Electronic copy available at: http://ssrn.com/abstract=1712765

Where is the Value in High Frequency Trading? ∗

Álvaro Cartea† and José Penalva‡

First Version: November 21, 2010, This version: December 21, 2011

Abstract

We analyze the impact of high frequency (HF) trading in financial markets based on a

model with three types of traders: liquidity traders (LTs), professional traders (PTs), and

high frequency traders (HFTs). Our four main findings are: i) The price impact of liquidity

trades is higher in the presence of the HFTs and is increasing with the size of the trade. In

particular, we show that HFTs reduce (increase) the prices that LTs receive when selling

(buying) their equity holdings. ii) Although PTs lose revenue in every trade intermediated

by HFTs, they are compensated with a higher liquidity discount in the market price. iii)

HF trading increases the microstructure noise of prices. iv) The volume of trades increases

as the HFTs intermediate trades between the LTs and PTs. This additional volume is a

consequence of trades which are carefully tailored for surplus extraction and are neither

driven by fundamentals nor is it noise trading. In equilibrium, HF trading and PTs coexist

as competition drives down the profits for new HFTs while the presence of HFTs does not

drive out traditional PTs.∗We would like to thank Harrison Hong for his valuable comments and discussions. We are also grateful to

Andrés Almazán, Gene Amromin, Michael Brennan, Max Bruche, Celso Brunetti, Guillermo Caruana, Terrence

Hendershott, Andrei Kirilenko, Pete Kyle, Albert Menkveld, Rafael Repullo, Eduardo Schwartz and Enrique

Sentana for their comments. We also thank seminar participants at the SEC, CFTC, Federal Reserve Board,

CEMFI, University of Toronto, and Universidad Carlos III. The usual caveat applies. This article benefited

from financial support from the Comunidad de Madrid-UC3M 2011/00093/00, 2009/00138/00, and Ministerio

de Ciencia e Innovación 2010/00047/001. We welcome comments, including references we have inadvertently

missed.†alvaro.cartea@uc3m.es, Universidad Carlos III de Madrid.‡jpenalva@emp.uc3m.es, Universidad Carlos III de Madrid.

1

Electronic copy available at: http://ssrn.com/abstract=1712765

Keywords: High frequency traders, high frequency trading, flash trading, liquidity

traders, institutional investors, market microstructure, microstructure volatility, execution

costs, market quality

1 Introduction

Around 1970, Carver Mead coined the term “Moore’s law” in reference to Moore’s statement

that transistor counts would double every year. There is some debate over whether this “law”

is empirically valid but there is no discussion that the last forty years have seen an explosive

growth in the power and performance of computers. Financial markets have not been immune

to this technological advance, it may even be one of the places where the limits of computing

power are tested every day. This computing power is harnessed to spot trends and exploit profit

opportunities in and across financial markets. Its influence is so large that it has given rise to a

new class of trading strategies sometimes called algorithmic trading and others high frequency

trading. We prefer to use algorithmic trading (AT) as the generic term that refers to strategies

that use computers to automate trading decisions, and restrict the term high frequency (HF)

trading to refer to the subset of AT trading strategies that are characterized by their reliance

on speed differences relative to other traders to make profits based on short-term predictions

and also by the objective to hold essentially no asset inventories for more than a very short

period of time.1

The advent of AT has changed the trading landscape and the impact of their activities is at

the core of many regulatory and financial discussions. The explosion in volume of transactions

we have witnessed in the last decade, and the speed at which trades are taking place, is highly

suggestive that AT is very much in use and that these strategies are not being driven out of the

market as a result of losses in their trading activities. Indeed, different sources estimate that

annual profits from AT trading are between $3 and $21 billion (Brogaard [2010] and Kearns1This definition is consistent with the one used in Kyle [Kirilenko et al. [2010] p3]: “We find that on May

6, the 16 trading accounts that we classify as HFTs traded over 1,455,000 contracts, accounting for almost athird of total trading volume on that day. Yet, net holdings of HFTs fluctuated around zero so rapidly thatthey rarely held more than 3,000 contracts long or short on that day.” See also the definition of HFT providedby CFTC Comissioner O’Malia [2011].

2

et al. [2010]). These strategies have supporters and detractors: on one side we find trading

houses and hedge funds who vigorously defend their great social value, whilst being elusive

about the profits they make from their use; and on the other hand there are trading houses

that denounce high frequency traders (HFTs) as a threat to the financial system (and their

bottom line). Although AT in general and HF trading in particular have been in the market

supervisors’ spotlight for quite some time and efforts to understand the consequences of HF

trading have stepped up since the ‘Flash Crash’ in May 6 2010 (SEC [2010], Commission et al.

[2010], Kirilenko et al. [2010], and Easley et al. [2011]) there is little (but increasing) academic

work that addresses the role of these trading strategies. In this paper, we provide a model of

HFTs, where they profit exclusively from extracting rents from other traders. With this model

we can isolate the effects of HFTs on market aggregates and how the burden implied by HFTs

in our model is shifted between different market participants.

To analyze the impact of HF trading in financial markets we develop a model with three

types of traders: liquidity traders (LTs), professional traders (PTs), and HFTs. In this model

LTs experience a liquidity shock and come to the market to unwind their positions which are

temporarily held by the PTs in exchange for a liquidity discount. HFTs are introduced as traders

who, thanks to their rapid information processing and quick execution ability, can position

themselves in the order book so as to profit from intermediating between the equity investor

(LT) and the PTs. HFTs work in a much finer time scale than other traders. Their strategy

of keeping inventories close to zero together with their speed leads them to act essentially as

“instantaneous” intermediaries, whose economic profits must come at the expense of those of

others, but whose actions have additional effects on market conditions. To identify the impact

of HFTs we use as benchmark the same model without HFTs, which corresponds to that of

Grossman and Miller [1988].

We highlight five main findings. First, HFTs’ additional intermediation increases the volume

of trade. The additional volume is neither driven by fundamentals (only the original trades,

without the HFTs, are driven by fundamentals) nor is it noise trading. Far from it, the extra

volume is a consequence of trades which are carefully tailored for surplus extraction. Moreover,

HF trading strategies introduce “microstructure noise”: in order to profit from intermediation

3

HFTs buy shares from one trader at a cheap price and sell it more dearly to another trader,

generating price dispersion where before there was only a single price. These properties, which

are built into the model, closely correspond to behavior reported in Kirilenko et al. [2010] and

are also found by Brunetti et al. [2011].

Second, the presence of HFTs exacerbates the price impact of the initial liquidity trades that

generate a temporary order imbalance, imposing a double burden on liquidity demanders: the

direct cost from the trading surplus extracted by the HFT, and the indirect cost of a greater

price impact. Furthermore, this effect is increasing in the size of the liquidity need, and is

consistent with the results in Zhang [2010].

Third, we find that the higher initial price impact arises as traders anticipate the future

additional trading costs from the presence of HFTs, which generates an increase in the liquidity

discount. Thus, PTs suffer countervailing effects from HFTs: increased trading costs from

HFT surplus extraction versus increased expected returns from higher liquidity discounts. In

our benchmark model these two effects cancel each other, leaving expected profits for PTs

unchanged. However, competition between HFTs introduces additional uncertainty for PTs,

and, in equilibrium, distorts their number.

Fourth, we find that standard measures of market liquidity, specially volume-based ones,

may lead to erroneous conclusions: in our model, HFTs do not increase liquidity and yet we

observe increased trading volumes. In fact, LTs face overall lower sales revenue and higher costs

of purchase, suggesting that liquidity is better measured through total cost of trade execution.

Finally, we consider competition between HFTs and the decision for PTs to become HFTs.

We find that profits from HF trading attract entrants who are willing to invest in acquiring the

skills necessary to compete for these profits, but that competition is limited. As the number of

HFTs increases, the expected profits of HF trading falls until the expected skills of an entrant

(relative to those of existing HFTs) are insufficient to generate enough profits to cover the

initial investments required to become an HFT. Thus, in equilibrium traditional PTs with low

expected skills as HFTs will continue in their traditional role, coexisting with others acting as

skilled and profitable HFTs.

4

Our analysis focuses on the effect of HFTs’ surplus extraction on trades initiated by liquidity

needs that generate temporary trading imbalances. Nevertheless, our analysis of the role and

effect of HFTs also applies to a broader set of circumstances. In particular it applies to trading

by mutual fund managers, hedge funds, insurance companies and other large investors, and

trading motivated not only by immediate liquidity needs, but also trading to build up or

unwind an asset position, for hedging, etc. The view taken in this paper that HFTs act as a

rent-extracting intermediary which increases the volume of trades without increasing the flow of

information, captures a widespread negative opinion of HFTs held by many market participants

(including North American and European regulators).

There are a number of empirical studies that look at the effect of AT and HF trading

on market quality. Although these studies employ different approaches and look at different

markets, one important difference between them is the depth and breadth of the data they

employ. In particular, only three studies have access to audit-trail data which includes trader

identity, something that all other empirical studies lack. The works of Kirilenko et al. [2010],

SEC [2010] and, very recently, Brunetti et al. [2011] study the role of HFTs in the E-mini

S&P 500 futures contracts. The fact that they can identify traders individually allows them to

isolate and study the impact of the trading activities of HFTs on other market participants and

the market as a whole. Interestingly, the results from these contemporaneous empirical studies

lend strong support to the stylized features that our theoretical model captures as well as the

implications concerning the impact that HF trading has on financial markets.

Kirilenko et al. [2010] study the impact of HF trading during the Flash Crash on May 6 2010.

They find that HFTs have the highest price impact among all types of traders and that “HFTs

are able to buy right as the prices are about to increase. HFTs then turn around and begin

selling 10 to 20 seconds after a price increase.” Moreover, they find that “The Intermediaries

[our Professional Traders] sell when the immediate prices are rising, and buy if the prices 3-9

seconds before were rising. These regression results suggest that, possibly due to their slower

speed or inability to anticipate possible changes in prices, Intermediaries buy when the prices

are already falling and sell when the prices are already rising.” Their findings strongly support

our assumption that HFTs (due to their speed advantage) can for the most part effectively

5

anticipate and react to price changes as a key part in their strategies for surplus extraction.

The recent work of Brunetti et al. [2011] shows that HFTs are responsible for a large volume

of trades in the E-Mini S&P 500 futures contracts, but this “high volume does not necessarily

imply a substantive flow of information. To the contrary, high volume may contribute to the

noise, at least, at high frequency sampling.” These results are consistent with our theoretical

model in which HFTs increase microstructure noise and volume but in a way that is not based

on market fundamentals.

Zhang [2010] studies changes in volatility over time and concludes that HF trading increases

stock price volatility and that this positive correlation between volatility and HF trading “is

stronger for stocks with high institutional holdings, a result consistent with the view that high-

frequency traders often take advantage of large trades by institutional investors”. This, again, is

consistent with our theoretical result that price impact is increasing in the size of the liquidity

imbalance.

