A limit order book model for latency arbitrage.

A limit order book model for latency arbitrage.

Samuel N. Cohen Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, UK (samuel.cohen@maths.ox.ac.uk).    Lukasz Szpruch Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, UK (szpruch@maths.ox.ac.uk).
Abstract

We consider a single security market based on a limit order book and two investors, with different speeds of trade execution. If the fast investor can front-run the slower investor, we show that this allows the fast trader to obtain risk free profits, but that these profits cannot be scaled. We derive the fast trader’s optimal behaviour when she has only distributional knowledge of the slow trader’s actions, with few restrictions on the possible prior distributions. We also consider the slower trader’s response to the presence of a fast trader in a market, and the effects of the introduction of a ‘Tobin tax’ on financial transactions. We show that such a tax can lead to the elimination of profits from front-running strategies. Consequently, a Tobin tax can both increase market efficiency and attract traders to a market.

Key words: limit order book, latency arbitrage, high-frequency trading, Tobin tax.

2000 Mathematics Subject Classification: 91B26, 91G99

1 Introduction

The study of investment decisions in the presence of a market based on a limit order book poses various challenges in the fields of mathematical finance and financial economics. Many phenomena which are classically overlooked, in particular the existence of a bid/ask spread and the price impact of trading, appear naturally in this context. These phenomena take on a particular importance when we consider decisions over a short time horizon, where the effects of the limit order book naturally outweigh long-term price fluctuations.

In this paper, we present a model in the spirit of [22, 33, 30, 18, 17, 1] that combines classical financial modelling with a limit order book. Inspired by recent work of Jarrow and Protter [23], we investigate whether technological developments, in particular the advent of ‘high-frequency trading’, may lead to some forms of risk-free profit, commonly called latency arbitrage. By linking this trading to its effects on the limit order book, we are able to give an economically justifiable explanation for these abnormal profits, including the prevention of unbounded profit.

We build a model of the interactions of a fast and a slow trader with the limit order book. By considering the fast trader’s competitive advantage in terms of order execution, and by allowing her to predict the slow trader’s actions, we can easily see that she can create a risk-free profit by using a ‘front-running’ strategy. From this we can give precise estimates on the various profits that the fast and slow traders are able to make, in terms of the underlying limit order book. We also model their interaction at equilibrium and the consequent market efficiency, and the effects of imposing a tax on transactions (a Tobin tax).

1.1 Instantaneous trading

In order to focus on the effects of high-frequency trading, rather than on the other complexities of markets with limit order books, we shall focus our attention on trades occurring at a single moment. This allows us to keep our attention on the observable qualities of the limit order book (its depth, the number of stocks available at a price, and tightness, the cost of turning around a position), and to avoid building a model for its resilience (the rate at which new orders enter the book, c.f. Kyle [26]).

From a mathematical perspective, focussing our attention on a single moment allows us to easily consider more complex models for the instantaneous behaviour of the limit order book. In particular, the bid and ask prices we observe will naturally jump whenever trades occur, which causes no mathematical difficulties. Our fast-trader’s strategies can also be modelled at a point, without assuming either right- or left-continuity. In some ways, this can be seen as similar to the model in [23], where prices are assumed to be continuous, except at a countable number of jump times. In their model, the high-frequency trader only acts as a high-frequency trader at the jump times (albeit with right-continuous actions), and so the key details follow from the analysis of this countable set of stopping times. Our approach is to reduce this analysis to a single time, restoring the model of [23] is then primarily a question of notation.

On the other hand, by focussing on a single moment, we avoid consideration of the theory of optimal execution, as in [1, 28, 2, 35, 21, 34, 13]. The majority of this theory revolves around the resilience of the limit order book, and how to exploit this resilience to minimise price impact. In the time scales considered for high-frequency trading, the resilience of the limit order book is a less important consideration, as the competitive advantage of the high-frequency trader disappears faster than the limit order book returns to equilibrium.

Our main attention will be on ‘predatory’ trades made by a high-frequency trader using a front-running strategy, in such a way that they have no net position before or after the trades are complete. Such a sequence of trades is often called a ‘round-trip’. Studying round-trips is a common feature of many models of limit order books (eg [1, 20, 16]), and is also practically significant for high-frequency traders. For example, Kirilenko et al [25] describe the ‘Hot Potato Effect’ before the flash-crash of May 6, 2010, when in a 14 second period, high-frequency traders exchanged over 27000 contracts while only changing their net position by about 200 contracts. By studying these transactions, we can model our high-frequency trader as having a high temporal-risk-aversion, they wish to make a profit by exploiting their speed advantage, and do not wish to expose themselves to price risk through time.

