# Beyond the Bid–Ask: Strategic Insights into Spread Prediction and the Global
Mid-Price Phenomenon

Yifan He Correspondence<EMAIL_ADDRESS>Department of Mathematics and
Statistics, Texas Tech University, Lubbock, TX 79409, USA Abootaleb Shirvani
Department of Mathematical Sciences, Kean University, Union, NJ 07083, USA
Barret Shao  XBTO Group, New York, NY 10019, USA  Svetlozar Rachev  Department
of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA
Frank Fabozzi  Carey Business School, Johns Hopkins University, Baltimore, MD
21202, USA

(September 1, 2024)

###### Abstract

This study introduces novel concepts in the analysis of limit order books
(LOBs) with a focus on unveiling strategic insights into spread prediction and
understanding the global mid-price (GMP) phenomenon. We define and analyze the
total market order book bid–ask spread (TMOBBAS) and GMP, showcasing their
significance in providing a deeper understanding of market dynamics beyond
traditional LOB models. Employing high-frequency data, we comprehensively
examine these concepts through various methodological lenses, including tail
behavior analysis, dynamics of log-returns, and risk–return performance
evaluation. Our findings reveal the intricate behavior of TMOBBAS and GMP
under different market conditions, offering new perspectives on the liquidity,
volatility, and efficiency of markets. This paper not only contributes to the
academic discourse on financial markets but also presents practical
implications for traders, risk managers, and policymakers seeking to navigate
the complexities of modern financial systems.

Keywords: Limit order book; bid–ask spread; mid-price; option pricing; implied
volatility; Rachev ratio

##  1 Introduction

The intricacies of financial markets, particularly the mechanisms underlying
the formation of prices and the dynamics of spreads, have long fascinated
researchers and practitioners alike. At the heart of these mechanisms lies the
limit order book (LOB), a pivotal structure that records the buy and sell
orders for a specific asset at different price levels. Traditionally, studies
of LOBs have concentrated on understanding the bid–ask spread and mid-price
dynamics, offering critical insights into the liquidity and efficiency of the
markets. However, with the advent of high-frequency trading (HFT), along with
the increasing complexity of financial markets, there is a need to take a
closer examination of these dynamics and identify the strategies that govern
spread prediction and the global mid-price (GMP) phenomenon.

Recent surveys, such as those compiled in (Gould et al., , 2013), provide a
comprehensive review of the empirical and theoretical advancements in our
understanding of LOBs. These reviews have been instrumental in shaping current
methodologies and highlighting areas where further inquiry is required,
particularly in the context of high-frequency data and modern trading
algorithms.

In this context, we introduce two novel concepts: the total market order book
bid–ask spread (TMOBBAS) and the GMP. These concepts extend beyond the
traditional bid–ask spread and mid-price measures by considering the entire
depth of the LOB, thereby providing a more holistic view of the market
dynamics. Our study is grounded in the rich academic lineage of LOB research,
drawing upon foundational work such as (Abergel et al., , 2016) offering a
comprehensive overview of LOB structures and their implications for market
analysis. Furthermore, our exploration of TMOBBAS and GMP is informed by the
theoretical underpinnings of market microstructure theory, as discussed in
(O’Hara, , 1998), and the empirical findings on the impact of HFT on the
liquidity and volatility of a market, as in (Brogaard et al., , 2014).

The introduction of TMOBBAS and GMP allows us to explore new dimensions of
market behavior, particularly in relation to liquidity and volatility. By
employing high-frequency data from prominent market instruments, we undertake
a meticulous analysis of these concepts through various lenses, including tail
behavior analysis, dynamics of log-returns, and risk–return performance
evaluation. Our methodology is informed by a blend of theoretical insights and
empirical analyses, ensuring a robust examination of the phenomena under
study. The work of (Cont, , 2001) on the empirical properties of asset returns
and the stylized facts of financial markets provides a foundational basis for
our analysis, while the methodologies for the tail behavior analysis draw on
the advanced statistical techniques described by (McNeil et al., , 2015).

The rest of this paper is organized as follows, Section 2 lays the groundwork
with a discussion on LOBs and introduces the foundational concepts of TMOBBAS
and GMP. Section 3 describes in detail the high-frequency data used for the
analysis. Then Section 4 examines the tail behavior of the financial returns.
The dynamics of log-returns are examined in Section 5 through advanced
statistical models. Option pricing and implied volatility analyses for TMOBBAS
and GMP are presented in Section 6. Section 7 evaluates the risk–return
performance of these indices, employing the Rachev ratio for the evaluation.
Section 8 presents a summary of our findings, their implications, and
directions for future research.