Further empirical papers find that HFTs improve market quality, may dampen volatility

and reduce price impact, Brogaard [2010], Hendershott et al. [2011], Hendershott and Riordan

[2009], and Hendershott and Riordan [2011]. Most of these studies find that AT and HFTs

have a positive impact on market quality but the data employed are not as comprehensive as

audit-trail data. For example, Brogaard [2010] and Hendershott and Riordan [2011] use data

provided by NASDAQ with an HFT identifier, instead of trader identity, but the identifier is

narrowly defined for specialized HFT trading operations and explicitly excludes HFT activity

from major traders, such as Goldman Sachs, Morgan Stanley, and other large integrated firms.

Finally, one of the early papers that looks at the impact of AT in financial markets is that

of Chaboud et al. [2009]. The authors study the impact of AT in the foreign exchange market.

One of their findings is that the presence of more AT is associated with lower volatility of the

‘fundamental value’ of exchange rates. This interesting finding is not in contradiction with

our theoretical prediction that HFTs increase microstructure noise of the fundamental value of

the asset because in their study they employ minute-by-minute observations which dilutes the

contribution of microstructure noise to the variance of the exchange rate.

6

The rest of the paper proceeds as follows. In Section 2 we develop our framework and

analysis that describes how the HFTs compete to win trades and how the winner of the trade

mediates between LTs and PTs. Within this framework, we establish the properties of equilib-

rium prices, quantities, and optimal HFT behavior. In Section 3 we develop the general model,

which allows for competition between HFTs and entry–in the form of an PT who considers

setting up an HF trading desk. Section 4 discusses the effects of HFTs on market aggregates,

and Section 5 concludes with a discussion of some key features about HFTs that require further

research (and quality data).

2 The Model

We study the role of HF trading in a normative setting, where the stock market has social

value because it facilitates the financing of economic activity, adding value to equity holders

by providing a way for them to convert their equity into cash (and viceversa) quickly and

at a reasonable price. Grossman and Miller [1988] (GM) provide a highly stylized model of

such a stock market. In the GM model, equity investors or LTs quickly find counterparties for

their trading needs.2 These counterparties are PTs who are willing to take the other side of

investor trades and hold those assets temporarily–until another investor enters the market to

eliminate the temporary order imbalance.3 Holding these assets entails price risk, which PTs

are willing to bear in exchange for a small discount on the asset’s price (the liquidity discount).

We introduce HFTs into the GM model, where an HFT is a trader who, thanks to her rapid

information processing and quick execution ability, can position herself in the order book in2The other interpretation of the model is intended for the futures market. Everything we say for the stock

market is also equally valid for trading in the futures market in exactly the same terms. We refer the interestedreader to the original article for details of how to reinterpret the model.

3In GM, our PTs are called “market makers”. This is because the GM model was designed with specialistmarkets in mind. Nevertheless, the same idea of markets providing liquidity for a premium is valid in moregeneral settings, such as those with an open limit order book, where liquidity is not provided by “specialists”but by a group of PTs that are in the business of providing liquidity by exploiting temporary price deviationsfrom “fundamental” prices. A model where PTs arise endogenously can be obtained by, for example, including acost for monitoring the order book. In such a model, PTs are traders who are permanently monitoring the orderbook and who, despite having no privileged information, are willing to assume trading positions for a possiblysignificant period of time when the price deviates from its fundamental value.

7

such a way that she can profit from intermediating between the equity investor (LT) and the

PTs.

2.1 The Basic Structure

The setting for our model is a world with three dates, t = 1, 2, 3 in which a temporary order

imbalance of size i affects conditions in a stock market with a cash asset (with a return normal-

ized to zero) and a risky asset. There are two “outside investors”, LT1 and LT2. At time 1, LT1

is endowed with cash and an amount i of shares which due to a liquidity shock he would like to

sell (that is, he wants to trade −i shares). On the other hand, the other trader, LT2, wants to

buy i > 0 units (trade i shares) but only reaches the market at date 2.4 If both traders met at

the same time they would exchange the asset at the current “fundamental” price, which is the

expected future cash value of the asset. But, as LT1 arrives to the market much earlier than

LT2, his only option is to sell to the rest of the market, represented by M professional traders

(PTs). These act as intermediaries and will accept LT1’s order at date 1 at the “right price”

and hold the shares until LT2 enters the market at date 2. The “right price”, which we call the

market price of the asset, is determined by the fundamental price minus a discount commanded

by the PTs for holding the asset until date 2. PTs bear the risk of having to sell the asset at a

loss after the release of public news that might adversely change the fundamental price of the

asset, and they require this discount to compensate them for assuming this risk. The “liquidity

discount” is calculated as the difference between the “fundamental” value of the asset and the

price at which the transaction takes place, the market price.

All these traders (LT1, LT2, and PTs) are price-taking and risk averse. They have expected

utility from wealth at date 3 and they choose their asset positions so as to maximize E [U (W3)],

where U (W ) = − exp (−aW ) is the utility function, W is wealth and a is the risk aversion

parameter. The future cash value of the (non-dividend paying) asset is P3 = µ+ε2+ε3, where µ

is constant, and ε2 and ε3 are normally and independently distributed with mean 0 and variance

σ2. The random variables, ε2 and ε3, represent public information that is announced between4We make the assumption that LT1 wants to sell and LT2 wants to buy to streamline the presentation. The

analysis is equally valid if LT1 wants to buy i shares and LT2 wants to sell them.

8

dates t = 1 and t = 2, and between t = 2 and t = 3 respectively and σ2 represents the variance

of the shocks of the fundamental value of the asset. Let µ2 = E [P3|F2] = µ + ε2 denote the

expectation of P3 conditional on information available at t = 2. We use the notation θxt (Pt) to

describe the asset holdings of trader x after trading at date t given price Pt. A positive asset

position, θxt > 0, implies holding a positive (long) number of shares.

2.2 High Frequency Trading

The GM model identifies the relevant “liquidity” time scale as the time between two liquidity-

driven and mutually balancing trades, those by LT1 and LT2, and this time period is sufficiently

long that news can produce significant changes in the fundamental value of the asset (hence

the compensation for risk demanded by PTs). HFTs, on the other hand, are characterized by

having very short holding periods–so short that the probability of substantive changes in the

fundamental value of the asset during the period is insignificant. Thus, a single HFT cannot,

by definition, provide “true liquidity” in the GM model: her holding period is not long enough

for her to be exposed to uncertainty arising from changes in the fundamental value of the asset,

and hence does not bridge the time period between LT1 entering the market and the arrival

of LT2. This does not mean that market prices are constant during the HFTs holding period.

In fact, they cannot be constant, otherwise the HFT could not profit from buying and selling

the asset (they could even lose money from paying the spread net of exchange fees). Our

assumption, which is consistent with the evidence described in the contemporaneous empirical

studies by Kirilenko et al. [2010] and Brunetti et al. [2011], is that HFTs can make use of the

microstructure details of the price setting process (the dynamics of the order book) together

with their ability to transact at millisecond speeds to make profits by intermediating between

the LT and the rest of the market (the PTs). Thus we model HFTs as an additional (and

profitable) intermediation layer between LT and PTs.5 Note however, that our model of HFT

participation minimizes their negative impact as profits for this additional intermediation layer

come from the extraction of existing trading surplus in a way that is frictionless–in the sense5As pointed out by Kirilenko and Tuzun (personal communication), the way we capture HFT behavior may

also be useful to model other market frictions. In particular, they suggest that our model may help understandthe (historical) figure of “scalpers” on the trading floor.

9

that it does not alter current economic supply and demand conditions. However, they do have

a contemporaneous impact both on prices, traded volumes, and a dynamic impact, as other

traders anticipate future surplus extraction by HFTs.

That HFTs intermediate profitably is evidenced by their activity and the fact that they

report positive profits. Only HFTs know the exact details of the strategies they use, but from

our discussions with market participants we gather that HF trading is implemented using a

variety of strategies. A component of these strategies, which we believe drives the profitability

of HF trading, is that HFTs can exploit their speed to alter market conditions (their positions

in the order book) in a way that makes buyers accept a slightly higher price and sellers a slightly

lower one than they would have done otherwise. The simplest such strategies are associated

with the term “price anticipation”, and the idea is relatively simple and works in a setting where

LTs split their trades in small packages and PTs are outcompeted by HFTs who are able to

advantageously position themselves ahead in the limit order book.

To illustrate how this could work consider the following stylized numerical example: Suppose

a trader (LT) needs liquidity and wants to sell a block of shares, splitting them in several

packages. As the first package comes into the system (say at the best buy price of $5.50 per

share), the HFT identifies an increase in selling pressure and cancels her outstanding posted

buy offers that have not been filled. She then posts additional sell offers, adding to the increased

selling pressure, in order to help clear the remaining posted buy orders in the book. Once the

book is clear, the HFT quickly reposts a significant number of offers at lower prices (say $5.47)

so that she is first in the buying queue. This is only possible if she can move quickly enough

that by the time the PT reacts to the increased selling pressure, new offers posted by the PT

sit behind those posted by the HFT at $5.47 per share. The LT finds that the market around

$5.50 has dried up and can only sell at $5.47. The following share packages are bought by the

HFT, who is at the front of the queue. Being at the front of the queue allows the HFT not only

to be the first to buy at this low price, but also to be the first to notice when the selling pressure

eases. At that moment, the HFT cancels her posted buy offers and starts posting sell offers at

slightly higher prices (say $5.49 first and then at $5.48). The PT sees the selling pressure shift

and the price rebound. Although the PT is sorry to see the lower prices disappear, he is still

10

willing to buy at $5.49, which allows the HFT to sell her earlier purchases and end up with a

zero net position. During this process, the HFT has been able to make profits on the shares

bought at $5.47 and sold at $5.48 or $5.49, while taking a loss on the shares executed at the

beginning, bought at $5.50 and sold at $5.48 or $5.49. An HFT who is fast enough will have

bought many more shares at $5.47 than at $5.50 by selectively canceling and reposting her

offers to make the strategy profitable. As for the other traders: The PT bought some shares

below the initial best buy price so he is satisfied; the other LT is also satisfied from having been

able to sell his shares even though for some of them he received a cent less than what the PT

paid for them.

Note that the HFT in this example has used her speed to cancel and repost at different

levels of the order book so as to (a) allow the selling pressure to drive prices lower than they

would have been in the absence of HFTs, and (b) position herself at the front of the queue in

the order book so as to be the first to exploit the temporary drop in prices, buy at low prices

and repost so that the rest of the market absorbs her trades at slightly higher prices.

