To understand this project, we first need to talk about what is a Limit Order Book (LOB) and how it relates to the price formation process.

An introduction to Limit Order Books (LOB)

The evolution of trading markets has progressed considerably in the last few decades. One of the main drivers of such evolution has been the creation and usage of fast-paced technological developments. In the past, liquidity was provided by the so-called market makers, which collected buy and sell orders from all market participants to set bid and ask quotes. While this traditional method was generally accepted, mainly, because of the lack of alternatives, it was also criticized due to the bias and questionable conflict of interest from the market maker. Nowadays, most exchanges use completely automated platforms called Electronic Communication Networks (ECN). These ECN enables a continuous double auction trading mechanism, which eliminates the need of a market maker or an intermediary that matches the opposite parties in a trade. The auction that the ECN manages is called “continuous double” since traders can submit orders in form of bids (i.e., buy orders) as well as asks (i.e., sell orders) at any point in time. ECNs increase significantly the speed of trading, taking only a few milliseconds from sending an order to its execution.

A bid limit order (resp. ask limit order) specifies the quantity and the price at which a trader wants to buy (resp. sell) certain asset. The limit order book consists of all the collection of limit orders from every trader. Outstanding limit orders are stored in different queues inside the order book. These queues are ordered by the price and type (bid or ask). The difference in price between the lowest ask price and the highest bid price is called the spread.

The counterpart of limit orders are market orders, which allow traders to buy and sell at the best available price. While limit orders will not trigger an immediate transaction, market orders are immediately executed. In this sense, limit orders accumulate, create, and extend the size of queues at both sides of the LOB, while market orders remove limit orders from the best available price. Sometimes informed traders are associated with traders that place market orders, while uninformed traders are associated to the ones that place limit orders, but this goes against the fact that many of the most successful hedge funds make extensive use of limit orders.

In addition to limit orders and market orders, cancellation of limit orders is another common operation. The basic idea of a cancellation is that a trader is no longer willing to buy or sell at the specified price. Cancellations account for a large fraction of the operations on an order book, partly due to the introduction and evolution of high frequency trading, in which the inter-arrival times of limit orders and cancellations, occur at a millisecond time scale.

A pictorial representation of the LOB can be seen in the following figure.

To better understand the dynamic behaviour of the LOB the following example, taken from (Abergel & Jedidi, 2011), is presented.

As illustrated from the figure, the different types of operations in a LOB may or may not affect the prices. Understanding how these operations affect these best prices is crucial in developing a parsimonious model for the price dynamics. Consider, for example, a buy (resp., sell) limit order arriving at a certain time $t$ with price $p_t$. Let $a_t$ and $b_t$ denote the current price of the best ask and bid at time $t$. Moreover, let also $s_t = a_t - b_t$ denote the bid-ask spread at time $t$ and assume, for illustration purposes, that $s_t > 1$. If the arriving order has price $p_t < b_t$ (resp., $p_t>a_t$) then it will increase the amount of outstanding limit orders at price $p_t$. However, if $b_t < p_t < a_t$, that order will become the best bid (respectively, best ask), hence, increasing (resp. decreasing) the mid-price. On the other hand, if the order is a market order to sell (resp., buy), this will decrease the amount of orders at the best bid (resp., ask). In this case, the price should match $b_t$ (or $a_t$, resp.) since traders will not sell something cheaper if they can sell it more expensive (resp. pay more for a stock that can get for a cheaper price). Moreover, if such order is large enough it may deplete several bid levels of the order book decreasing the best bid price (resp. depleting several ask levels of the orderbook increasing the best ask price).

An important feature of a LOB is that traders can choose between submitting limit and market orders. The biggest advantage of limit orders is the possibility of matching better prices than the ones they can obtain with market orders, but as drawback, there is a risk of never being executed. Conversely, market orders never match at prices better than the best bid or the best ask, but the execution is certain and immediate. Usually, the bid-ask spread can be considered as a measure of how expensive is the certainty and immediacy of buying or selling the underlying asset.

A plausible model of the LOB

From a modeling perspective, sometimes it is important to identify the different types of traders that are able to participate in the market. LOBs allowed traders to immediately obtain liquidity, but at the same time, they also allow other traders to supply liquidity to those who require it later. On the exchanges, most traders combine limit orders and market orders to create a trading strategy according to their needs and the current state of the order book. However, broadly speaking, traders with short-horizon strategies, as arbitrageurs, technical traders, and indexers, prefer to post market orders, while, traders with long-horizon strategies, as portfolio managers, place limit orders.

There are many practical advantages in understanding LOB dynamics. Examples of these are: gaining clearer insight into how best to act in a given market situation, devising optimal order execution strategies or optimal placement strategies, minimizing the market impact; designing better electronic trading algorithms, and assessing market stability.

In terms of modeling, we can follow a classical approach of a toy model as presented in (Cont & de Larrard, 2013). In here, we can do an overarching model as presented below:

• Let ${}^{a/b}L_t^{(n)}$ be the process that define the arrival or Limit Orders at the $n-$th level of the ask/bid side of the orderbook.
• Similarly, ${}^{a/b}M_t^{(n)}$ will denote the processes determining the arrival of Market Orders at the $n-$th level of the ask/bid side of the orderbook.
• Also, let ${}^{a/b}C_t^{(n)}$ be the process driving the cancellations at the the $n-$th level of the ask/bid side of the orderbook.

In terms of the volumes, we can denote by

• ${}^{a/b}V_t^{(L\;;\;n)},{}^{a/b}V_t^{(M\;;\;n)}$ and ${}^{a/b}V_t^{(C\;;\;n)}$ the volume of the corresponding limit order, market order or cancellation arriving at the ask or bid side at level $n$.