##  2 Preliminaries

From (Abergel et al., , 2016), we know that a LOB is essentially a file in a
computer that contains all orders sent to the market, with their
characteristics, such as the sign of the order, its price, quantity, and a
timestamp. In academia, the usual approach involves simplifying the structure
of the LOB to facilitate applying it more efficiently. Figure 1 is an example
of an LOB with a simplified structure.

Figure 1: Example of an LOB with a simplified structure

In Figure 1, the blue side depicts the bid prices (5, 6 and 7), representing
the prices of limit buy orders, while the red side depicts the ask prices (10,
11 and 12), namely, the prices of the limit sell orders. In the bid prices and
ask prices, we also encounter the corresponding best bid price and best ask
price. The best bid price is the highest among the limit buy orders, while the
best ask price is the lowest among the limit sell orders. Hence, in this
example, the best bid price is 7 and the best ask price is 10.

When discussing the best bid price and best ask price, there are two concepts
we cannot ignore. One is the bid–ask spread (BAS), the other is the mid-price.
The BAS is defined as the difference between between the best ask price and
the best bid price, while the mid-price is defined as the average of the bid
price and the ask price. In the example in Figure 1, the BAS is $10-7=3$ and
the mid-price is $(10+7)/2=8.5$.

In this paper, we define two new concepts, which contain the BAS and mid-price
as special cases. Before delving into these new concepts, let us introduce a
new definition:

######  Definition 1.

(TMOBAP and TMOBBP). The total market order book ask price (TMOBAP) is the
quotient of the value of a market order that is sufficient to purchase the
entire volume of shares in the ask limit order book by the total number of
shares available in the ask limit order book. Similarly, the total market
order book bid price (TMOBBP) is the quotient of the value of the market order
that is sufficient to purchase the entire volume of shares in the bid limit
order book by the total number of shares available in the bid limit order
book.

Referring to Definition 1, the two new concepts we define are referred to as
the total market order book bid–ask spread (TMOBBAS) and the global mid-price
(GMP), respectively. They are defined as follows:

TMOBBAS $\displaystyle:=\textrm{TMOBAP - TMOBBP},$ (1) GMP
$\displaystyle:=(\textrm{TMOBAP + TMOBBP})/2.$ (2)

Let us continue to use Figure 1 to provide a clearer explanation of these new
terms. Based on Definition 1, Equations (1) and (2), and referencing Figure 1,
we can derive the following:

TMOBAP $\displaystyle=\frac{10\times 200+11\times 150+12\times
250}{200+150+250}\approx 11.08,$ (3) TMOBBP $\displaystyle=\frac{5\times
100+6\times 150+7\times 200}{100+150+200}\approx 6.22,$ TMOBBAS
$\displaystyle:=\textrm{TMOBAP - TMOBBP}\approx 4.86,$ GMP
$\displaystyle:=\textrm{(TMOBAP + TMOBBP)/2}\approx 8.65.$

As observed, it is evident that both TMOBBAS and GMP are functions of the
LOB’s depth. The results in Equations (3) give TMOBBAS and GMP a depth of 3.
The reason we say that BAS and mid-price are a special case of TMOBBAS and
GMP, respectively, is that BAS and mid-price are actually the TMOBBAS and GMP
with a depth of 1, which is easy to verify.

##  3 The Data

Obtaining the high-frequency LOB data with which to compute the TMOBBAS and
GMP for real-world analysis is a challenging endeavor due to the value and
sensitivity of this information. Fortunately, Limit Order Book System - The
Efficient Reconstructor (LOBSTER)111The official website of LOBSTER is
https://lobsterdata.com/. provided us with free high-frequency data samples
with which to do academic research.

In our analysis, we adhere to the guidelines outlined in (Huang and Polak, ,
2011) to select HFT data samples for Amazon (AMZN), Apple (AAPL) and Google
(GOOG) from 9:30:00 AM to 4:00:00 PM on June 21, 2012. In order to eliminate
the impact of the previous trading day, we remove the first 5% of the
timestamps and the corresponding data in the original data set to obtain the
new dataset for us to analyze. Table 1 summarizes the dataset that we will
use.