In order to keep the model tractable we do not introduce the dynamics of HFT strategies

into the model. Instead, we focus on the key outcomes of HFT activity: profitable trading at a

much finer timescale, and zero net positions. To do this, we allow several actions to take place

at liquidity trading times, t = 1, 2, when in reality they take place sequentially at several points

in time around, but very close to, those liquidity trading times. Thus, by executing different

actions in a very short time span, HFTs can exploit the microstructure of trade execution

and the dynamics of the order book, both at t = 1, as the price moves from the pre-trade

fundamental level (µ) to its post-trade market price (P1), and at t = 2–as it moves from P1 +ε2

to P2.

Instead of modeling the details of how HFTs profitably exploit their speed advantage, we

use the following abstract characterization: [(i)] HFTs use their speed to position themselves

between LTs and PTs. In the benchmark model (this section) we start by assuming that a single

HFT intermediates all trades. This is profitable because in the process of intermediation, HFTs

can extract trading surplus. In particular, [(ii)] we assume the HFT takes over other traders’

positions in two blocks: she obtains a first block of shares at an advantageous price: the market

11

price Pt plus or minus a haircut ∆ (depending on whether the HFT is selling or buying), and

obtains the second block of shares at the market price, Pt. Thus, the HFT profits from each

trade twice: once, when acquiring the position from one side of the market, and a second time,

when transferring the position to the other side of the market.

As we discuss below (in Section 3) the ability of HFTs to profit from their speed and

knowledge of the dynamics of trade execution depends on a number of factors and in general

HFTs will not be able to intermediate all trades, and even those they intermediate may not

be profitable. It could also happen that LTs break their trades into blocks, some of which are

matched with one or more HFTs during the trading process, while others are matched to PTs

directly, or the trade may become a “hot potato” moving from one HFT to another. As adding

these extra properties to the model makes it less tractable, we introduce them later, when we

include competition amongst HFTs in the model (Section 3).

The assumption that the HFT captures the trade in two blocks is mathematically convenient

for two reasons: one, it allows us to avoid having to model the time interval around liquidity

trades, and two, it allows us to capture the notion that HFTs exploit the microstructure of

price setting without having to model the exact (and unknown) details of how they actually do

it.

2.3 The GM model with an HFT: a numerical example

The following numerical example helps illustrate the mechanics of HFT intermediation profits

in our stylized, two-block trading model, and how those profits are determined by market

conditions. The example is immediately followed by the description of the general benchmark

model.

Suppose that an LT wants to sell 1,000 shares, the expected price of the asset is $5.50

and there are nine PTs. Suppose further that all traders are risk averse and they have linear

demands for assets given by

θT (P ) = θLT (P ) = 5,000 (5.50− P ) .

12

Hence, they will only buy shares if the price is below $5.50.

In equilibrium the holdings of all traders (one LT and nine PTs) have to add up to the

supply of LT (1,000 shares) so that

9θT (P ) + θLT (P ) = 1,000

⇒ 9× 5,000 (5.50− P ) + 5,000 (5.50− P ) = 1,000

⇐⇒ P = 5.48 .

The price drop to $5.48 per share reflects how the market reacts to the LT’s selling pressure.

At the lower price each trader (including the LT) will hold 100 shares. This implies that the

(price sensitive) LT who initially wanted to sell 1,000 shares (at $5.50) ends up only selling

900 shares at a price of $5.48 (the fundamental price $5.50 minus a 2 cent liquidity discount).

Thus, the liquidity trade involves fundamental traders (LT and PTs) exchanging assets at the

market price, $5.48.

In a market with HFTs, the bid stack is populated with HFT bids that compete to capture

the trade and unwind it at a profit. Buying bids disappear as soon as the market feels the

increased selling pressure. In our simplified benchmark model, the increased selling pressure is

entirely absorbed by a single HFT who successfully captures the trade at a discount, say she

takes a 1 cent per share haircut and buys at $5.47. Having bought the shares from LT, the

HFT unloads it on the market (PTs), and again takes a haircut, selling to PTs at the slightly

higher price of $5.49.

However, the HFT is an intermediary and her profits are determined by the quantities she

trades, which are determined by “fundamental” traders’ asset demands. At a price of $5.47, LT

will want to hold 5,000 (5.50− 5.47) = 150 shares, thus selling only 850 shares. At a price of

$5.49, PTs will hold 5,000 (5.50− 5.49) = 50 shares each, which in total amounts to 450 shares.

The trades from LT and the PTs do not cancel so that the HFT is left holding 400 shares. This

is not satisfactory for an HFT who wants to hold zero inventory at the end of trading. Thus,

the HFT makes use of her ability to acquire blocks at different prices and modifies her trading

13

strategy slightly: After offering to buy 850 shares from LT at $5.47, she offers to buy another

batch of 50 shares from the LT at $5.48. The LT finds these trades acceptable and executes

them. Having acquired all LT’s shares, the HFT turns around and first offers to sell 50 shares

to each of the PTs at $5.49. Then, she offers to sell them another 50 shares each at $5.48.

Again, this combination of offers is acceptable to the PTs and they agree to execute them. At

the end of these trades, the HFT has made 2 cents on each of 450 shares sold at $5.49 (and

bought at $5.47) and another cent on 400 shares bought at $5.47 and sold at $5.48 for a total

of $13.00. She also has sold 50 shares at $5.48 at no profit to clear her inventory, and is left

with no shares, only profits. The market has seen a doubling of the volume of shares traded,

plus some transactions taking place at prices above and below the “market price” of the asset

($5.48)–the fundamental price ($5.50) minus the equilibrium liquidity discount ($0.02).

This numerical example captures how we model the HFT’s profit extraction strategy, and

how the HFT behavior affects market variables, such as prices and trades. Furthermore, note

that the HFT’s strategy is frictionless in the sense that the quantity demanded by the market

and supplied by the LT (900 shares) is the same that would have been traded without the HFT

(at the constant market price of $5.48). We now proceed formally with the benchmark model.

2.4 The benchmark model

The timeline is as follows (Figure 1): at date 1, the price sensitive trader LT1 comes to the

market wanting to sell i shares. He is price sensitive so that his date 1 excess demand is given

by θLT11 (P1)− i, where we recall that θxt (Pt) is the asset holding after trading at date t. The

HFT trades with LT1 in two blocks, she buys the first block at P1 − ∆ and the other at P1.

Immediately, and still at date 1, the HFT turns around and sells to the PTs in two blocks: one

block at P1 + ∆ and the other at P1. As time passes between dates 1 and 2, there are public

announcements of news about the future value of the asset, P3, which generates fluctuations in

the expected value of the asset. At date 2, the other price sensitive liquidity trader LT2 decides

to enter the market to acquire i shares. His excess demand is θLT22 (P2) + i. The HFT quickly

purchases all the shares that the PTs had bought in date 1, plus those that LT1 could not sell

14

in date 1, in two blocks: one block at P2−∆ and the other at P2. Then the HFT turns around

and sells all the shares to LT2 in two blocks: one at P2 + ∆ and the other at P2.6

Figure 1: Timeline of trading in the presence of the HFT

We now proceed by solving the general model by backward induction, starting at date 2

and then moving one period back to the problem at date 1.

Trading at t = 2

The second liquidity trader, LT2, enters at t = 2 and given price P2, this price sensitive trader’s

excess demand is θLT22 (P2) + i shares, but he knows that there is an HFT who will extract

some surplus. In the benchmark model, any trader who wants to buy θ (P ) + i units at price P

will have to go through the HFT who will split the order in two blocks at quantity θ̃: the HFT

will sell θ̃+ i units at price P + ∆ and the remaining θ (P )− θ̃ units at P . To ensure trades are

mutually acceptable at all prices the quantity θ̃ corresponds to the trader’s excess demand at

P + ∆, that is θ̃ = θ (P + ∆). We assume traders accept trading in two blocks and paying the

haircut as part of the cost of doing business, and that they take the quantity θ̃ as exogenously6The time sequence of trades is not important, as the HFT can acquire or sell short shares in anticipation

of, or in reaction to liquidity trades. The key is that the HFT’s net position coming in and out of the liquidityshocks is essentially zero.

15

given.7 We use the notation θ̃xt to describe the quantity that the HFT uses to split trader x’s

desired trades at date t into the two blocks. The consistency condition, θ̃ = θ (P + ∆), coupled

with the fact that the HFT splits the orders into only two blocks, limits the amount of surplus

the HFT can extract.

We also limit the advantage gained by the HFT in terms of the information she can use to set

haircuts. In principle, the HFT could distinguish between different traders and different periods

of time. But, being able to make such fine grained distinctions between traders would require

extracting an amount of information from the order flow that we consider unrepresentative

of what goes on in stock markets today. Nevertheless, we do believe that, depending on the

amount of liquidity available, HFTs may be able to extract sufficient information from the

order flow at a sufficient speed to allow them to distinguish a large liquidity trade coming in

(one that will move the market price substantially) from regular trades posted by other market

participants. The more liquid the market, the harder it will be to make this distinction. In the

basic analysis, we allow the HFT to distinguish “large” trades (the initial trade by LT1 at date

1 and LT2’s trade at date 2) from “small” trades (trades by PTs and LT1 at t = 2). Thus, the

HFT sets two haircuts, one for large trades ∆L, and one for small trades ∆S .8

Since LT2’s excess demand is θLT22 (P2) + i, the HFT applies the large haircut to him and

sells i+ θ̃LT22 shares at the (higher) price, P2 + ∆L, and θLT2

2 (P2)− θ̃LT22 , at the market price

P2. Thus, given an initial wealth of WLT22 , LT2’s final wealth is

WLT23 = WLT2

2 + (P3 − P2)(θLT2

2 + i)− P3i−

(θ̃LT2

2 + i)

∆L,

where we are omitting the functional dependence of asset holding on prices, using θLT22 instead

of θLT22 (P2).

7It is possible to introduce a strategic element whereby the trader alters his bidding behavior in anticipationof the HFT. This can greatly complicate the model. Basically, it affects how much of the trading surplus isextracted by the HFT and imposes an additional distortion on trade.

8In the Appendix we have included more general formulas and additional results with slightly differentassumptions on how the HFT applies haircuts across dates and traders.

16

Substituting for wealth in the utility function, we obtain:

E [U (W3) |F2] = − exp(−a((θLT2

2 + i)

(µ2 − P2)− iµ2 −WLT22 −

(θ̃LT2

2 + i)

∆L

)+

12a2σ2 (θ2)2

).

(1)

LT2’s final excess demand, θLT22 + i, is such that it maximizes (1). Hence,

θLT22 + i =

µ2 − P2

aσ2+ i . (2)

Similarly, LT1’s and PTs’ excess asset demands at t = 2 are given by

θLT12 − i =

µ2 − P2

aσ2− i and θT2 =

µ2 − P2

aσ2.

These three expressions reflect the frictionless introduction of HFTs, as asset demand sched-

ules (at t = 2) are unaffected by the known presence of HFTs (relative to the original GM

model).