1.2 Foreknowledge and high-frequency trading

For clarity of exposition, suppose that there are traders Alice (fast) and Bob (slow). Alice and Bob may both submit their orders effectively at the same time, and based on the same information. However, Alice’s superior speed (typically due to Alice’s server being located closer to the exchange) leads to Alice’s order being executed prior to Bob’s.

Alice, our high-frequency trader, will wish to make profit by exploiting her higher trading speed. Her primary effect on Bob is that, as her orders are executed first, she will have an impact on the price at which Bob can trade. This effect is a form of slippage (or temporary price impact), and is also modelled in [10, 1]. While this gives Alice a comparative advantage, it does not allow her to obtain a risk-free profit at a single instant.

To allow Alice to make such a profit, we will suppose that Alice, as a high-frequency trader, is able to know the quantity Bob intends to trade trade instantaneously before Bob does so. In this setting, Alice is able to manipulate the limit order book in such a way as to make a profit from Bob.

If we assume that Bob is an algorithmic trader, then Bob’s behaviour is predictable if the algorithm is known. Hence, if Alice has a good approximation of Bob’s algorithm, Bob’s trades are susceptible to the strategies that we analyse. From an example of Jarrow and Protter, [23, Example 1], by placing and cancelling small orders and watching Bob’s reactions, Alice can ‘learn’ Bob’s algorithm for a very low cost, and can then front-run it. From an industry perspective, Arnuk and Saluzzi [3] also assume that a high-frequency trader knows the trades coming in to the market, supporting our basic model.

As emphasized by Moallemi et al. [27] the optimal execution strategies derived by Bertsimas and Lo [4] and others lead to this foreknowledge, as the optimal strategy for liquidating a position involves an equipartitioning through time, and is therefore predictable, and can be detected in the market. Moallemi et al [27] then construct Nash equilibrium strategies with Bayesian updating, under certain assumptions on the form of optimal strategies (it must be Gaussian and linear in certain parameters). In Section 6, we extend our model to encompass one side of this decision making, as we determine Alice’s behaviour when she only has distributional knowledge of Bob’s actions.

Such a foreknowledge assumption is also implicit in the model of predatory trading considered in Brunnermeier and Pedersen [8]. In their model, the predatory trader knows that the other party must liquidate their position, and trades accordingly. In contrast to [8], by modelling a limit order book, we see that predatory trading of the type we consider will not lead to overshooting of the price, in fact, that the limit order book before and after trades are completed is the same whether the high-frequency trader is present or not. This is fundamentally because our high-frequency trader cannot, in an instant, increase the bid price by purchasing stocks, as only the ask price will be affected. Therefore, in our model, a single trader cannot create profitable ‘momentum’, and can only front-run the slow trader’s exogenously defined trades.

By making this foreknowledge assumption, and by considering only behaviour at an instant, our model differs markedly from Cvitanić and Kirilenko [10]. In [10], the high-frequency trader makes profits by taking orders away from the front of the limit order book, rather than by exploiting the shape of the book. This implies that the high-frequency trader must take on some temporal risk, and does not make a risk-free profit.

Our work also serves as a partial justification for the approach of Jarrow and Protter [23], where it is assumed that the high-frequency trader is able to obtain profits from using an optional integral. We show that this is indeed the case in our model (see Section 4.3).

We note at this point that we do not model the possibility that, if Bob is an algorithmic trader, Alice can trade so as to trigger Bob’s actions. We shall generally allow Bob’s trades to be determined exogenously (or possibly by giving Bob a demand function, for the purposes of determining his response to Alice). This assumption prevents Alice from being able to independently generate profit from Bob, which could lead to Alice obtaining unrealistically large profits from exploiting Bob’s algorithmic nature (see Remark 5.7.)

1.3 Market equilibrium

As we shall see, our model does lead to a high-frequency trader being able to make a risk-free profit. However, this profit is bounded, as Alice can only profit from adjusting the limit order book up to the quantity that Bob is planning to trade. In this way, we see that while we may violate the principle of ‘no free lunch with vanishing risk’ (c.f. Delbaen and Schachermayer [11]), due to nonlinearities Alice’s payoff, we sill still satisfy the principle of ‘no unbounded profit with bounded risk’ (c.f. Karatzas and Kardaras [24]). In this sense, our market is still economically reasonable, even though Alice is able to make a risk-free profit.

On the other hand, we shall also examine the behaviour of Alice and Bob when Bob becomes aware of Alice (Section 7). In this setting, we shall see that, even if Bob is unable to avoid the costs imposed by Alice, he will continue to trade, albeit in smaller quantities. Therefore, Alice’s presence does not render the market unworkable, even though it does introduce a deadweight loss in the market.