Ticker | Number of Timestamps | Maximum Value of Depth
---|---|---
AMZN | 248,202 | 10
AAPL | 365,113 | 10
GOOG | 132,411 | 10
Table 1: Summary of our datasets

To analyze TMOBBAS or GMP at different depths, we must truncate our dataset to
include a specific number of ask prices, bid prices, ask volumes and bid
volumes. For instance, if we intend to examine TMOBBAS and GMP of AAPL at a
depth of 2, we should truncate the dataset for AAPL to include two ask prices,
two bid prices, two ask volumes and two bid volumes. Table 2 is the part of
the dataset for AAPL. It is evident that to explore TMOBBAS and GMP at a depth
of 2, we need to truncate nine columns, including the timestamp, two ask
prices, two ask volumes (sizes), two bid prices, and two bid volumes.

Time | Ask | Ask | Bid | Bid | Ask | Ask | Bid | Bid
---|---|---|---|---|---|---|---|---
| price 1 | size 1 | price 1 | size 1 | price 2 | size 2 | price 2 | size 2
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$
35159.318815640 | 5865700 | 100 | 5863100 | 100 | 5866100 | 1900 | 5862900 | 100
35159.428838154 | 5865700 | 100 | 5863100 | 100 | 5865900 | 200 | 5862900 | 100
35159.432266639 | 5865700 | 100 | 5863100 | 100 | 5865900 | 200 | 5862900 | 100
35159.439130585 | 5865700 | 100 | 5863200 | 100 | 5865900 | 200 | 5863100 | 100
35159.450166122 | 5865700 | 100 | 5863200 | 100 | 5865900 | 200 | 5863100 | 100
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$
57599.913117637 | 5776700 | 300 | 5775400 | 410 | 5776800 | 200 | 5775300 | 1400
Table 2: Part of dataset for AAPL at depth 2

The values in the “Time” column are the time difference between the current
time and midnight in seconds. For example, “34200” means 34,200 seconds from
midnight, which, when converted to hours, is 9.5 hours, namely, 9:30:00 AM.
Similarly, “57600” indicates 57,600 seconds from midnight, corresponding to 16
hours or 4:00:00 PM.

Regarding the price columns, “Ask price 1” is the best ask price, i.e., the
lowest ask price, while “Ask price 2” is the second-best ask price. Similarly,
“Bid price 1” is the best bid price, i.e., the highest bid price, and “Bid
price 2” is the second-best bid price. The values in the price columns are in
US dollars (USD) multiplied by 10,000. For example, “5865700” is 586.57 USD.

Lastly, the values in the size columns are the corresponding number of shares
for the given ask price or bid price.

##  4 Tail Behavior Analysis

###  4.1 Non-Gaussianity of Log-Return

Let $S_{t,d}$ denote TMOBBAS or GMP at timestamp $t$ with a depth of $d$,
where $34,200\textrm{ s}\leq t\leq 57,600\textrm{ s}$ and $1\leq d\leq 10$. If
we regard $S_{t,d}$ as the price of a specific “asset” and we are interested
in exploring the return of this asset, then we can define the log-return
$r_{t,d}$ as the following:

$\displaystyle r_{t,d}:=\log\left(\frac{S_{t+\Delta t,d}}{S_{t,d}}\right)$

where $t+\Delta t$ ($\Delta t>0$) is the timestamp for the next time step
after timestamp $t$. The rationale for employing log-returns in this paper is
grounded in the consideration that in HFT, the $\Delta t$ should be very
small, requiring our study to be based on a continuous-time framework.
Furthermore, in a later section, we will formulate a continuous-time option
pricing model, and (Shreve, , 2004) emphasizes that continuous-time option
pricing typically employs log-returns.

Figure 2 displays the evolution of TMOBBAS, GMP and their log-returns over
time at all ten different depths for AAPL222To conserve space, our primary
focus in the main text will be on AAPL from this section on, due to its having
the largest market capitalization up to 2024 among these three stocks.
Comparable analyses for AMZN and GOOG can be found in Appendix A and Appendix
B, respectively. Since the analyses for AMZN and GOOG are similar to that for
AAPL, we will only provide related graphs but ignore the explanatory text..

(a)

(b)

Figure 2: The evolution of (a) TMOBBAS, (b) GMP and their log-return over time
at all 10 different depths for AAPL (Each graph contains two $y$-axes. The
left $y$-axis indicates the size of TMOBBAS or GMP, and the right $y$-axis
indicates the size of log-return. The evolution of TMOBBAS or GMP is
represented by the blue line while the evolution of the log-return is denoted
by the orange line.)

