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High-frequency trading in a limit order book

Marco Avellaneda & Sasha Stoikov

April 24, 2006

Abstract

We study a stock dealer’s strategy for submitting bid and ask quotes in a

limit order book. The agent faces an inventory risk due to the diﬀusive nature of

the stock’s mid-price and a transactions risk due to a Poisson arrival of market

buy and sell orders. After setting up the agent’s problem in a maximal expected

utility framework, we derive the solution in a two step procedure. First, the

dealer computes a personal indiﬀerence valuation for the stock, given his current

inventory. Second, he calibrates his bid and ask quotes to the market’s limit

order book. We compare this ”inventory-based” strategy to a ”naive” best

bid/best ask strategy by simulating stock price paths and displaying the P&L

proﬁles of both strategies. We ﬁnd that our strategy has a P&L proﬁle that

has both a higher return and lower variance than the benchmark strategy.

1 Introduction

The role of a dealer in securities markets is to provide liquidity on the exchange by

quoting bid and ask prices at which he is willing to buy and sell a speciﬁc quan-

tity of assets. Traditionally, this role has been ﬁlled by market-maker or specialist

ﬁrms. In recent years, with the growth of electronic exchanges such as Nasdaq’s Inet,

anyone willing to submit limit orders in the system can eﬀectively play the role of a

dealer. Indeed, the availability of high frequency data on the limit order book (see

www.inetats.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 submission

strategies of bid and ask orders in such a limit order book.

The pricing strategies of dealers have been studied extensively in the microstruc-

ture literature. The two most often addressed sources of risk facing the dealer are

(i) the inventory risk arising from uncertainty in the asset’s value and (ii) the asym-

metric information risk arising from informed traders. Useful surveys of their results

can be found in Biais et al. [1], Stoll [13] and a book by O’Hara [10]. In this paper,

we will focus on the inventory eﬀect. In fact, our model is closely related to a paper

by Ho and Stoll [6], which analyses the optimal prices for a monopolistic dealer in

1

a single stock. In their model, the authors specify a ’true’ price for the asset, and

derive optimal bid and ask quotes around this price, to account for the eﬀect of the

inventory. This inventory eﬀect was found to be signiﬁcant in an empirical study of

AMEX Options by Ho and Macris [5]. In another paper by Ho and Stoll [7], the prob-

lem of dealers under competition is analyzed and the bid and ask prices are shown to

be related to the reservation (or indiﬀerence) prices of the agents. In our framework,

we will assume that our agent is but one player in the market and the ’true’ price is

given by the market mid-price.

Of crucial importance to us will be the arrival rate of buy and sell orders that

will reach our agent. In order to model these arrival rates, we will draw on recent

results in econophysics. One of the important achievements of this literature has

been to explain the statistical properties of the limit order book (see Bouchaud et al.

[2], Potters and Bouchaud [11], Smith et al. [12], Luckock [8]). The focus of these

studies has been to reproduce the observed patterns in the markets by introducing

’zero intelligence’ agents, rather than modeling optimal strategies of rational agents.

One possible exception is the work of Luckock [8], who deﬁnes a notion of optimal

strategies, without resorting to utility functions. Though our objective is diﬀerent to

that of the econophysics literature, we will draw on their results to infer reasonable

arrival rates of buy and sell orders. In particular, the results that will be most useful

to us are the size distribution of market orders (Gabaix et al. [3], Maslow and Mills

[9]) and the temporary price impact of market orders (Weber and Rosenow [14],

Bouchaud et al. [2]).

Our approach, therefore, is to combine the utility framework of the Ho and Stoll

approach with the microstructure of actual limit order books as described in the

econophysics literature. The main result is that the optimal bid and 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 calibrates

his bid and ask quotes to the limit order book, by considering the probability with

which his quotes will be executed as a function of their distance from the mid-price.

In the balancing act between the dealer’s personal risk considerations and the market

environment lies the essence of our solution.

The paper is organized as follows. In section 2, we describe the main building

blocks for the model: the dynamics of the mid-market price, the agent’s utility ob-

jective and the arrival rate of orders as a function of the distance to the mid-price. In

section 3, we solve for the optimal bid and ask quotes, and relate them to the reserva-

tion price of the agent, given his current inventory. We then present an approximate

solution, numerically simulate the performance of our agent’s strategy and compare

its Proﬁt and Loss (P&L) proﬁle to that of some benchmark strategies.