This equilibrium is different to that considered in [5], as we do not study the formation of the limit order book at equilibrium, but rather the equilibrium interactions of Alice and Bob given the limit order book. It is also not the same as those in [6, 10], as we do not look at dynamic equilibrium behaviour, instead focussing on a single instant.

One method of preventing Alice from interacting with the market is to impose a ‘Tobin’ tax on financial transactions. Such a tax is currently being proposed by the European Commission (see [9]). We shall see that such a tax will either completely prevent Alice from participating in the market, or will not alter her behaviour at all. However, we also show that there exists a small range of possible tax rates which will prevent Alice from participating without imposing larger costs on the rest of the market. That is, unlike in most microeconomic models of taxation, there is a small band of tax rates which may improve the market efficiency. Such a result largely agrees with the analysis of Tobin taxes in [29, 12, 36], however here for the setting of high-frequency trading. On the other hand, we shall see that outside this band, the tax does nothing to prevent Alice’s predatory trading, and will impose further inefficiencies on the market.

Furthermore, we show that for an individual slow investor, the negative effects of imposing the tax can be completely outweighed by the prevention of high-frequency trading, making a market more attractive to trade. This suggests that the ‘flight of capital’ historically seen when transaction taxes are imposed (see, for example, Wrobel [37]) may be less pronounced or even reversed in the modern era of high-frequency trading.

2 A Model of a Limit Order Book market.

2.1 Basic trades

We consider two types of orders: market and limit orders.

A limit order is an order to sell or buy a certain number of shares of an asset at a specified price, some time in the future. A trader submits his limit order to the exchange, where it is added to the limit order book.

A market order is an order to immediately buy or sell a certain number of shares at the most favorable price available in the limit order book. A market order to buy (to sell) is executed against the limit order to sell (to buy). The lowest specified price in the limit order book for a sell order is called the ask price, the highest price of a buy order in limit order book is called the best bid price.

We assume that at any time traders can observe the limit order book against which market orders at time will be executed. We represent the limit order book by two functions and (we use a subscript to indicate that this is the limit order book before any trades have occurred). Intuitively,

so tells us how many stocks can be purchased at the price at time . Formally, we shall take as the density of the sell orders on the limit order book, (against which market buy orders will be executed). Similarly, represents the density of the buy orders on the limit order book (against which market sell orders will be executed). Naturally, and are nonnegative, however we shall not assume that they are strictly positive.

Therefore, as and are densities, we define

so denotes the number of stocks offered for sale on the limit order book for a price less than , while is the number of stocks offered to buy on the limit order book for a price at least . Both these quantities determine the cost of trading using market orders.

Suppose a market order to buy stocks arrives. The right quasi-inverse function,

(where, ) tells us the ask price after stocks have been bought using market orders, that is, the price needed to purchase one more stock (formally, an infinitesimal number more stocks) than . For convenience, we denote this quantity

(2.1)

Similarly, the left quasi-inverse tells us the bid price after stocks have been sold, and we write .

Using this notation, we can naturally define the ask and bid prices by

Lemma 2.1.

The total cost to buy stocks using a market order is

similarly revenue from selling stocks using a market order is

Proof.

We prove only the equality of the final two terms in the first equation. As is differentiable, its right quasi-inverse is differentiable except at its discontinuities, that is, when . These points do not contribute to the first integral, as on such a set. Hence, we can exclude all such points from the first integral and use the classical change of variables result to obtain the desired equality. ∎

Lemma 2.2.

We have the identities

If we assume that also has an upper derivative, we also obtain

Similarly, with lower derivatives,

Proof.

These all follow from the definition of and the chain rule (on the set ). ∎

Remark 2.3.

In particular, we note that

Lemma 2.4.

If has an upper derivative at , then for ,

and similarly for and . If we assume that the limit order book has an affine density above the ask price (that is, is constant for ), then the remainder terms vanish. If the limit order book has a constant density above the ask price (that is, , as modelled in [22, 30, 31], and empirically studied in [7]), then

For simplicity, in this case we shall say that the limit order book has constant density.

Proof.

This is simply an application of Taylor’s theorem. ∎

Remark 2.5.

From these equations, we can clearly see the effect in of the ask price (first order), the simple trading impact (second order) and the change in the limit order book height, or equivalently, the curvature of (third order).

3 Fast trading and slippage

From here onwards, we shall consider trades at a given time . We shall therefore omit the from , etc… whenever this does not lead to confusion.