Market clearing requires that the excess demand of LT1 (who arrived at date 1), plus PTs’,

plus LT2’s sum to zero:

θLT12 − i+MθT2 + θLT2

2 + i = 0 ,

so that the equilibrium price is P2 = E [P3|F2] = µ2 , and the asset holdings after trade in

date 2 are θLT22 = θLT1

2 = θT2 = 0. The quantities used by the HFT for extracting surplus are

obtained from the corresponding demand functions θ̃LT12 = θ̃T2 = θLT1

2 (P −∆S) and θ̃LT22 =

θLT22 (P + ∆L):

θ̃LT12 =

∆S

aσ2, θ̃M2 =

∆S

aσ2, θ̃LT2

2 = −∆L

aσ2.

The HFT will extract trading surplus equal to9

Π2 (i) =∣∣∣θ̃LT2

2 + i∣∣∣∆L +

∣∣∣θ̃LT12 − θLT1

1

∣∣∣∆S +M∣∣∣θ̃T2 − θT1 ∣∣∣∆S . (3)

9We are assuming that the PTs and LT1 are net sellers of the asset. This is confirmed as in equilibrium PTswill be net buyers in the first period and have zero net final holdings (and θLT1

2 − θLT11 has the same sign as

PTs’ changes in asset holdings at date 2).

17

The benchmark model: Trading at t = 1

Traders at date t = 1 anticipate what will happen at t = 2. They also know that there will be

public information revealed prior to date 2 trading and that P2 = µ2 = µ + ε2 is random and

normally distributed with mean µ and variance σ2.

Traders’ future wealth can be written as

WLT13 = WLT1

0 +(P2 − P1)(θLT1

1 − i)+(P3 − P2)

(θLT1

2 − i)+P3i−

(θLT1

1 − θ̃LT12

)∆S−

(i− θ̃LT1

1

)∆L

and

W T3 = W T

0 + (P2 − P1) θT1 + (P3 − P2) θT2 −(θT1 − θ̃T2

)∆S − θ̃T1 ∆S ,

where θLT1t − i and θTt are LT1’s and PT’s excess demands respectively in dates 1 and 2.

Using E [P2|F1] = E [µ+ ε2] = µ, simplifies the expression for traders’ wealth, and it is

straightforward to derive optimal excess demands:

θLT11 − i =

1aσ2

(µ− P1 −∆S)− i and θT1 =1aσ2

(µ− P1 −∆S) .

Notice that traders anticipate that their current asset demand (and hence positions at the

end of trading at t = 1) will affect future trading and hence the (date 2) haircuts they will

have to pay. Thus, the HFT’s strategy, though statically frictionless, does generate a dynamic

distortion, as future haircuts affect current asset demands.

The date 1 market clearing condition is

(θLT1

1 − i)

+MθT1 = 0 .

From this we obtain: (i) the market clearing price

P1 = µ−∆S −iaσ2

M + 1, (4)

18

(ii) traders’ asset holdings after trading in date 1,

θLT11 =

i

M + 1and θT1 =

i

M + 1,

and, using θ̃LT11 = θLT1

1 (P −∆L) and θ̃T1 = θT1 (P + ∆S), (iii) the quantities that are subject

to haircuts

i− θ̃LT11 = i− i

M + 1− ∆S

aσ2and θ̃T1 =

i

M + 1− ∆S

aσ2.

Again, the HFT is able to extract trading surplus and generate profits of:

Π1 (i) =∣∣∣i− θ̃LT1

1

∣∣∣∆L +M∣∣∣θ̃T1 ∣∣∣∆S . (5)

In summary, from the above analysis we extract the following conclusions:

Theorem 2.1. For a given order imbalance of magnitude i > 0:

1. Market clearing prices are

P1 = E [P2|F1]− iaσ2

M + 1−∆S , (6)

P2 = E [P3|F2] = µ2 ,

and the liquidity discount at t = 1 is

E [P2|F1]− P1 =iaσ2

M + 1+ ∆S . (7)

2. Asset trading is described on Table 1:

19

Table 1: Prices and Volume of Trades

Price LT1 LT2 PT (total)

P1 −∆L − MM+1

i+ ∆Laσ2

P1 −∆Laσ2 M ∆S

aσ2

P1 + ∆SMM+1

i−M ∆Saσ2

P2 + ∆L i− ∆Laσ2

P2 − ∆Saσ2

∆Laσ2 −M ∆S

aσ2

P2 −∆S − 1M+1

i+ ∆Saσ2 − M

M+1i+M ∆S

aσ2

3. The HFT acts as counterparty to all these trades and makes profits equal to

Π (i) = Π1 (i) + Π2 (i) , (8)

where Π1 (i) and Π2 (i) are described in Equations (5) and (3) respectively.

The presence of HFT intermediation affects the market clearing price at t = 1, when the

order imbalance is initiated. Traders, anticipating future haircuts imposed by HFTs, amplify

the selling pressure of the liquidity trader initiating the order imbalance (LT1), driving the

date 1 market clearing price down further than it would have been without HF trading (that

is, with ∆L = ∆S = 0), and increasing the liquidity discount paid by LT1 to encourage PTs to

buy the asset from him (Equation (7)).

Table 1 illustrates a number of the properties and implications of our stylized model. For

example, our model exhibits (additional) “microstructure volatility” induced by HFT interme-

diation. We can observe how the HFT’s surplus extraction, which involves buying and selling

at different prices, introduces additional prices at which transactions take place. These price

movements around the market clearing price have no informational content and may lead a

casual outside observer to erroneous conclusions about what the “true” market price of the

asset should be at each date. These properties and implications are supported by the empirical

findings in Brunetti et al. [2011].

20

Naturally, having allowed the HFT to intermediate all trades, the volume of trade doubles.

This assumption is weakened in Section 3. Nevertheless, these additional trades (and hence

HFTs) do not add liquidity–in fact, the presence of HFTs results in an increase in the liquidity

discount, thus liquidity is reduced. Hence, rebate schemes based on volume are likely to dis-

proportionately benefit HFTs who are not providing liquidity, as the following simplistic rebate

scheme illustrates:

Corollary 2.2. If there is a rebate of c cents per share, the HFT gets half of all rebates, and

the M professional traders split a quarter of all rebates between them.

2.5 Optimal Haircuts

Having captured a trade, the HFT realizes that her profits depend on the size of the haircut

and the constraint imposed by the minimum tick size. We have little to say about the latter

constraint, other than it may be binding, in which case the optimal haircut is the minimum

tick size. Without this constraint, the HFT faces a trade-off between a larger haircut, and

a smaller number of assets traded using that haircut. Thus, she will adjust her behavior by

setting haircuts that maximize the profits of her trading activity. As the HFT distinguishes

the haircut for large trades (∆L) from the one for the small trades (∆S), the HFT’s profits

described in Equation (9) can be expressed as:10

Π (i) = ∆L

(M

M + 1i− ∆L

aσ2

)+∆S

(M

M + 1i−M ∆S

aσ2

)+∆L

(i− ∆L

aσ2

)+∆S

(i− (M + 1) ∆S

aσ2

),

(9)

and by maximizing (9) with respect to ∆L and ∆S we obtain:

Lemma 2.3. The HFT maximizes profits by choosing

∆S =12

1(M + 1)

iaσ2 ,

∆L =14

2M + 1M + 1

iaσ2 = ∆S

(M +

12

).

10In the Appendix we discuss another case (where the HFTs make no distinctions and apply the same haircutto all trades).

21

The size of the haircuts is increasing in i, the size of the liquidity need, because of the

pressure from LTs. This implies that the corresponding additional microstructure noise and

liquidity discounts will increase for larger order executions. If we interpret the number of PTs,

M , as a proxy for the amount of “real liquidity” available for the traded asset, then the haircut,

microstructure noise and liquidity discount decrease with available liquidity.

Furthermore, since the HFT can distinguish between large and small trades, she will impose

bigger haircuts on large trades than on small trades, and the increase in haircut for large

trades is proportional to the amount of “true” liquidity. This seems counterintuitive, but as we

mentioned above and will see in more detail below, in the competition section, “true” liquidity

has the additional effect of making it easier for LTs to remain undetected, and hence making it

harder for the HFT to distinguish large from small trades thereby counterbalancing this effect

on haircuts.

Figure 2 below illustrates how the haircut for the large (small) trades increases (decreases)

with M , and Figure 3 shows the optimal asset holdings, θxt and θ̃xt , for the liquidity traders

and PTs.

Moreover, we see that in a market with more PTs, the HFT increases the haircut imposed

on large trades from liquidity seekers, LT1 at date 1 and LT2 at date 2, and lowers it on small

trades. When an HFT can identify large trades, in the limit as the number of PTs goes to

infinity,

limM→∞

∆S = 0 and limM→∞

∆L =12iaσ2 ,

we see that the haircut on small trades disappears and all the trading surplus extracted by the

HFT comes from large trades. In addition, as the number of PTs goes to infinity, the liquidity

discount (Equation (7)) also goes to zero, so that LT1 and LT2 behave in the same way: their

liquidity needs are immediately satisfied (LT1 sells i in date 1), but they trade half their shares

at a price of µ ± 12 iaσ

2 and the other half at the asset’s fundamental value, µ. Then, overall

the HFT then makes profits equal to 12 i

2aσ2 .

22

Figure 2: Optimal Haircuts for large and small trades

1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Number of PTs

De

ma

nd

s a

ssu

min

g i =

a =

σ

2 =

1

∆LT1

1 = ∆LT22

∆T1 = ∆T

2 = ∆LT12

3 Competition in the expanded model

In the previous section we analyzed the benchmark model with a single HFT who intermediated

all trades and set the haircut. We saw how the haircuts affected trading behavior and market

prices. In particular, we saw how it affected trades, prices, and the liquidity discount. In this

section we expand the model to incorporate additional interactions between HFTs and between

HFTs and LTs, so as to capture the tempering effect of competition on the ability of HFTs to

intermediate “fundamental” trades and profit from doing so.

3.1 Modeling competition amongst HFTs and entry decisions

Our discussions with regulators and practitioners on both sides of the HFT fence identified a

number of issues regarding the way in which HFTs profit from financial intermediation (and

incorporate issues raised in the previous section): (A) Becoming an HFT, whether it is a new

23

Figure 3: Optimal asset holdings when the HFT sets a haircut for large trades and anotherhaircut for small trades.

1 2 3 4 5 6 7 8 9 10−0.5

0

0.5

1

Number of PTs

Dem

ands

ass

umin

g i

= a

= σ

2 = 1

θT11 = θT

1

θ̃LT11

θT1 = θLT1

1

θ̃T1

θ̃LT12 = θ̃T

2

θ̃T2 = θ̃LT1

2

θ̃LT22

market entrant or a PT that decides to upgrade, requires high initial sunk costs in hardware,

co-location, software and data, but, most importantly, it demands skills specific to HF trading,

all of which pose barriers to entry. (B) Traders, both LTs and PTs, have reacted to changes in

market conditions. One of these changes is that they break up their trades into small blocks

to try to mitigate their price impact when they take significant positions; a proportion of these

trades are captured directly by PTs (bypassing HFTs). (C) An HFT’s ability to position herself

at the front of the queue (and capturing an incoming trade) is no guarantee that the HFT will

be able to close the trade at a profit. (D) An HFT cannot know whether an incoming trade is

a liquidity trade or if it is coming from another trader, specially another HFT.

