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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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