2

2 The model

2.1 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 is modeled as a Brownian motion

St=s+σWt.(2.1)

We choose this model over the standard geometric Brownian motion because it yields

simpler solutions and ensures that the utility functionals introduced in the sequel

remain bounded.

This mid-price will be used solely to value the agent’s assets at the end of the

investment period. He may not trade costlessly at this price, but this source of

randomness will allow us to measure the risk of his inventory in stock. In section 2.4

we will introduce the possibility to trade through limit orders.

2.2 The optimizing agent with ﬁnite horizon

The agent’s objective 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

and simply holds an inventory of qstocks until the terminal time T. This passive

strategy will later prove to be useful in the case when limit orders are allowed. The

agent’s value function is

v(x, s, q, t) = Et[−exp(−γ(x+qST)]

where xis the initial wealth in dollars. This value function can be written as

v(x, s, q, t) = −exp(−γx) exp(−γqs) exp γ2q2σ2(T−t)

2(2.2)

which shows us directly its dependence on the market parameters.

We may now deﬁne the reservation bid and ask prices for the agent. The reserva-

tion bid price is the price that would make the agent indiﬀerent between his current

portfolio and his current portfolio plus one stock. The reservation ask price is deﬁned

similarly below. We stress that this is a subjective valuation from the point of view

of the agent and does not reﬂect a price at which trading should occur.

Deﬁnition 1.Let vbe the value function of the agent. His reservation bid price rbis

given implicitly by the relation

v(x−rb(s, q, t), s, q + 1, t) = v(x, s, q, t).(2.3)

3

The reservation ask price rasolves

v(x+ra(s, q, t), s, q −1, t) = v(x, s, q, t).(2.4)

A simple computation involving equations (2.2), (2.3) and (2.4) yields a closed-

form expression for the two prices

ra(s, q, t) = s+ (1 −2q)γσ2(T−t)

2

and

rb(s, q, t) = s+ (−1−2q)γσ2(T−t)

2

in the setting where no trading is allowed. We will refer to the average of these two

prices as the reservation or indiﬀerence price

r(s, q, t) = s−qγσ2(T−t),

given that the agent is holding qstocks. This price is an adjustment to the mid-

price, which accounts for the inventory held by the agent. If the agent is long stock

(q > 0), the reservation price is below the mid-price, indicating a desire to liquidate

the inventory by selling stock. On the other hand, if the agent is short stock (q < 0),

the reservation price is above the mid-price, since the agent is willing to buy stock at

a higher price.

2.3 The optimizing agent with inﬁnite horizon

Because of our choice of a terminal time Tat which we measure the performance of

our agent, the reservation price (2.2) 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

be liquidated at the mid-price ST. In order to obtain a stationary version of the

reservation price, we may consider an inﬁnite horizon objective of the form

¯v(x, s, q) = EZ∞

0

−exp(−ωt) exp(−γ(x+qSt))dt.

The stationary reservation prices (deﬁned in the same way as in Deﬁnition 1) are

given by

¯ra(s, q) = s+1

γln 1−

(2q+ 1)γ2σ2

2ω−γ2q2σ2

and

¯rb(s, q) = s+1

γln 1−(2q−1)γ2σ2

2ω−γ2q2σ2,

where ω > 1

2γ2σ2q2.

The parameter ωmay therefore be interpreted as an upper bound on the inventory

position our agent is allowed to take. The natural choice of ω=1

2γ2σ2(qmax + 1)2

would ensure that the prices deﬁned above are bounded.

4

2.4 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 by (2.1). The agent quotes the bid price pband the ask

price pa, and is committed to respectively buy and sell one share of stock at these

prices, should he be ”hit” or ”lifted” by a market order. These limit orders pband pa

can be continuously updated at no cost. The distances

δb=s−pb

and

δa=pa−s

and the current shape of the limit order book determine the priority of execution

when large market orders get executed.

For example, when a large market order to buy Qstocks arrives, the Qlimit

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 pQis the price of the highest limit order executed in this trade, we deﬁne

∆p=pQ−s

to be the temporary market impact of the trade of size Q. If our agent’s limit order

is within range of this market order, i.e. if δa<∆p, his limit order will be executed.