We wish to study the effects of Alice’s priority to Bob, that is, the consequences of the fact that, when Alice and Bob both trade at the same time, Alice’s order will be executed before Bob’s. The first and simplest effect of Alice’s priority is an increase in ‘slippage’, where Bob’s trade is executed further along the limit order book than if Alice was absent. This occurs when both Alice and Bob trade at the same time, in the same direction. For simplicity, we shall assume that they both try to purchase stock, the analysis if they both try to sell stock is perfectly analogous.

Let Alice attempt to buy a quantity , and Bob a quantity . Then Bob’s order will not be executed at the front of the limit order book, but is affected by being executed after Alice’s order. That is, Alice will pay the usual total cost , but Bob will pay the higher cost . Bob’s loss due to slippage can then be measured by the difference between this quantity and , the amount Bob would usually pay.

Lemma 3.1.

The total cost of Bob’s trade is given by

(3.1)

If the limit order book has a constant density

Proof.

Simply expand and using Lemma 2.4. ∎

In the absence of Alice, Bob would expect to pay the quantity . From this approximation we can see that, provided the density of the limit order book at the ask price is sufficiently high, or the density is approximately constant at the ask price (so the third term disappears), for small trades, Bob’s loss due to slippage is proportional to the covariation of Alice and Bob’s trades (as measured by ), with the proportion given by the inverse of the height of the limit order book density at the ask price.

4 Fast Limit orders and Latency Arbitrage

In the situation considered in Section 3, Alice has a clear competitive advantage over Bob. However, she has not realised an arbitrage profit, as she has a entered into a net position in the stock. To allow pure arbitrage, where Alice starts and ends with no net position in the stock, but obtains a profit through trade, we also need to consider how Alice can place orders within the limit order book.

In Jarrow and Protter [23], it has been shown that unequal access to to stock exchange may lead to some types of arbitrage. They have assumed that the fast trader invest according to an optional (rather than predictable) strategy, and realises the gains from the optional Ito integral (see section 4.3). This mathematical structure allows a high speed trader to obtain abnormal profits, as they can capture an additional term due to the quadratic variation of the stock.

In [23], there is no significant discussion of how, in a real market, the mathematical formalism of an optional stochastic integral could be realised, or through what economic means these additional gains can be obtained. In this section we will show that, by allowing Alice to also make fast limit order trades, if Alice foreknows Bob’s trading strategy, these additional profits can be achieved in our model.

Suppose that Alice foreknows that Bob will purchase stocks. (Again, the analysis if Bob will sell stock is perfectly analogous.) Then Alice can make a profit with zero risk using the following recipe, where Alice front-runs Bob’s trades. Recall that all of Alice’s actions will be executed before Bob’s trade.

Strategy 1.

Suppose Alice knows that Bob will purchase stocks. Then Alice acts as follows:

  1. Purchase stocks using market orders. This will clear the limit order book up to the point .

  2. Place a limit order to sell stock at the price (or distributed infinitesimally below, so that these limit orders will be executed first). This is the limit order against which Bob will trade.

  3. Sell remaining stock using market orders.

Lemma 4.1.

For general , Alice’s profit under this strategy is given by

(4.1)

If , we say that Alice has realised a latency arbitrage opportunity. We call the latency profit (or, from Bob’s perspective, the latency cost).

Proof.

Alice’s purchase of stocks will cost her , and will clear the limit order book up to the price . Alice then places her limit order to sell stocks for a price , so that these orders are the lowest in the order book. Bob’s trade to purchase stock is executed, and as all of Alice’s limit order will be executed with Bob. This gives Alice revenue of . Finally, Alice sells any excess stock using market orders. As no previous trades have been executed against the lower side of the limit order book, the revenue from this is given by . ∎

Lemma 4.2.

If the limit order book has a constant density, Alice’s profit is maximised by trading the volume .

Proof.

As the limit order book has a constant density, is independent of and is linear, as in Lemma 2.4. Hence, if the optimal value of is greater than , by a first order condition,

giving a contradiction. Therefore . On the other hand, for we have

and so, by a first order condition, is optimal. ∎

Remark 4.3.

When the limit order book does not have a constant density, in particular if , then Alice’s impact on the ask price is increasing. As she is able to sell stock to Bob for , this yields higher revenues, and if these revenues are increasing sufficiently fast, they may compensate for the loss of having to sell excess purchased stock on the lower side of the limit order book.

As discussed in [19], newer exchange systems may require ‘market’ orders to specify both a quantity and a maximum acceptable price (for buy orders, a minimum for sell orders), rather than simply a quantity. This forms a protection against extreme fluctuations in price, as were seen in the ‘flash crash’ of May 6, 2010. If we were to model a market of this type, then the above result would be valid independently of the shape of the limit order book, as long as Bob specified that he wished to buy stocks for a maximum price of . For this reason, and for mathematical tractability, we shall hereafter assume that whenever Alice knows Bob’s actions perfectly.