Another imnportant quantity that we need to keep track of is the current volume at the order book. With this is mind,

• Let ${}^{a/b}Q_t^{(n)}$ be the amount of outstanding limit orders at time $t$ at level $n$ on the ask/bid side.

Finally, we need to account for the time the price changes occur and understand their statistical properties. Indeed,

Let $\sigma_1^{(a/b)}$ be the first time that a price change occurs from the ask or bid side. That is,

• let $ \sigma_1^{(a/b)} = \inf\left\{ t \geq 0 \mid {}^{a/b}Q_0^{(1)} + {}^{a/b}L_t^{(1)} - {}^{a/b}M_t^{(1)} - {}^{a/b}C_t^{(1)} = 0 \right\}$.

Notice that $\sigma_1^{(a/b)}$ is indeed the firt time a price change from the ask/bid occurs as it becomes the first moment that Market Orders and Cancellations from the ask/bid at level 1 deplete the available number of original Limit Orders at level 1 plus the further Limit Orders at level 1 that have arrived at the corresponding side of the book. Then, naturally, the first time a price change occurs is

• $ \tau_1 = \min\left\{ \sigma^{(a)}_1, \sigma^{(b)}_1 \right\}$

In this fashion, we can continue quantifying when price changes occur on the LOB. Define recursively

• $\sigma_{n+1}^{(a/b)} = \inf\left\{ t \geq \sigma_n^{(a/b)} \mid {}^{a/b}Q_{\sigma_n^{(a/b)} }^{(1)} + {}^{a/b}L_t^{(1)} - {}^{a/b}M_t^{(1)} - {}^{a/b}C_t^{(1)} = 0 \right\}$

and

• $\tau_{n+1}=\min\left\{\sigma^{(a)}_n,\sigma^{(b)}_n \right\}$.

So, if we denote by $X_n$ the amount by which the price changed right after the $n-$th price change $\tau_n$ and by $\{S_t\}_{t\geq0}$ the price process, then it is clear that

• $S_t=S_0+\sum\limits_{k=0}^{N_t} X_k$,

where $N_t=\max\{n \mid \tau_1+\tau_2+\ldots+ \tau_n\leq t\}$. Indeed, $N_t$ just counts how many price changes have occurred up to time $t$ and the price at time $t$ is the starting proce $S_0$ plus all the changes that have occurred so far.

Notice that in essence, this is a very open model and our specifications on the dynamics of the Volumes, the Arrival processes, the distribution and statistical properties of Price Changes and their dependencies among themselves can be broadly specified, and what we would like is to understand how these inter-dependencies play a role in generating the different price process models used on a diffusion-limit scale.

What are some relevant questions regarding this model?

There are many interesting questions regarding this generic family of models. For example

• How does the statistical properties of the processes determining the amount by which the price changes and the processes determining when those orders arrive influence important quantities such as the volatility of the asset?
• How does the interactions and dependencies between the different quantities affect the price dynamics?
• What type of limiting models can you expect to get and under what conditions?

The last question is, in essence, one of the main questions I have been actively pursuing:

That is, what conditions on the processes generating the dynamics of the price in a Limit Order Book market are needed so that after the correct rescaling we can get, say, a Geometric Brownian motion, or even more complicated, a Stochastic Volatility model.

What my work on this project has been?

The resulting published work from this project on my side can be found in (Chávez-Casillas et al., 2019), (Swishchuk et al., 2019), (Chávez-Casillas, 2023) and (Chávez-Casillas & Figueroa-López, 2017).

<!--
  See https://www.debugbear.com/blog/responsive-images#w-descriptors-and-the-sizes-attribute and
  https://developer.mozilla.org/en-US/docs/Learn/HTML/Multimedia_and_embedding/Responsive_images for info on defining 'sizes' for responsive images
-->

  
    <source
      class="responsive-img-srcset"
      
        srcset="/assets/img/1-480.webp 480w,/assets/img/1-800.webp 800w,/assets/img/1-1400.webp 1400w,"
        type="image/webp"
      
      
        sizes="95vw"
      
    >
  

<img
  src="/assets/img/1.jpg"
  
    class="img-fluid rounded z-depth-1"
  
  
    width="100%"
  
  
    height="auto"
  
  
  
  
    title="example image"
  
  
  
    loading="eager"
  
  onerror="this.onerror=null; $('.responsive-img-srcset').remove();"
>

</picture>

</figure>

</div>
<div class="col-sm mt-3 mt-md-0">

</div>
<div class="col-sm mt-3 mt-md-0">

</div>

</div>

You can also put regular text between your rows of images, even citations (missing reference). Say you wanted to write a bit about your project before you posted the rest of the images. You describe how you toiled, sweated, bled for your project, and then… you reveal its glory in the next row of images.

The code is simple. Just wrap your images with <div class="col-sm"> and place them inside <div class="row"> (read more about the Bootstrap Grid system). To make images responsive, add img-fluid class to each; for rounded corners and shadows use rounded and z-depth-1 classes. Here’s the code for the last row of images above:

<div class="row justify-content-sm-center">
  <div class="col-sm-8 mt-3 mt-md-0">
    {% include figure.liquid path="assets/img/6.jpg" title="example image" class="img-fluid rounded z-depth-1" %}
  </div>
  <div class="col-sm-4 mt-3 mt-md-0">
    {% include figure.liquid path="assets/img/11.jpg" title="example image" class="img-fluid rounded z-depth-1" %}
  </div>
</div>

–>