We assume that market buy orders will ”lift” our agent’s sell limit orders at

Poisson rate λa(δa), a decreasing function of δa. Likewise, orders to sell stock will

”hit” the agent’s buy limit order at Poisson rate λb(δb), a decreasing function of δb.

Intuitively, the further away from the mid-price the agent positions his quotes, the

less often he will receive buy and sell orders.

The wealth and inventory are now stochastic and depend on the arrival of market

sell and buy orders. Indeed, the wealth in cash jumps every time there is a buy or

sell order

dXt=padNa

t−pbdNb

t

where Nb

tis the amount of stocks bought by the agent and Na

tis the amount of stocks

sold. Nb

tand Na

tare Poisson processes with intensities λband λa. The number of

stocks held at time tis

qt=Nb

t−Na

t.

The objective of the agent who can set limit orders is

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

δa,δbEt[−exp(−γ(XT+qTST)))].

Notice that, unlike the setting described in the previous subsection, the agent controls

the bid and ask prices and therefore indirectly inﬂuences the ﬂow of orders he receives.

Before turning to the solution of this problem, we consider some realistic functional

forms for the intensities λa(δa) and λb(δb) inspired by recent results in the econophysics

literature.

5

2.5 The trading intensity

One of the main focuses of the econophysics community has been to describe the laws

governing the microstructure of ﬁnancial markets. Here, we will be focussing on the

results which address the Poisson intensity λwith which a limit order will be executed

as a function of its distance δto the mid-price. In order to quantify this, we need to

know statistics on (i) the overall frequency of market orders (ii) the distribution of

their size and (iii) the temporary impact of a large market order. Aggregating these

results suggests that λshould decay as an exponential or a power law function.

For simplicity, we assume a constant frequency Λ of market buy or sell orders.

This could be estimated by dividing the total volume traded over a day by the average

size of market orders on that day.

The distribution of the size of market orders has been found by several studies to

obey a power law. In other word, the density of market order size is

fQ(x)∝x−1−α(2.5)

with α= 1.53 in Gopikrishnan et al. [4] for U.S. stocks, α= 1.4 in Maslow and Mills

[9] for shares on the NASDAQ and α= 1.5 in Gabaix et al. [3] 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 and how to

measure it. Some authors ﬁnd that the change in price ∆pfollowing a market order

of size Qis given by

∆p∝Qβ(2.6)

where β= 0.5 in Gabaix et al. [3] and β= 0.76 in Weber and Rosenow [14]. Potters

and Bouchaud [11] ﬁnd a better ﬁt to the function

∆p∝ln(Q).(2.7)

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. λb(δb) for the arrival of sell orders and λa(δa) for the

arrival of buy orders. For instance, using (2.5) and (2.7), we derive

λ(δ) = ΛP(∆p > δ)

= ΛP(ln(Q)> Kδ)

= ΛP(Q > exp(Kδ))

= Λ R∞

exp(Kδ)x−1−αdx

=Aexp(−kδ)

(2.8)

where A= Λ/α and k=αK. In the case of a power price impact (2.6), we obtain

an intensity of the form

λ(δ) = B(δ)−α

β.

6

Alternatively, since we are interested in short term liquidity, the market impact

function could be derived directly by integrating the density of the limit order book.

This procedure is described in Smith et al. [12] and Weber and Rosenow [14] and

yields what is sometimes called the ”virtual” price impact.

3 The solution

3.1 Optimal bid and ask quotes

Recall that our agent’s objective is given by the value function

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

δa,δbEt[−exp(−γ(XT+qTST)))].(3.1)

This type of optimal dealer problem was ﬁrst studied by Ho and Stoll [6]. One of the

key steps in their analysis is to use the dynamic programming principle to show that

the function usolves the following Hamilton-Jacobi-Bellman equation

ut+1

2σ2uss + maxδbλb(δb)u(s, x −s+δb, q + 1, t)−u(s, x, q, t)

+ maxδaλa(δa) [u(s, x +s+δa, q −1, t)−u(s, x, q, t)] = 0

u(S, x, q, t) = −exp(−γ(x+qS)).

The solution to this nonlinear PDE is continuous in the variables s,xand tand

depends on the discrete values of the inventory q. Due to our choice of exponential

utility, we are able to simplify the problem with the ansatz

u(s, x, q, T ) = −exp(−γx) exp(−γθ(s, q, t)).(3.2)

Direct substitution yields the following equation for θ

θt+1

2σ2θSS −1

2σ2γθ2

S

+ maxδbhλb(δb)

γ[1 −eγ(s−δb−rb)]i+ maxδahλa(δa)

γ[1 −e−γ(s+δa−ra)]i= 0

θ(s, q, T ) = qs.