Lemma 4.4.

In the case , we have the following equation for Alice’s profit

(4.2)

As before, if the limit order book has an affine density, then this equation is exact, and if the limit order book has a constant density, the term can be omitted.

Proof.

Simply expand (4.1) using Lemma 2.4. ∎

Lemma 4.5.

If , then Alice’s trades have no impact on the shape of the limit order book following Bob’s trades.

Proof.

We will only prove the lemma for the ask-side of the limit order book, the bid-side can be proved by symmetry. For clarity, we denote all quantities before any trades have occurred with a subscript , those which are after Alice but before Bob with a subscript , and those after both Alice and Bob with a subscript .

Suppose that the Alice is not present at the market. Before any trades are executed, at time , the limit order book is described by the function

Then Bob’s market order is executed and a new ask is given by , and the ask-side limit order book has the form

Now suppose that Alice is present. As Alice purchases stocks using market orders, then places a market order at to sell stocks, after Alice’s trades, the ask-side limit order book has the form

where denotes a Kronecker delta.

Then Bob’s market order arrives, and buys stocks at the price and stocks from the remainder of the limit order book. This implies that the net purchase from the limit order book is of precisely stocks. As a result, the ask-side limit order book after Alice and Bob’s transactions has the form

the same as in the absence of Alice. ∎

The Lemma shows that in the case where , the overall price dynamics are not changed due to the presence of Alice. We shall see (Section 5) that Alice’s presence is still detectable, by considering the overall volume traded.

4.1 Limits of arbitrage

In the setting we have been considering, it is clear that Alice can obtain a risk-free profit, as soon as Bob trades. We can now answer two questions

  1. Where is Alice’s profit coming from?

  2. What are the restrictions on Alice’s profit?

From an economic perspective, Alice’s profit comes from an opportunity cost faced by Bob. Presuming , if Bob was able to trade against the original limit order book, his cost of entering his position would be . Instead, he faces the cost , which implies an opportunity cost of , Alice’s profit. If Bob is trading because he believes he has determined a mispricing, then, presuming Bob is correct, Alice has acted as a classical arbitrageur, and Bob faces trading at an efficient market price. On the other hand, if Bob is a noise trader, then one can consider Bob as creating mispricing opportunities, which Alice is able to exploit.

In this situation, we note that Alice is making a risk-free profit. However, Bob does not face as large a price penalty as he would if Alice was simply copying Bob’s strategy. This is easily seen by comparing the approximations (3.1) and (4.2). In (3.1), with we see that Bob’s penalty is (to second order) given by , whereas in (4.2) the second order term is halved.

In [23], the fact Alice can obtain an arbitrage profit led Jarrow and Protter to conclude that an equivalent martingale measure may not exist, as there is a ‘Free Lunch with Vanishing Risk’ in such a market, contradicting the fundamental theorem of Delbaen and Schachermayer [11]. This is clearly true in our setting. As the abnormal profit grows proportionally to , the square of the amount traded by Alice, but they placed no bounds no Alice’s strategies, in [23], this may lead to Alice obtaining an unbounded profit.

In our approach, Alice is not able to make an unbounded profit. As Alice trades the amount , and is chosen by Bob, not Alice, we see that Alice’s profit cannot be scaled beyond this point. Therefore, in this simple model, even though Alice makes a bounded profit with zero risk, we still have no unbounded profit with bounded risk. This connects our analysis with weaker concepts of no-arbitrage, as considered by Karatzas and Kardaras [24].

This situation is not economically unreasonable, as we have only considered whether Alice is capable of making an arbitrage profit on the market, without considering extra-market costs associated with doing so. In order to realise this profit, Alice needs to invest significant quantities in the development of fast trading systems, and in arranging for these systems to be collocated with the exchange servers. These costs form a significant barrier to entry in this market, preventing the arbitrage opportunity from being universally exploited. Furthermore, as Alice must continue to be the fastest trader in this section of the market, a form of the Red Queen Effect111From Lewis Carrol’s Through the Looking Glass, where the Red Queen states “It takes all the running you can do, to keep in the same place.” will force Alice to continually improve her systems, thereby continuing to incur such costs.

4.2 Transaction costs and portfolio valuation

Suppose that Alice is not purely attempting to make an arbitrage profit, but also wishes to take up a position in the stock. In this section, we shall see that Alice can use her speed advantage to shift her liquidity cost to Bob, whenever Bob’s trades are of a similar size in the same direction as Alice’s.