In order to incorporate competition into our model we make two assumptions: that there

are entry barriers, and that competition centers around intermediating trades and not around

price discounts.

The existence of entry barriers can take two forms as mentioned in (A), both of which

we include. The first is entry barriers in terms of initial costs of entering the HFT market.

The publicity surrounding HF trading profits attracts competition, specially from PTs who

24

understand the usefulness of the additional speed and processing advantages. These would

be the first to be able to value the costs and benefits from becoming HFTs, and whether the

investments required for entry are worthwhile or not. The second is skill. There is a steep

learning curve to become a profitable HFT, and it is not clear that differences with early

entrants can be reduced. The joint effect of these barriers is to limit the number of entrants,

something which is consistent with available evidence: In the literature and discussions with

market participants we find that the number of HFTs has not increased in some markets and

increased in others, but the potential profits for the latest and future entrants have dwindled

to the point of being essentially negative. Furthermore, despite the increase in the number

of HFTs, profits from early entrants seem to remain stable. Empirically, the precise number

of HFTs operating in different markets and venues, and how this figure has evolved over the

last few years, is largely unknown. Exceptions are the study by Kirilenko et al. [2010] which

identifies 16 HFTs, out of the almost 12,000 active traders, during the Flash Crash, Brunetti

et al. [2011] identify 11 HFTs, out of approximately 15,000 active traders, and that of Menkveld

[2011] which discusses the role of a single monopolist HFT that operates in Chi-X. What we

conclude from this is that competition amongst HFTs is present though limited.

The assumption on the nature of competition is based on three arguments. First, price

competition is naturally restricted by the minimum tick size. Second, price competition should

have wiped out HFT profits, something that is inconsistent with the information received from

financial markets. A standard Bertrand price competition model tells us that you only need

two competitors to drive HFT profits to zero, and that is not happening. And third, HFTs’

relative advantage is in exploiting their edge in speed and information processing. It is more

about skill and technology (in capturing and closing a trade, and positioning trades in the order

book) than about offering the best execution price.

We implement our first assumption by imposing an entry cost and incorporating skill dif-

ferences into our model of competition. The second assumption is incorporated by fixing prices

(we keep the haircuts used above, ∆S and ∆L, as given), and focusing on the effects of compe-

tition between HFTs on quantities (trades intermediated). In particular, we assume that HFTs

(as a block) do not profitably intermediate all trades between LT and PTs, but only a fraction

25

of those, denoted by ϕ. Equivalently, ϕ is the probability that HFTs capture the whole trade,

and does so profitably. Skill differences is what determines how trades intermediated by HFTs

are distributed amongst competing HFTs. In particular, conditional on the HFTs capturing an

incoming liquidity trade, we assume that HFT j = 1, . . . , n, obtains expected profit pjΠ (i),

where pj is the probability that HFT j is the one to capture the trade.11

We focus on two factors that determine how skills and competition affect ϕ and p: the

number of competing HFTs, n, and each HFT’s (relative) skill which is parametrized by ηj .

The skill parameter, ηj , determines the success of HFT j in capturing a round-trip trade. In

our model a high ηj means that HFT j’s probability of capturing a trade that yields profit Π (i)

is higher; thus we let pj = p (ηj) be an increasing function of ηj . The individual skill parameter

ηj allows us to capture differences between early and late entrants, and in particular, to be

able to have successful early HFTs coexisting with marginally profitable new entrants. It also

captures the idea that entrants can continue to extract profits as long as they can retain their

skill advantage at capturing rents from HF trading both relative to other HFTs, and other

traders in general.

The probability of HFT intermediation, ϕ, is affected by two factors. First, there is a direct

effect of competition: a higher number of competitors, n, leads to a lower ϕ. This captures

the effect that more HFTs will generate noisier trading (related to issues (C) and (D) above),

and hence reduce the overall effectiveness of HFT profit extraction strategies. As ϕ is also the

fraction of trades that are profitably intermediated, it reflects the effect that with a greater

number of HFTs a trade, even if captured, may be intermediated at a zero profit (excluding

rebates).

A second key factor is the relative technological gap between HFTs and other traders, as

suggested by (B) above. The individual skill parameter allows us to calculate an average skill11This assumption is consistent with the following anecdotal description of HF trading behavior: HFTs

position orders in the queue so that their bids and offers are hit or lifted whilst maximizing the probabilitythat they close the position at a profit over a very short-term horizon. These strategies require the HFTs toconstantly cancel and send orders to both sides of the book as well as orders that could be passive or aggressiveto jockey for position in the limit order book. Thus, once a market order hits a bid or lifts an offer, it is notsurprising to see a large number of orders disappear from both sides of the order book. The HFTs who areunable to capture the trade or are not adequately positioned, cancel their orders and withdraw, leaving the bestpositioned HFT to intermediate the incoming trade.

26

parameter: η̄ = 1n

∑nj=1 ηj . A greater technological gap between HFTs and other professional

traders (a higher η̄) increases the proportion of trades that are profitably captured by HFTs.

These two factors are incorporated into the probability that HFTs (as a block) profitably

intermediate trades, as described by the function ϕ (n, η̄), where ϕ (n, η̄) is decreasing in n and

increasing in η̄.

3.2 The model, expanded to include competition

We incorporate competition in HF activities in two stages. First, we describe the outcome of

competition when there is a fixed number, n, of HFTs who compete to extract surplus when a

liquidity trade of size i is brought to the market. Second, we consider the entry decision of a

potential new HFT—for example a PT who considers whether or not to become an HFT—who

must make an initial investment (sunk cost) to compete with other HFTs.

Suppose that an LT enters the market with a liquidity need of size i. Now, market partic-

ipants face uncertain HFT intermediation. Overall, this increase in realism comes at the cost

of analytical tractability. In particular, asset demands at t = 1 are obtained as the solution to

θ1 − q (θ1) ∆S =1aσ2

(µ− P1) ,

where

q (θ1) =1

1 + 1−ϕϕ exp

(a(θ1 − θ̃2

)∆S

) . (10)

These demands correspond to an intermediate case between the GM model (where q = 0)

and the benchmark model (where q = 1) and, as expected q ∈ [0, 1] is increasing in ϕ. We

lose analytical tractability because q is not a constant, but a function of θ1. The additional

uncertainty requires a relatively greater liquidity discount, nevertheless, the qualitative effects

remain unchanged.

A trade captured by an HFT represents a potential profit of Π (i), as approximated by those

described in Equation (8) above. As a block the HFTs compete for the expected profit ϕΠ (i),

27

but the individual success of HFT j in profiting from a particular trade is determined by the

probability that once captured, the profits from intermediation go her, pj .

To make the effect of relative skill differences on the profitability of being HFT analytically

tractable we introduce additional assumptions. For each trade and each HFT j, there is an

i.i.d. realization θj of a random variable Θj . We assume that Θj is exponentially distributed

with parameter ηj (the HFT’s relative skill), i.e. Θj ∼ exp (ηj). Given the realizations of Θ for

all HFTs, (θj)nj=1, the trader with the smallest realization “wins the trade”. Thus, given ϕ, the

probability that trader j extracts profits from the trade is given by

ϕ (n, η̄) p (ηj) := ϕPr {Θj = min {Θ1, . . . ,Θn}} = ϕηj∑nu=1 ηu

.

Hence, the expected profit for trader j from a trade of size i is ϕp (ηj) Π (i). The reader

can readily verify that if one were to introduce the parameters ϕp (ηj) in the optimal haircut

problem in the benchmark model of Section 2 the results do not change.12

Now we analyze the decision of a potential entrant. Suppose a PT observes the activities of

n HFTs and considers becoming an HFT himself. The PT identifies the irreversible investment

costs K of setting up an HF trading desk (hardware, software, human capital, co-location,

etc.), including the opportunity cost of abandoning current activities such as traditional market

making. The potential entrant does his research and determines that if he were to obtain a skill

level η from his investment, the expected profits would depend on p (η) and on the expected

number of liquidity trades N(i) of size i. Thus PT’s expected utility from future HFT is

ϕp (η) E [U (Π (i) ;N (i))]. Then, a PT with skill η compares the utility from wealth K, U (K),

with the expected utility from future profits, and becomes an HFT if

k ≤ ϕp (η) ,12With the possible exception of the size of the optimal haircut. Nevertheless, the optimal haircut as described

in the previous section could still be valid if the aggregate HFTs’ price setting behavior emulates that of amonopolist–maximizing the sum of profits from both the incoming liquidity shock and the subsequent one,canceling the first (by LT2). This could occur in a context with a small number of major players who are able toreach tacit collusion agreements. Alternatively, the minimum tick size could act as an effective barrier to pricecompetition. In the latter case, the haircut would not be optimal, but fixed endogenously, and hence unaffectedby competition.

28

where k = U (K) /E [U (Π (i) ;N (i))] denotes the relative “utility” cost of a trade.

The entrant has a reasonable estimate of k, but faces uncertainty with respect to the effec-

tiveness of the investment, η and ϕ, though uncertainty on the latter is relatively insignificant

for large n (where the relative importance of ηj on η̄ is small). For simplicity we assume that the

skill of the entrant, η ∈ (1,∞), is random and with a distribution such that its inverse, 1/η, is

uniformly distributed, that is 1η ∼ U (0, 1), and the effect on η̄ is negligible. As discussed above,

the probability of becoming the HFT that intermediates a liquidity trade depends not only on

the HFT’s skill but more importantly, on her skill relative to those of existing HFTs. Thus,

any potential entrant needs to take into account that there are n HFTs with skills described by

parameters {ηu}nu=1. Then, an entrant with skill η becomes the monopolist with probability

p (η) =η

η +∑n

i=1 ηi=(

1 + n

∑ni=1 ηiη

)−1

.

Let αj =∑n

i=1 ηi > 1 denote the total skill pool of other existing HFTs (excluding j)–we

drop the j subscript until the final part of this section when it becomes useful. Then, the

cdf and pdf of p (η) are Gp (z) = 1α

(1 + α− 1

z

)and gp (z) = 1

α1z2 respectively, and E [p (η)] =

1α log (1 + α).13

Given a relative cost of entry of k, and letting α̂ denote the total skills of incumbent HFTs,

there will be entry until

k ≥ ϕ

α̂log (1 + α̂) .

This condition establishes the relationship between entry costs, relative skill levels, and

the probability of successful intermediation by HFTs. But, as our model does not incorporate

positive effects from HFT, we cannot use it to make a reasonable welfare analysis. Nevertheless,

allowing for positive effects from increased execution speed, or fewer welfare-enhancing trades

being lost, as in Foucault et al. [2011], we would find externalities in the entry decision by a13See the Appendix for details.