(3.3)

Applying the deﬁnition of reservation bid and ask prices (given in Section 2.2) to the

ansatz (3.2), we ﬁnd that rband radepend directly on this function θ. Indeed,

rb(s, q, t) = θ(s, q + 1, t)−θ(s, q, t) (3.4)

is the reservation bid price of the stock, when the inventory is qand

ra(s, q, t) = θ(s, q, t)−θ(s, q −1, t) (3.5)

7

is the reservation ask price,when the inventory is q. From the ﬁrst order optimality

condition in (3.3), we obtain the optimal distances δband δa. They are given by the

implicit relations

s−rb(s, q, t) = δb−

1

γln 1−γλb(δb)

∂λb

∂δ (δb)!(3.6)

and

ra(s, q, t)−s=δa−1

γln 1−γλa(δa)

∂λa

∂δ (δa)!.(3.7)

In summary, the optimal bid and ask quotes are obtained through an intuitive,

two step procedure. First, we solve the PDE (3.3) in order to obtain the reservation

bid and ask prices rb(s, q, t) and ra(s, q, t). Second, we solve the implicit equations

(3.6) and (3.7) and obtain the optimal distances δb(s, q, t) and δa(s, q, t) between the

mid-price and optimal bid and ask quotes. This second step can be interpreted as a

calibration of our indiﬀerence prices to the current market supply λband demand λa.

3.2 Approximations and numerical simulations

We now test the eﬀectiveness of our strategy by running simulations focussing pri-

marily on the shape of the P&L proﬁle and the ﬁnal inventory qT. We will refer to

our strategy as the ”inventory” strategy, and compare it to some benchmark strate-

gies that are symmetric around the mid-price, regardless of the inventory. One such

strategy, the ”best bid/best ask” strategy, consists in quoting bid and ask prices equal

to s−M

2and s+M

2respectively, where Mis the market spread (i.e. the diﬀerence

between the best bid and best ask prices in the limit order book). Another such

strategy, which we refer to as the ”symmetric” strategy, uses the same spread as the

inventory strategy, but centers it around the mid-price, rather than the reservation

price.

In order to simplify computations, we assume exponential arrival rates

λ(δ) = Ae−kδ (3.8)

which are consistent with the results of Potters and Bouchaud [11], as described in

Section 2.5. Once we have chosen our time step dt, we we obtain the value for A

through the relation

λ−

M

2dt =AekM

2dt = 1.(3.9)

Equation (3.9) says that if we post a limit order at a distance δ=−M

2from the

mid-price, this limit order is essentially a market order that will get executed with

probability one in a dt time interval.

In the ﬁrst step of our algorithm, rather than solving (3.3) numerically to obtain

the indiﬀerence bid and ask prices given by (3.4) and (3.5), we use the average of

8

the suboptimal bid and ask prices derived in section 2.2 for the static version of the

problem. In other words we will be using the proxy

r(s, q, t) = s−qγσ2(T−t) (3.10)

for both rband ra. For the second step of our algorithm, we substitute the exponential

order arrival function (3.8) into (3.6) and (3.7) and use the reservation price in (3.10)

to obtain

δb=γqσ2(T−t) + 1

γln(1 + γ

k) (3.11)

and

δa=−γqσ2(T−t) + 1

γln(1 + γ

k).(3.12)

Our inventory strategy therefore quotes a ﬁxed spread of

δa+δb=2

γln(1 + γ

k)

centered around the reservation price (3.10).

To illustrate the behavior of the inventory strategy over time, we simulate one mid-

price path, and calculate our agent’s indiﬀerence (reservation) price and the optimal

bid and ask quotes. The paths in ﬁgure 1 were simulated with the parameters s= 100,

T= 1, σ= 2, dt = 0.005, q= 0, γ= 0.1, k= 1.5 and M= 0.5. For this choice of

parameters, the optimal spread is δa+δb= 1.29.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

98.5

99

99.5

100

100.5

101

101.5

102

102.5

103

Time

Stock

Price

Mid−market price

Price asked

Price bid

Indifference Price

Figure 1. The mid-price, indiﬀerence price and the optimal bid and ask quotes

9

Note that when the indiﬀerence price is lower than the mid-price, the relation

(3.10) indicates that the inventory position qtmust be positive. This can be observed

at time t= 0.5 for example. Since our agent is anxious to sell due to a positive

inventory, the bid and ask quotes are low relative to the mid-price. This, in turn,

encourages a higher intensity of market buy orders, until time t= 0.55, where the

inventory is back to zero and the mid-price and indiﬀerence prices overlap.