Theorem 4.6.

Suppose Alice and Bob both wish to purchase stock at the same time. Alice’s desired net trade is denoted by , Bob’s by . For simplicity, assume (as always, the analogous result holds if ).

By exploiting her foreknowledge and increased speed, the cost to Alice of entering into this position can be reduced to

This implies that, if , then Alice will pay no more than the ask price for each unit of stock (ignoring the term). Bob’s total cost is given by

That is, Bob pays both the slippage cost , as in Lemma 3.1, and the latency arbitrage cost , as in Lemma 4.4.

Proof.

As Alice wishes to enter into a position , she can follow the following strategy.

Strategy 2.

Alice uses a modified version of her earlier strategy.

  1. Purchase a total of stocks using market orders, this will clear the limit order book up to .

  2. Place a limit order to sell stocks at the price . These limit orders are then executed against Bob’s incoming market buy order.

Alice’s total cost from these trades is given by , and she is left with a net position of stocks. Expanding and using Lemma 2.4, we have

If Alice follows this strategy, Bob will pay the cost . The expansion of Bob’s cost again follows from Lemma 2.4.

In the case this means that Alice can completely avoid the liquidity cost associated with her price impact. Furthermore, this leads to the strange situation where Bob and Alice hold the same portfolio, but due to differences in execution, they purchased it at a different cost.

It is a simple exercise to value a portfolio over given time interval in the same spirit as Roch [31]. This will allow us to see that Alice and Bob will hold the same portfolios with same liquidation value, but Bob’s liquidity costs due to price impact have quadrupled (from the term in in Lemma 2.4 to the total cost of here), while Alice can trade as if she has no price impact whatsoever.

When Alice and Bob wish to trade in the opposite directions, then it is clear that they can both benefit from trading simultaneously. When their trades are of the same size, if Bob wishes to buy stock, Alice places the quantity she wishes to sell as a limit order at the front of the limit order book, and Bob trades against this. Bob benefits by reducing his price impact, and Alice benefits by being able to sell at the ask price, rather than at the bid (and by avoiding price impact).

4.3 Optional integration

We now discuss how Alice’s profits can be seen to come from an optional integral, more formally linking our approach with [23].

In [23], our high-frequency trader Alice obtains a profit from being able to use an optional, rather than predictable, integral. This could be interpreted as a backwards integral in the sense of [32], however in [23] the optional integral is only needed at a countable number of jump points, simplifying the mathematical analysis.

To be more precise Protter and Jarrow considered the following price process

where and are semimartingales with respect to such that ( and have no common jumps). can be understood as the fundamental value of the stock and is price impact of the high-frequency trader (Alice). The portfolio value for an ordinary trader who is using a predictable strategy is given by

Since Alice’s strategy is assumed to be a predictable process except at its jump, her portfolio value is given by

Hence the increased profit available to a fast trader (but unavailable to a slow trader) is

That is, considering only a single time where , if denotes the change in excess profit at , we have the relations

(4.3)

Now consider our model, as in Section 4.2. Assuming a constant limit order book density above the ask price (equivalently, up to an approximation of appropriate order), we can calculate Alice’s profit at a jump in the price. From Theorem 4.6, Alice’s increased profit at an upward jump in the price (that is, including her transaction costs, but not including the profits from a prior position) is given by

where and are the changes in Alice and Bob’s positions respectively, and is the value at which Alice can value each stock she possesses. (Note that in [23] there is no bid-ask spread, so there is no ambiguity about the appropriate book value of a stock, that is, .)

Suppose Alice’s trade is of a constant size relative to Bob’s trade, for some , so that

The impact of this trade on the ask price is given by

hence, when Alice and Bob both purchase stock, Alice’s profit is

Considering the other side of the limit order book, we obtain the general equation for Alice’s profit,

Now assume that that , the ‘mid-price’, and that before trades occur, there is no bid-ask spread (a strong assumption of resilience in the market, but consistent with the lack of a Bid-Ask spread in [23]). Then when Alice and Bob both purchase stock,

and similarly when they sell stock. We can then write the profit as

Assuming the symmetry

for some price impact factor , we finally have

which is, for , precisely the abnormal benefit available to a high-frequency trader in (4.3).

5 Churning of trades and total volume

Remark 5.1.

From here to the end of this paper, we always assume that the limit order book has a constant density above the ask price and below the bid price, as this allows us to give analytically simple values for the optimal trading levels for Alice and Bob. Therefore, for notational simplicity, we write simply write and .

In the previous sections, we have described a possible strategy that will lead to Alice making a latency arbitrage profit. From Lemma 4.5 we see that although Alice momentarily trades on the limit order book, as her limit orders are immediately executed by Bob, the overall dynamics of the limit order book on any larger timescale are unchanged.

