reaching our agent is small. On the other h, dt must e larger than the typical tick time, otherwise the agent s quotes will e updated so frequently that he will not see any orders (particularly if his quotes are outside the market id/ask spread). As far as our simulation is concerned, we chose the following parameters: s ¼ 100, T ¼ 1, ¼, dt ¼ 0.005, q ¼ 0, ¼ 0.1, k ¼ 1.5 A ¼ 140. The simulation is otained through the following procedure: at time T, the agent s quotes a are computed, given the state variales. At time t þ dt, the state variales are updated. With proaility a ( a )dt, the inventory variale decreases y one the wealth increases y s þ a. With proaility ( )dt, the inventory increases y one the wealth decreases y s. The p mid-price is updated y a rom increment ffiffiffiffi dt. Figure 1 illustrates the id ask quotes for one simulation of a stock path. Notice that, at time t ¼ 0.15, the id ask quotes are relatively high, indicating that the inventory position must e negative (or short stock). Since the id price is aggressively placed near the mid-price, our agent is more likely to uy stock the inventory quickly returns to zero y time t ¼ 0.. As we approach the terminal time, our agent s id/ask quotes look more like a strategy that is symmetric around the mid-price. Indeed, when we are close to the terminal time, our inventory position is considered less risky, since the mid-price is less likely to move drastically. We then run 1000 simulations to compare our inventory strategy to the symmetric strategy. This strategy uses the average id/ask spread of the inventory strategy over the time period, ut centres it around the mid-price. For example, the performance of the symmetric strategy that quotes a id/ask spread of $1.49 (corresponding to the average spread of the optimal agent with ¼ 0.1) is displayed in tale 1. This symmetric strategy has a higher return higher stard deviation than the inventory strategy. The symmetric strategy otains a slightly higher return since it is centred around the mid-price, therefore receives a higher s p_a p_ Figure 1. The mid-price the optimal id ask quotes. Strategy Tale simulations with ¼ 0.1. Average spread Profit (Profit) Final q (Final q) Inventory Symmetric Inventory strategy 160 Symmetric strategy Strategy Figure. ¼ 0.1. Tale simulations with ¼ Average Spread Profit (Profit) Final q (Final q) Inventory Symmetric volume of orders than the inventory strategy. However, the inventory strategy otains a P&L profile with a much smaller variance, as illustrated in the histogram in figure.

7 Feature 3 Downloaded By: [Mt Sinai School of Medicine, Levy Lirary] At: 3:19 April 00 Strategy Inventory strategy Symmetric strategy Tale simulations with ¼ 1. Average spread 10 Inventory strategy Symmetric strategy Figure 3. ¼ Profit (Profit) Final q (Final q) Inventory Symmetric to avoid accumulating an inventory. This strategy produces low stard deviations of profits final inventory, ut generates more modest profits than the corresponding symmetric strategy (see figure 4). References Biais, B., Glosten, L. Spatt, C., Market microstructure: a server of microfoundations, empirical results policy implications. J. Financ. Markets, 005,, Bouchaud, J.-P., Mezard, M. Potters, M., Statistical properties of stock order ooks: empirical results models. Quant. Finance, 00,, Gaaix, X., Gopikrishnan, P., Plerou, V. Stanley, H.E., Institutional investors stock market volatility. Quart. J. Econ., 006, 11, Gopikrishnan, P., Plerou, V., Gaaix, X. Stanley, H., Statistical properties of share volume traded in financial markets. Phys. Rev. E, 000, 6, R4493 R4496. Ho, T. Macris, R., Dealer id ask quotes transaction prices: an empirical study of some AMEX options. J. Finance, 194, 39, Ho, T. Stoll, H., On dealer markets under competition. J. Finance, 190, 35, Ho, T. Stoll, H., Optimal dealer pricing under transactions return uncertainty. J. Financ. Econ., 191, 9, Luckock, H., A steady-state model of the continuous doule auction. Quant. Finance, 003, 3, Maslow, S. Mills, M., Price fluctuations from the order ook perspective: empirical facts a simple model. Phys. A, 001, 99, O Hara, M., Market Microstructure Theory, 1997 (Blackwell: Camridge). Potters, M. Bouchaud, J.-P., More statistical properties of order ooks price impact. Physica A: Stat. Mech. Appl., 003, 34, Smith, E., Farmer, J.D., Gillemot, L. Krishnamurthy, S., Statistical theory of the continuous doule auction. Quant. Finance, 003, 3, Stoll, H.R., Market microstructure. In Hook of the Economics of Finance, edited y G.M. Constantinides, et al., 003 (North Holl: Amsterdam). Weer, P. Rosenow, B., Order ook approach to price impact. Quant. Finance, 005, 5, Figure 4. ¼ 1. The results of the simulations comparing the inventory strategy for ¼ 0.01 with the corresponding symmetric strategy are displayed in tale. This small value for represents an investor who is close to risk neutral. The inventory effect is therefore much smaller the P&L profiles of the two strategies are very similar, as illustrated in figure 3. In fact, in the limit as! 0 the two strategies are identical. Finally, we display the performance of the two strategies for ¼ 1 in tale 3. This choice corresponds to a very risk averse investor, who will go to great lengths Appendix Herein, we consider the geometric Brownian motion ds u S u ¼ dw u with initial value S t ¼ s, the mean/variance ojective h Vðx; s; q; t ¼E t ðx þ qs T i ðqs T qs ; where x is the initial wealth in dollars. This value function can e written as Vðx; s; q; t ¼x þ qs q s e ðt t 1 :

8 4 Feature Downloaded By: [Mt Sinai School of Medicine, Levy Lirary] At: 3:19 April 00 This yields reservation prices of the form R a ðs; q; t ¼s þ ð1 q s e ðt t 1 R ð 1 q ðs; q; t ¼s þ s e ðt t 1 : These results are analogous to the ones otained in section..