On the other hand, at time t= 0.25, the indiﬀerence price is higher than the

mid-price, indicating that the inventory position must be negative (or short stock).

Since the bid price is aggressively placed near the mid-price, the inventory quickly

returns to zero at time t= 0.3. Notice that as we approach the terminal time, the

indiﬀerence price gets closer to the mid-price, and our agent’s bid/ask quotes look

more like a symmetric strategy. 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 ”best

bid/best ask” strategy. The average and standard deviations of the proﬁt and ﬁnal

inventory are indicated in Table 1. The standard deviation of qTfor our inventory

Strategy Spread Proﬁt std(Proﬁt) Final q std(Final q)

Inventory 1.29 62.94 5.89 0.10 2.80

Best bid/best ask 0.5 48.43 14.57 0.72 9.56

Symmetric 1.29 67.21 13.43 -0.018 8.66

Table 1: 1000 simulations with γ= 0.1

strategy is 2.80 and the standard deviation of the best bid/best ask strategy is 9.56.

This was to be expected, since our strategy has a tendency to sell when the inventory

is positive and buy when the inventory is negative. What is surprising, is that our

inventory strategy has a P&L proﬁle that has both a higher return and lower standard

deviation than the best bid/best ask strategy. This is illustrated in ﬁgure 2, which

displays the histogram of the two P&L proﬁles. The reason why the best bid/best ask

strategy performs so poorly is that its bid/ask spread of $0.50 is signiﬁcantly lower

than the optimal of $1.29. Intuitively, the best bid/best ask strategy receives more

orders than the inventory strategy, but the inventory strategy makes signiﬁcantly

more proﬁt each time it receives an order.

We then compare our ”inventory” strategy to a more sophisticated strategy, which

we call the ”symmetric” strategy. This strategy uses the bid/ask spread of the in-

ventory strategy, i.e. δa+δb=2

γln(1 + γ

k), but centers it around the mid-price. For

example, the performance of the symmetric strategy that quotes a bid/ask spread of

$1.29 (corresponding to the optimal agent with γ= 0.1) is displayed in Table 1. This

symmetric strategy has a higher return (of $67.21 versus $62.94) and higher standard

deviation ($13.43 versus $5.89) than the inventory strategy. The symmetric strategy

10

obtains a slightly higher return since it is centered around the mid-price, and there-

fore receives a higher volume of orders than the inventory strategy. However, the

inventory strategy obtains a P&L proﬁle with a much smaller variance, as illustrated

in the histogram in Figure 3.

Figure 2. γ= 0.1 Figure 3. γ= 0.1

The results of the simulations comparing the ”inventory” strategy for γ= 0.01

with the corresponding ”symmetric” strategy are displayed in Table 2. This small

value for γrepresents an investor who is close to risk neutral. The inventory eﬀect is

therefore much smaller and the P&L proﬁles of the two strategies are very similar, as

illustrated in Figure 4. In fact, in the limit as γ→0 the two strategies are identical.

Strategy Spread Proﬁt std(Proﬁt) Final q std(Final q)

Inventory 1.33 66.78 8.76 -0.02 4.70

Symmetric 1.33 67.36 13.40 -0.31 8.65

Table 2: 1000 simulations with γ= 0.01

Finally, we display the performance of the two strategies for γ= 0.5 in Table 3.

This choice corresponds to a very risk averse investor, who will go to great lengths to

avoid accumulating an inventory. This strategy produces low standard deviations of

proﬁts and ﬁnal inventory, but generates more modest proﬁts than the corresponding

symmetric strategy (see Figure 5).

Strategy Spread Proﬁt std(Proﬁt) Final q std(Final q)

Inventory 1.15 33.92 4.72 -0.02 1.88

Symmetric 1.15 66.20 14.53 0.25 9.06

Table 3: 1000 simulations with γ= 0.5

11

Figure 4. γ= 0.01 Figure 5. γ= 0.5

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