Therefore, from an econometric perspective, this type of arbitrage may not be noticed if we were to only analyse the shape of the limit order book through time. For that reason we introduce a new quantity , the total volume traded using market buy orders at time . The volume traded is the actual number of assets that have changed ownership through market buy orders at time , and is readily observable on many markets.

As it will be important for us to keep track of whether we are referring to the shape of the limit order book before or after trades have occurred, we shall again subscript all quantities which refer to the limit order book before either Alice or Bob has traded with , all those which refer to after Alice’s trades and before Bob’s by , and all those which are after both Alice and Bob’s trades with .

If each stock changed ownership at most once at time , there would be a natural relationship between the ask price after a trade and , given by (2.1). That is, if market buy orders have been executed, and no new limit orders have been placed on the market at the moment , we would expect the ask price after all trades are executed to satisfy

or equivalently, .

If Alice follows Strategy 1, the volume traded through market buy orders will increase to . However, as we have seen in Lemma 4.5, if we know that remains unchanged. This breakdown of the relation between and the resultant ask price is due to Alice trading with both market and limit orders simultaneously. Using this observation, we can precisely define the degree of churning (that is, of instantaneous round-trip transactions) present in the market.

Definition 5.2.

Define the quantity

which measures the volume traded at time in excess of that which is implied by the change in the limit order book. We will say that the limit order book has been churned at time if .

We note that this definition does not work in the presence of cancellations of limit orders, as these may affect the ask price without requiring any transaction volume.

As is readily observable in many markets, as is the limit order book, the quantity provides a ready econometric quantity for the study of this type of high-frequency trading.

5.1 Regulation and Tobin tax

From a regulatory perspective, it may be advisable to discourage Alice’s latency arbitrage trading, as it increases trade volumes (leading to higher administrative costs) and segregates markets according to access to high-frequency trading. A simple method of regulating this trading is for the market to charge Alice for access to high-frequency trading, either through an access charge, or through increased collocation costs. This is problematic, as the market (as a private corporation), then faces a conflict of interest, as they receive revenue from Alice’s actions. This also does not assist Bob, as he will still only be able to trade at the higher price.

Another possible type of regulation, which is be suggested by the increased traded volumes, is to impose a transaction cost on all market participants, in the form of a financial transaction tax, or Tobin tax. This type of taxation is currently being proposed for certain transactions within the European Union (see [9]). We shall here propose a simple model of the tax, and study its affects on Alice’s behaviour. As Alice is trading twice, it is natural to assume that she will face a larger tax burden than Bob, thereby discouraging her from attempting latency arbitrage.

In this section we simply consider Alice’s behaviour in the presence of the tax, without looking for the effects of the tax on equilibrium behaviour of Bob, Alice and ‘market makers’ (who provide the initial limit order book). The equilibrium analysis will be considered in Section 7.2.

Definition 5.3 (Tobin tax).

Suppose a market buy order is executed with a nominal monetary value . The party placing a market order must pay tax on this transaction at a rate , (that is, they must pay total cost). The limit-order counterparty must pay tax on this transaction at a rate , (that is, they only receive payment).

We say the overall rate of taxation is

and that are chosen such that .

Remark 5.4.

In some markets, there may be a premium paid to those placing limit orders, as a means of encouraging liquidity. This corresponds to , and poses no problem in our setting, provided market orders are taxed at a rate such that .

Let us analyse the impact of a Tobin tax on the market.

Theorem 5.5.

In the presence of a Tobin tax with overall rate , if Bob trades a quantity and

Alice’s profit is maximised by not trading, that is, . Otherwise, Alice’s profit is maximised by trading as if the tax were not present, that is, .

Proof.

That can be verified as in Lemma 4.2.

If Alice makes a trade of size , she faces the cost

Her trade changes the ask price to

She then submits a limit order of size (since ) that is executed against Bob’s market order, giving her revenue

Hence her profit is

As this is a quadratic with positive coefficient of , and , is negative until the point , where

and is thereafter increasing and positive.

Therefore, we see that a Tobin tax will only act to prevent Alice from exploiting small trades, and is completely ineffectual at preventing Alice from trading large quantities, particularly when the market is illiquid (i.e. is small). Furthermore, we have the following corollary.

Corollary 5.6.