29

PT. A PT’s entry calculation of the value of becoming an HFT does not account for the fact

that it affects the total amount of HFT intermediated trades (unmodeled, but the effect could

be positive) and the amount of trades profitably intermediated by HFTs. The latter effect is

ambiguous, as an increase in n reduces ϕ, but the skill of the entrant can be above or below η̄,

and a sufficiently high η entrant could have a positive net effect on ϕ. Overall, as discussed in

Foucault et al. [2011], the role for regulation depends on whether these externalities bias PTs

towards excessive or insufficient entry.

Beyond the externalities of becoming an HFT, our model also allows for an analysis of the

(unmodeled) dynamics of skill differences. In particular, if skill differences between early and

later entrants are not eroded over time, differences in profitability can become permanent. This

would lead to a situation in which, as HFTs’ relative skills stabilize, the market settles around

a core set of players who extract profits from a large fraction of trades, while less skilled, later

entrants compete for the remaining trades and scrape enough to cover their operating (and

opportunity) costs. An additional dynamic factor to consider is that, as other non-HF traders

upgrade and adapt their trading strategies and technology to the presence of HFTs, the overall

technological gap (η̄) decreases and the overall profitability of HF trading is eroded.

4 Measuring the impact of HFTs on financial markets

Empirical studies of HFT offer seemingly conflicting conclusions on the effect of HFTs. On

the one hand, Hendershott et al. [2011] find that AT improves liquidity and narrows effective

spreads and Brogaard [2010] finds that HFTs contribute to price discovery and reduce volatil-

ity. Although the study of Brogaard [2010] identifies a subset of HFTs, his data set cannot

differentiate how much of the activity of the 26 firms in his study is AT and how much is HF

trading and there might be large market participants that use HF strategies which are not

identified in the datase as HFTs; the same applies to the results of Hendershott et al. [2011].

On the other hand, there are the studies that employ proprietary audit-trail data, SEC [2010],

Kirilenko et al. [2010], and Brunetti et al. [2011]. They find evidence that HFTs increase price

impact, increase volume of trading without increasing the flow of information, and increase

30

noise around the fundamental value of the asset. Our abstract model allows us to identify

the effect of HFTs and how they may be affecting different aggregate financial metrics such as

volume, liquidity, price impact, and price volatility. This we do here, and find that our model

is more consistent with the results from the latter line of empirical research.

Throughout this section we focus on the two extreme cases where the HFTs capture all

incoming trades and the benchmark GM model where there are no HFTs; i.e. we assume

that q = 1 and q = 0 in (10) respectively. The general case, with competition and imperfect

intermediation, falls in between these two.

4.1 Liquidity, Volume and Price Impact

Our stylized model is designed to capture the value of a stock market as a forum where equity

holders can convert their equity into cash (and viceversa) quickly and at a reasonable price. In

practice it is usually hard for an outside observer to determine both the time and the overall cost

of execution of significant blocks of shares, as share blocks are split in several (and increasingly

many) parts executed separately, and the identity of traders is not available for empirical

analysis.14 Thus, most studies, by necessity, focus on indirect measures of liquidity such as

bid-ask spreads, execution speed, and fill rates, for small trades (less than 10,000 shares).

In our stylized model, liquidity (speed and cost of execution) is essentially measured in two

ways: (i) how many shares can the liquidity trader sell when he originates a trade imbalance,

that is, how many shares does LT1 sell at date t = 1, relative to his desired total sale of i units;

and (ii) what is the difference between the revenue obtained by liquidity traders when selling

(the cost paid when buying) their total trading blocks (−i and i) relative to executing them at

the asset’s “fundamental” value.14This is recognized by the SEC in their report [SEC concept release Jan10] “Measuring the transaction costs

of institutional investors that need to trade in large size can be extremely complex. These large orders often arebroken up into smaller child orders and executed in a series of transactions. Metrics that apply to small orderexecutions may miss how well or poorly the large order traded overall.” (p38-39)

31

In our model we are able to identify two ways in which the HFT affects the amount of

shares the liquidity trader can sell at t = 1: a direct and an indirect way. Although, having

identified them we find that in both ways the net effect of HFTs on quantities is zero.

The direct way is the effect of the HFT’s surplus extraction strategy on trader’s demand for

assets (both on the shape of the demand function and on the equilibrium quantity demanded).

Direct effects are limited. First, by assumption we have ruled out strategic effects on the part

of traders vis-a-vis the HFT, which would be a natural source of distortions of asset demand

functions and reduced trading (and hence liquidity). Second, the HFT’s surplus extraction

process is faultless. This, together with her extreme aversion to holding any inventory leads

to no distortions in the final quantity of assets demanded: LT1 continues to carry over θLT11 =

i/(M + 1) to date 2—the same amount as if there were no HFT.

The indirect way is through the effect of the HFT on the number of PTs, which in turn

affects liquidity (in terms of the number of shares sold at t = 1). Below (and in more detail

in Section 4.3) we find that HFTs have both a negative and a positive effect on PT’s revenue,

which, when q = 1, exactly offset each other. Thus, in the benchmark model HFTs have neither

a direct nor an indirect effect on the amount of LT1’s initial liquidity need i that is immediately

executed–leaving “fill rates” (the percentage of the desired trade executed) and “trade delays”

(the time to completion of desired trade) unaltered.

Liquidity can also be measured in terms of the cost of executing trades. We first look at

average price paid and received; and then we look at the total revenue and total cost for each

type of trader when selling and buying shares. As benchmark we use market clearing prices in

the absence of an HFT. In that case the market clearing prices (which are the same for buyers

and sellers) are:

P∆=01 = µ− iaσ2

M + 1, P∆=0

2 = µ2 .

This implies that without the HFT, LT2 faces infinite liquidity (buys at the asset’s fundamental

value), while LT1, the originator of the trade imbalance, faces a liquidity discount at date t = 1

of iaσ2

M+1 (given to PTs).

32

When the HFT is present (q = 1) we observe two effects on average execution price: traders

pay the HFT a haircut on a fraction of their trades, and the liquidity discount increases. We

start with LT2’s simpler problem, as he buys shares at their fundamental value. LT2 wants to

buy i units (trade i) and ends up buying θ̃LT22 at P2 +∆L per share, followed by i+θLT2

2 − θ̃LT22

at P2 per share. Then, using the equilibrium values of P2, θLT22 and θ̃LT2

2 , the average price

paid by LT2 is:

P̄LT22 = P2 −

θ̃LT22 ∆L

i= µ2 +

(∆L)2

iaσ2,

= P∆=02 +

(14

(1− 1

M + 1

))2

iaσ2 . (11)

From Equation (11), we can see that LT2 pays more to acquire his position than when there is

no HFT. The price uplift payed by LT2 is given by the second term on the right-hand side of

equation (11) which is the surplus per share lost to the HFT.

PTs trade twice. At date 1 they buy the asset, receive the liquidity discount and pay the

haircut to the HFT; at date 2 they sell and pay the haircut. The average purchase price at

date 1 is:

P̄ T1 = P1 +θ̃T1θT1

∆S = µ− iaσ2

M + 1− 1

4iaσ2

M + 1

= P∆=01 − 1

4iaσ2

M + 1, (12)

from which we see that PTs receive an even better price than without HFT—they manage a 25%

higher liquidity discount, even after accounting for the haircut paid to the HFT. Nevertheless,

at date 2, the average sale price they obtain is:

P̄ T2 = µ−∆S +θ̃T2

θT1∆S = P∆=0

2 − 14iaσ2

M + 1, (13)

so they end up selling back at a lower price (relative to the case without HFT), and losing the

initial extra liquidity discount garnered from the first transaction.

33

Finally, LT1 offers a liquidity discount and pays a haircut at date 1. In exchange he receives

P̄LT12 when he completes his liquidity trade at t = 2. The average price received by LT1 at

date 1 is:

P̄LT11 = P1 −

i− θ̃LT11

i− θLT11

∆L

= P∆=01 − 1

4

(M2 + 5M + 5

4

M2 + 2M + 1

)iaσ2 , (14)

and at date 2:

P̄LT12 =

µ2θLT11 −

(θLT1

1 − θ̃LT12

)∆S

θT11

,

= P∆=02 − 1

4iaσ2

M + 1. (15)

This implies that LT1 receives less money for liquidating his position at both dates (relative

to the case without HFT) as can be seen in Figure 4.

From equations (11-15), it is clear that LTs, LT1 and LT2, bear the brunt of the haircut

imposed by the HFTs, and that this impact is increasing in i. Although PTs suffer the loss of

consumer (date 1) and supplier (date 2) surpluses, the liquidity discount received by the PTs

for shares at date 1 is greater (the equilibrium price is lower) than in the absence of the HFT,

and the net effect on PT’s expected wealth is zero.

34

Figure 4: Average price for LT1 in period 1

1 2 3 4 5 6 7 8 9 101

1.5

2

2.5

Number of PTs

Ave

rage

price f

or

LT

1 in

perio

d 1

, i =

a =

1 a

nd µ

= 2

With HFTNo HFT

1 2 3 4 5 6 7 8 9 101.3

1.4

1.5

1.6

1.7

1.8

1.9

2

Number of PTs

Ave

rag

e p

rice

fo

r P

T in

pe

rio

d 1

, i =

a =

1 a

nd

µ =

2

With HFTNo HFT

35

Figure 5: Average price for LT1 (sells) and LT2 (buys) in period 2

1 2 3 4 5 6 7 8 9 101.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

Number of PTs

Ave

rag

e p

rice f

or

LT

1 a

nd

LT

2 in p

eriod

2, i =

a =

1 a

nd µ

= 2

LT1 (sells) with HFTNo HFT LT2 (buys) with HFT

1 2 3 4 5 6 7 8 9 101.8

1.85

1.9

1.95

2

2.05

2.1

Number of PTs

Ave

rag

e p

rice

fo

r P

T in

perio

d 2

, i =

a =

1 a

nd

µ =

2

With HFTNo HFT

36

We now consider the effect on total revenues and costs. Clearly, if PTs face no average price

impact, PTs’ total revenue from intermediation is unaffected by the presence of HFTs. On the

other hand, LT1’s total revenues decrease in the presence of the HFT due to an increase in the

liquidity discount and the loss of surplus. Similarly, LT2 faces a higher cost of acquiring his

desired position as a result of the loss of surplus. But, if we compare which of the two liquidity

traders is worse off we find that the effect of HFTs on LT1’s revenue from selling i is the same

as that on LT2’s costs from buying i.

From Equations (14) and (15) we can obtain LT1’s revenues with and without HFT:

RLT1 = µi− i2aσ2

(M + 1)2

116(4M2 + 24M + 3

)and R∆=0

LT1 = µi− M

(M + 1)2 i2aσ2 .

Then, the presence of the HFT reduces his revenue from selling i shares by the amount

R∆=0LT1 −RLT1 =

i2aσ2

(M + 1)2

116(4M2 + 8M + 3

), (16)

which increases, at an increasing rate, with the size of i and also increases in M .

Similar calculations give us LT2’s costs with and without HFT:

CLT2 = µi+2M + 1

4(M + 1)i2aσ2 − i2aσ2

(M + 1)2

116(4M2 + 4M + 1

)and C∆=0

LT2 = µi .

Interestingly, not only do LT2’s costs of buying i shares increases in M , and in the size of

the liquidity need, i (at an increasing rate), but the increase in cost from the HFT for LT2 is

exactly the same as the reduction in revenue for LT1:

CLT2 − C∆=0LT2 =

i2aσ2

(M + 1)2

116(4M2 + 8M + 3

)= R∆=0

LT1 −RLT1.

Finally, with a large number of PTs (as M tends to infinity), and using the fact that the

liquidity discount (Equation (32)) becomes zero, we can characterize what happens to costs

and revenues: CLT2 → µi + 14 i

2aσ2 and RLT1 → µi − 14 i

2aσ2 as M → ∞, as well as what

happens to HFT profits, which go to 12 i

2aσ2.

37

4.2 Price volatility and microstructure noise

We recall that in the GM model the volatility of the fundamental value of the asset is σ

and pricing pressures, due to a temporary liquidity imbalance, push market prices away from

their fundamental value but do not affect the ‘long-term’ volatility of prices. Put differently,

temporary pricing pressures increase the volatility of microstructure noise but do not affect

the (true) volatility of the value of the asset. Therefore, in a market with no HFTs, as that

described in the GM model, microstructure noise is caused by temporary selling and buying

pressures and this effect is exacerbated if HFTs that intermediate trades between LTs and PTs

are introduced. In our benchmark model we find that the HFT increases microstructure noise

in two ways. First, the price impact (i.e. selling pressure) of LT1 is larger than in the absence

of the HFT, and second, the instantaneous intermediation of the HFT around periods 1 and 2,

sees transactions fluctuate about the market prices P1 and P2 (see Table 1). In this section we

provide numerical examples where we show how the (short-term) volatility of prices increases

due to the microstructure noise induced by the HFT.

When there is no HFT intermediating transactions between LTs and PTs there is one trans-

action price at date 1, P∆=01 , and another transaction price at date 2, µ2. However, when HFTs

intermediate all transactions the tape will record four prices in date 1, {P1 −∆L, P1, P1 + ∆S , P1},

and four in date 2, {µ2 −∆S , µ2, µ2 + ∆L, µ2}. Therefore, it is inevitable to observe an in-

crease in microstructure noise within dates 1 and 2 which is solely caused by the HFT’s presence.

Figure 6 shows this microstructure volatility for prices in dates 1 and 2 that result from the HFT

intermediating all trades.15 Furthermore, Figure 7 depicts the standard deviation of prices at

both dates. We see that price volatility is much higher when an HFT operates in the markets.

Note however that in our model this increased volatility does not generate additional risk

for traders. Traders, in their evaluation of price risk (in their objective functions) recognize that

the microstructure noise generated by the HFT translates into a deterministic effect on their15To calculate the mean and variance of prices we also use the price at time t = 0 and assume it is the

fundamental value µ, thus the mean and variance of the price in date 1 are

P̄1 =µ+ 4P1

5and V [P1] =

1

5

{(µ− P̄1

)2+(P1 −∆L − P̄1

)2+ 2

(P1 − P̄1

)2+(P1 + ∆S − P̄1

)2}.

Similarly, to calculate the mean and variance of prices with date 2 we assume that the first observation is µ.

38

Figure 6: Microstructure volatility of prices in dates 1 and 2 induced by HF trading

1 2 3 4 5 6 7 8 9 100.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

Number of PTs

Sta

nda

rd D

ev o

f p

rices in

pe

rio

d 2

, i =

a =

1 a

nd

µ

= 2

Price volatility period 2Price volatility period 1

execution costs while leaving their final holdings unaffected. In the case of random execution, as

studied in Section 3, the additional uncertainty on future execution costs introduces an element

of risk which results in an increase in the liquidity discount.

4.3 Number of Professional Traders

Our model allows us to derive the impact of HFT’s surplus extraction activities on the number

of PTs, and hence on the “true” supply of liquidity in the market. Although we know that

when q = 1 HFTs have a zero net impact on the average price paid by PTs, and hence will not

affect their revenues, we proceed as in GM and assume that the cost of entry for the PT is c.

This allows us to analyze in detail the countervailing effects of the HFT on PTs. The entry

cost c is sunk before knowing the liquidity shock i which we assume to be normally distributed

and independent of the other shocks affecting prices. At time t = 0 the expected utility of an

39

Figure 7: Standard deviation of prices in dates 1 and 2 with and without HF trading

1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Number of PTs

Sta

nd

ard

Dev o

f price

s in b

oth

period

s, i =

a =

1 a

nd µ

= 2

With HFTNo HFT

individual PT is E[U(W T

3 − c)∣∣F0

], where W T

3 is given by (24). Free entry of PTs will occur

until

E[U(W T

3 − c)∣∣F0; q = 1

]= E

[U(W T

0

)∣∣F0 ; q = 1], (17)

and recall that θT2 = 0. To calculate the expectations in equation (17) we require the expected

value and variance of wealth in period three. The expected value of wealth can be decomposed

into a liquidity discount and a loss of surplus:

E[W T

3

∣∣F1; q = 1]

= W T0 − c

+θT1iaσ2

M + 1+ θT1 ∆S , liquidity discount, (18)

−θT1 2∆S +1aσ2

(2 (∆S)2

), loss of surplus. (19)

40

From equation (18) we observe that the PT receives a liquidity discount for buying θT1 shares

from period 1 to 2. This discount consists of two terms. The first term iaσ2/(M + 1) is the

liquidity discount (per share) that a PT receives in the absence of an HFT; and the second term

(θT1 ∆S) is the extra discount per share that a PT receives if there is an HFT intermediating

all trades. On the other hand, the term in equation (19) identifies the loss of wealth from the

HFT’s surplus extraction strategies.16

As for the variance of wealth, it remains unaffected by the presence of the (faultless) HFT:

Var[W T

3

∣∣F1; q = 1]

= σ2(θT1)2. (20)

The extra liquidity discount in the presence of an HFT (second term in equation (18)) exactly

offsets the haircuts (19) paid to the HFT in the round-trip trade (buy from LT1 in period 1

and sell to LT2 in period 2), and the number of PTs will be the same with or without HFT.

The exact canceling of the two effects occurs because of the particular information assump-

tions we have made which allows the HFT to distinguish only large from small trades. In the

Appendix we discuss the case where the HFT does not distinguish trades and charges the same

haircut to all participants. In that case (with q = 1), the number of PTs increases, as the extra

liquidity discount effect is greater than the haircuts.17

5 Conclusions

We use the GM model as benchmark to introduce an HFT who due to her rapid execution

and information processing ability is able to intermediate trades between LTs and PTs. In our

model the HFTs devise strategies which are carefully tailored to exploit the microstructure of16Note that there are two terms in equation (19). The first term is surplus extraction if the HFT took a

haircut in periods 1 and 2 for all shares that the PT buys and sells, but recall from above that in every periodthe HFT breaks PT’s order in two batches and only applies a haircut to one batch. Therefore, the second termin equation (19) is a correction term that accounts for the quantities which are traded in both periods withoutpaying a haircut.

17We expect this alternative case to be more relevant for liquidity trades that are small relative to the standardtrade size. This is because such trades would be more difficult for the HFT to identify as liquidity trades asopposed to regular (PT) trades.

41

price setting and thereby extract trading surplus from market participants and hold essentially

zero inventories over time.

Our model shows that the presence of HFTs increases the price impact of liquidity trades

(in proportion to the size of the trade), and when HFTs successfully intermediate all trades

they have no effect on the number of PTs, although they increase microstructure volatility, and

double trading volume. Thus, the presence of HFTs distorts market conditions, not through the

amount of shares traded, but through prices. By exacerbating price impact, the HFTs induce

additional market impact costs on participants, especially the LT. This cost is proportional

to the size of the trade which implies that large LTs, such as institutional investors trading

to change the composition of a portfolio, are the most affected by the presence of HFTs–an

effect that is consistent with Zhang [2010]’s findings. It is also consistent with a growing

trend amongst such investors to move execution to off-exchange HFT-protected venues such as

crossing networks and dark pools.

We show that the introduction of HFTs increases the microstructure noise around the

fundamental value of the asset in two ways. First, the price impact of LTs is larger than

in the absence of the HFT, and second, the instantaneous intermediation of the HFTs sees

transactions fluctuate around market prices; this result is supported by the empirical work of

Brunetti et al. [2011].

In the particular case we have analyzed, it is the LTs who bear all the costs from the

presence of HFTs. LTs lose on two accounts: they lose trading surplus to the HFT, and they

must offer a higher liquidity discount to PTs to get them to buy their shares. Moreover, in the

extreme case where HTFs successfully intermediate all trades PTs find themselves unaffected

because the revenue they lose to the HFTs is compensated by a higher liquidity discount from

LTs. This implies that overall, the effect of HFTs is to reduce the value of the stock market

as a forum for providing a way for investors to convert their equity into cash (and vice-versa)

quickly and at a reasonable price. This is due primarily to the adverse effect of HFTs on prices

(costs of execution). This value reduction would, in a more general framework, be passed on

to the firms raising capital in equity markets. As equity buyers recognize the increased trading

42

execution costs from HFTs, they will require greater discounts from IPO issuers, resulting in

greater IPO underpricing, specially for shares sold to large institutional investors.

By taking an extreme position on the role of HFTs, our model has allowed us to identify ways

in which an additional intermediation layer negatively affects trading. Two factors reduce this

negative effect: the first is competition. As discussed above, competition between HFTs helps

reduce unnecessary intermediation directly, though it may also lead to increased skill differences

between HFTs and PTs, and a corresponding increase in unnecessary intermediation. The

second, is the positive effects from HFT intermediation which we have not explicitly included

in the model.

Public discussions of HFT identifies several of these benefits. One is that HFTs have reduced

the time to execution faced by LTs. Our model does not explicitly include this as execution

around liquidity dates is collapsed to a single date. Nevertheless, if the HFT is to intermediate

between LTs and PTs, it must act before PTs and hence execution time for LTs must be

lower. SEC [2010] reports a reduction in average execution time from 10.1 seconds in January

2005 to 0.7 seconds in October 2009. Whether the increased execution speed compensates

for the additional trading costs is something that requires a more detailed analysis, but also

it is something that traders are facing little choice on. A second benefit is that increased

trading speed (exploited by cross-market and index arbitrage HFT strategies) has increased

the speed of adjustment of prices across markets and between similar assets, such as index

derivatives and ETFs. The advantage of such rapid adjustments and the implied increase in

price informativeness and efficiency has to be weighed against the increased microstructure

volatility. Overall, the discussion has to include these (unmodeled) benefits, but also other

(unmodeled) costs, such as the possibility of greater pressure on market institutions, regulators,

and additional challenges on market integrity.