In the absence of Alice, the tax revenue from Bob’s trade of will be

If Alice is present and trades optimally, and , the tax revenue will be

From this, we see that whether the tax is placed on the market or limit order side of the transaction is irrelevant for the calculation of total revenue (under the assumption that Bob’s trades and the original limit order book do not change, see Section 7.2). On the other hand, is more strongly affected by an increase in than , so in terms of deterrence of Alice’s trading, a tax on the limit order side of the transaction is more efficient. This effect is simply because, in our model, Alice trades a larger monetary value using limit orders than market orders (), and so responds more strongly to a tax on these orders.

Remark 5.7.

The fact that Alice will not profit from trading small quantities has implications for models when Alice can trigger Bob’s trades. Suppose that Bob is an algorithmic trader, and that Alice knows Bob’s algorithm. Suppose furthermore that Alice can trigger Bob to trade a small quantity for negligible cost. Without a Tobin tax, Alice can use this fact to generate profits from Bob, by continuously triggering and front-running small trades. However, in the presence of a Tobin tax, Alice will only profit when Bob trades a quantity . So, if Alice is not able to trigger such trades without incurring significant costs herself, the introduction of a Tobin tax may have a stabilising effect on the market.

6 Imperfect knowledge

In all the preceding analysis, we have assumed that Alice knows perfectly the actions that Bob will take. Some reasons for this are given in Section 1.2. When Alice does not perfectly know Bob’s actions, but only has some prior distribution for the size and direction of Bob’s trade, then her behaviour becomes more difficult to study. The main reason for this is that Alice must decide whether to take the risk of purchasing stock and placing it on the limit order book, not being sure whether Bob will purchase it. If Alice is unwilling to carry a position forward, any stock that Bob refuses to buy must be sold at a loss, using market orders executed against the bid side of the limit order book.

In an even more extreme case, if Alice is not sure of the direction of Bob’s trade, and trades on the wrong side of the book, then the resolution of her position will occur after Bob’s trades, that is, against the bid side of the limit order book with the first stocks removed. Alice then faces a slippage cost due to Bob, which will worsen her position further. Finally, it is conceivable that there are situations where Alice does not know the direction of Bob’s trade, but the shape of the limit order book is such that it is in her interest to instantaneously modify both sides of the book. These considerations lead to a significantly more complex analysis.

In Moallemi, Park and Van Roy [27], a dynamic model for a predatory trader with Bayesian updating is considered. Under the assumption that all prior distributions are Gaussian and all strategies are linear in the prior’s parameters, they derive a Nash equilibrium strategy for Alice and Bob. Here, we only consider Alice’s behaviour, and only in a static setting. On the other hand, we do not assume that Alice’s prior has any particular structure.

6.1 Size uncertainty

We now focus on one of the simplest forms of uncertainty, where Alice knows the limit order book perfectly, but is not able to perfectly predict Bob’s actions. For simplicity, we suppose that Alice knows the direction, but not the size, of Bob’s trade. We also assume that Alice is unable to carry a net position forward, and therefore will be forced to liquidate any excess purchased stock using a market order.

Theorem 6.1.

Suppose that Alice assigns some subjective atomless probability to Bob trading an amount . We assume that this probability already incorporates Alice’s risk aversion, so that she simply wishes to maximise the expectation of her profit, . Assume a constant density of the limit order book both above the ask price and below the bid price.

Then Alice’s optimal trade is either or a solution to the nonlinear equation in terms of the lower partial moment and the probability

Proof.

As the limit order book density is constant, Alice’s profit is given by

As is atomless, the expectation of this quantity under is

For , writing

we have (formally using the lower derivative, so as to ensure that we have sufficient regularity to exchange the order of integration and differentiation)

Setting this derivative to zero, we obtain a nonlinear equation for the optimal quantity in terms of the lower partial moment and the probability .

In many cases, this equation will not have a simple analytic solution, but is easy to solve numerically. The following example provides a case with a straightforward solution.

Example 6.2.

Suppose Alice believes Bob’s choice is uniformly distributed on for some . Then for , and . Therefore, Alice’s optimal value of is

This simple equation agrees with our intuition. First, Alice will take a larger position when is small, as in this situation she can exploit Bob to a higher extent. Second she will not take any position if the spread is too large, as if Bob does not take up a sufficiently large position, she must sell at the lower price . Conversely, when there is no spread, she will always take up a position. Finally, if is large, she will take up a larger position, as if she does need to sell excess stock, then she can do so without suffering from a significant price impact on the sale.

It is also interesting to note that, in this example, no matter what the shape of the limit order book, Alice will never take a position of more than . This is perhaps surprising, but is due to the fact that Alice’s gain from purchasing the last portion of the limit order book is quite slight, and her potential loss from each stock purchased increases with the size of the trade. On the other hand, it is interesting to see that, for some shapes of the limit order book, Alice will purchase more than Bob’s average trade