An additional issue that arises from our analysis is the question of how to measure (socially

valuable) liquidity, when the analyst, as an outside observer, has no access to the identity

of traders and hence cannot determine directly the market impact costs of trades. A clear

conclusion from our model is that the total number of trades seems like a poor measure of

the ability of the market to provide prompt and fair value to investors, in a market with rent-

43

seeking hyperfast algorithms whose asset positions are essentially zero most of the time. Also,

the presence of HFTs who generate additional microstructure noise at smaller intervals and

accelerate market transactions raises several questions regarding how to measure price impact,

whether we should adjust current measures to account for the increase in the speed of execution,

and whether these measures adequately capture an investor’s cost of executing a trade.

Finally, in our knife-edge benchmark case HFTs had no effect on the number of liquidity

providing PTs. Alternatively, in the Appendix we consider an another strategy where HFTs

always charge the same haircut per trade (regardless of the size of the trade, type of trader,

and date). The effect in that case is that for PTs, the increase in the liquidity discount is

greater that the loss of surplus to HFTs, and in equilibrium, their number increases. This

suggests that the overall effect of HFTs on the “true liquidity providers” in not at all clear,

and we need good empirical work to (a) determine the speed-cost effect on outside investors

and LTs, (b) determine if HFTs are raising the cost of business for PTs, and (c) if they do,

whether the liquidity they provide is a good substitute for the one that is being driven out.

HFTs clearly generate costs, but they also generate benefits, and the net effect requires more

precise empirical analysis.

References

Jonathan Brogaard. High frequency trading and its impact on market quality. SSRN Working

Paper, 2010.

Celso Brunetti, Andrei Kirilenko, and Shawn Mankad. Identifying high-frequency traders in

electronic markets: Properties and forecasting. 2011.

Alain Chaboud, Erik Hjalmarsson, Clara Vega, and Ben Chiquoine. Rise of the Machines:

Algorithmic Trading in the Foreign Exchange Market. SSRN eLibrary, 2009.

U.S. Commodity Futures Trading Commission, the U.S. Securities, and Exchange Commission.

Findings regarding the market events of may 6, 2010. Report, SEC, September 2010.

44

David Easley, Marcos Mailoc Lopez de Prado, and Maureen O’Hara. The microstructure of

the ‘Flash Crash’: Flow toxicity, liquidity crashes and the probability of informed trading.

The Journal of Portfolio Management, 37(2):118–128, November 2011.

Thierry Foucault, Sophie Moinas, and Bruno Biais. Equilibrium High Frequency Trading.

International Conference of the French Finance Association (AFFI), May 2011, 2011.

Sanford J. Grossman and Merton H. Miller. Liquidity and market structure. The Journal of

Finance, 43(3):617–37, July 1988.

Terrence Hendershott, Charles M. Jones, and Albert J. Menkveld. Does algorithmic trading

improve liquidity? The Journal of Finance, 66(1):1–33, 2011.

Terrence J. Hendershott and Ryan Riordan. Algorithmic trading and information. SSRN

eLibrary, 2009.

Terrence J. Hendershott and Ryan Riordan. High Frequency Trading and Price Discovery.

SSRN eLibrary, 2011.

Michael Kearns, Alex Kulesza, and Yuriy Nevmyvaka. Empirical limitations on high frequency

trading profitability. SSRN worjing papers, 2010.

Andrei A. Kirilenko, Albert (Pete) S. Kyle, Mehrdad Samadi, and Tugkan Tuzun. The Flash

Crash: The Impact of High Frequency Trading on an Electronic Market. SSRN eLibrary,

2010.

Albert J. Menkveld. High Frequency Trading and The New-Market Makers. SSRN eLibrary,

2011.

Scott O’Malia. Letter to the members of the technology advisory committee. CFTC, 2011.

SEC. Concept release on equity market structure. Concept Release No. 34-61358; File No.

S7-02-10, SEC, January 2010. 17 CFR PART 242.

Frank Zhang. The Effect of High-Frequency Trading on Stock Volatility and Price Discovery.

SSRN eLibrary, 2010.

45

A Appendix

Above we argued that the HFT can discriminate across order size when trades come to the

market which enables her to apply haircuts for large and small trades. If the HFT is not

able to discriminate by size or type of trader she can still apply one haircut to all trades she

intermediates. Hence we can repeat the analysis for the benchmark case above while setting

the following haircuts:

∆ = ∆T11 = ∆T1

2 = ∆T1 = ∆T

2 = ∆T22 .

Then, the optimal ∆ set by the HFT is:

∆ = iaσ2 (2M + 1)(M + 1) (3 + 2M)

. (21)

As in the case with two haircuts discussed in the body of the paper, LT1’s optimal holding is

θT11 = i/(M + 1), but as we show below, the equilibrium number of PTs in this case is higher

than without HFT when q = 1.

Other results are similar to those discussed above: First, the volatility of realized transaction

prices increases. Second, the price impact of the liquidity trades in both periods is substantial:

equilibrium sell (buy) prices are lower (higher) than the competitive price in the absence of

the HFT. Third, the expected returns that the PT face from buying in period 1 and selling in

period 2 increase (at the expense of traders), and the equilibrium number of PTs present in the

market increases when compared to the number of PTs that are present in a market without a

HFT. Fourth, the total volume of trades doubles relative to the number of trades observed in

the absence of the HFT.

In the interest of space we only discuss the equilibrium number of PTs. We proceed as

above where the entry condition (17) becomes

eacE[e−

12a2σ2( i

M+1)2β(M)

]= 1 where β(M) =

12M2 + 12M + 7(3 + 2M)2 . (22)

By inspecting (22) we know that the number of PTs with and without an HFT will be the same

only when β(M) = 1 and this occurs for M = 1/2. When M > 1/2 we have more PTs in the

46

presence of HFT. If we denote the number of PTs by M and the number of PTs in the absence

of HFT by M∆=0 we can show that

M∆ =√β (M) (M∆=0 + 1)− 1 ,

hence we have that M > M∆=0. (The function β(M) is increasing in M and we are interested

in values for M > 1).

A.1 General Haircuts

Π2 (i) =∣∣∣θ̃LT2

2

(P2 + ∆LT2

2

)+ i∣∣∣∆LT2

2 +∣∣∣θ̃LT1

2

(P2 −∆LT1

2

)− θLT1

1

∣∣∣∆LT12

+M∣∣∣θ̃T2 (P2 −∆T

2

)− θT1

∣∣∣∆T2 . (23)

WLT13 = WLT1

0 + θLT12 P3 +

(θLT1

1 − θLT12

)P2 −

(θLT1

1 − θ̃LT12

)∆LT1

2 − θLT11 P1 +

(i− θ̃LT1

1

)∆LT1

1 ,

W T3 = W T

0 + θT2 P3 +(θT1 − θT2

)P2 −

(θT1 − θ̃T2

)∆T

2 − θT1 P1 − θ̃T1 ∆T1 . (24)

θLT11 =

1aσ2

(µ− P1 −∆LT1

2

), (25)

θT1 =1aσ2

(µ− P1 −∆T

2

). (26)

P1 = µ− ∆LT12 +M∆T

2 + iaσ2

M + 1, (27)

47

θLT11 =

i

M + 1+

1aσ2

M

M + 1(∆T

2 −∆LT12

), (28)

θT1 =i

M + 1− 1aσ2

1M + 1

(∆T

2 −∆LT12

), (29)

θ̃LT11 =

i

M + 1+

1aσ2

M

M + 1(∆T

2 −∆LT12

)+

∆LT11

aσ2,

θ̃T1 =i

M + 1− 1aσ2

1M + 1

(∆T

2 −∆LT12

)− ∆T

1

aσ2.

Π1 (i) =∣∣∣i− θ̃LT1

1

(P1 −∆LT1

1

)∣∣∣∆LT11 +M

∣∣∣θ̃T1 (P1 + ∆T1

)∣∣∣∆T1 . (30)

For a given order imbalance of magnitude i > 0:

1. Market clearing prices are

P1 = E [P2|F1]− iaσ2

M + 1− ∆LT1

2 +M∆T2

M + 1, (31)

P2 = E [P3|F2] = µ2 ,

and the magnitude of the liquidity discount at t = 1 is

E [P2|F1]− P1 =iaσ2

M + 1+

∆LT12 +M∆T

2

M + 1. (32)

(a) Asset trading is described on Table 2:

48

Table 2: Prices and Volume of Trades

Price LT1 LT2 PT (total)

P1 −∆LT11 − M

M+1

(i− ∆T

2 −∆LT12

aσ2

)+

∆LT11aσ2

P1 −∆LT11aσ2 M

∆T1

aσ2

P1 + ∆T1

MM+1

(i− ∆T

2 −∆LT12

aσ2

)−M ∆T

1aσ2

P2 + ∆LT22 i− ∆LT2

2aσ2

P2 −∆LT12aσ2

∆LT22aσ2 −M ∆T

2aσ2

P2 −∆LT12 − 1

M+1

(i+M

∆T2 −∆LT1

2aσ2

)+∆LT1

2aσ2

P2 −∆T2 − M

M+1

(i− ∆T

2 −∆LT12

aσ2

)+M ∆T

2aσ2

(b) The HFT acts as counterparty to all these trades and makes profits equal to

Π (i) = Π1 (i) + Π2 (i) , (33)

where Π1 (i) and Π2 (i) are described in Equations (30) and (23) respectively.

A.2 Competition

Result: The distribution of p (η) has cdf Gp (z) = 1α

(1 + α− 1

z

)and pdf gp (z) = 1

α1z2 .

Furthermore, E [p (η)] = 1α log (1 + α).

Proof: As 1/η ∼ U [0, 1] then 1/p (η) ∼ U [1, 1 + α], and the support of the distribution of

p (η) is[

11+α , 1

]. To compute the cdf of p (η) we use the result that for a random variable X

49

with cdf FX (z) the cdf of the random variable Y = g(X), where g is a deterministic decreasing

function, is given by

Y ∼ G (z) where G (z) = 1− FX(g−1 (z)

).

Thus, letting g(x) = 1/x and using FX (k) = (k − 1) /α, the cdf and pdf of p(η), denoted by

Gp and gp respectively, are given by

Gp (z) = 1− FX(

1z

)= 1− z−1 − 1

α=

1α

(1 + α− 1

z

), (34)

and by differentiating (34) with respect to z we obtain

gp (z) =1α

1z2.

Finally, it is straightforward to calculate

E [p (η)] =1α

log (1 + α) .

50