To enhance our understanding of the log-returns at various depths, we aim to
examine their distributions. From Figure 3, we can directly observe that the
distributions of the log-returns of TMOBBAS and GMP are decidedly non-
Gaussian.

(a)

(b)

Figure 3: Comparison between the kernel density (represented by black solid
lines) of log-returns of (a) TMOBBAS, (b) GMP and the corresponding Gaussian
distribution with the same sample mean and standard deviation (depicted by the
red dashed lines) at all 10 depths for AAPL

Moreover, in order to quantitatively illustrate the non-Gaussianity of the
log-returns, we can compute the excess kurtosis of the log-returns. Suppose we
have an independently and identically distributed (i.i.d.) sample of $X$ with
a size of $n$, where $\mathbb{E}(X)=\mu$ and
$\bm{\mathrm{Var}}(X)=\sigma^{2}$. Then the formula to compute the excess
kurtosis is

$\textrm{excess kurtosis}:=\frac{\mathbb{E}[(X-\mu)^{4}]}{\sigma^{4}}-3.$ (4)

The reason for subtracting $3$ in Equation (4) is to make sure the excess
kurtosis of the Gaussian distribution is zero. In addition to the traditional
excess kurtosis, we also propose a robust measure of excess kurtosis in
detecting a distribution’s tails, as suggested by (Hogg, , 1972, 1974):

$\textrm{robust excess
kurtosis}:=\frac{U_{0.05}-L_{0.05}}{U_{0.5}-L_{0.5}}-2.59,$ (5)

where $U_{\alpha}:=\frac{1}{\alpha}\int_{1-\alpha}^{1}F^{-1}(y)\textrm{ d}y$,
$L_{\alpha}:=\frac{1}{\alpha}\int_{0}^{\alpha}F^{-1}(y)\textrm{ d}y$ and
$F^{-1}(y)$ is the $y$-th quantile of the sample distribution. Still, the
reason for subtracting $2.59$ in Equation (5) is to ensure that the robust
excess kurtosis of the Gaussian distribution is zero.

Figure 4 displays the evolution of the log-returns of TMOBBAS and GMP over
depths for AMZN, AAPL, and GOOG. From Figure 4, we can observe that both the
excess kurtosis and the robust excess kurtosis for each depth are
significantly greater than zero, indicating that the distribution of the log-
returns of TMOBBAS and GMP for these three stocks is leptokurtic.

(a)

(b)

(c)

(d)

Figure 4: The evolution of excess kurtosis for log-returns of (a) TMOBBAS, (b)
GMP and robust excess kurtosis for log-returns of (c) TMOBBAS, (d) GMP over
depth (AMZN, AAPL, and GOOG are represented by red, green, and blue line,
respectively)

In summary, to capture the entire density of the log-returns for TMOBBAS and
GMP, we need to take into account a non-Gaussian distribution that has tails
that asymptotically approach zero more slowly than a Gaussian distribution.
From previous work ((Rachev and Mittnik, , 2000), (Cont, , 2001), (Shimokawa
et al., , 2007), (Lux, , 2009), (Rogers and Zhang, , 2011), etc. among
others), we know that when facing such a problem, we typically examine the
tail behavior first to determine the nature of the distribution’s tail.
Moreover, tail estimation allows us to gain insights directly from the tail
statistics, embodying the principle of “letting the tail speak for itself”
((Embrechts et al., , 1997) and (McNeil et al., , 2015)).

In the subsequent subsections, we will utilize two methods to show that the
distribution of log-returns of TMOBBAS and GMP is heavy-tailed.

###  4.2 Generalized Pareto Distribution Fit

The initial method we propose involves utilizing the Generalized Pareto
Distribution (GPD) to model the tail data of the log-returns. As indicated by
(Balkema and De Haan, , 1974) and (Pickands III, , 1975), a broad category of
tail distribution functions can be well approximated by the GPD.

Let $F_{\textrm{GPD}}(x;\sigma,\xi)$ represent the cumulative distribution
function (CDF) of the GPD

$\displaystyle
F_{\textrm{GPD}}(x;\sigma,\xi)=\begin{cases}1-\left(1+\frac{\xi}{\sigma}x\right)^{-1/\xi}&\textrm{if
}\xi\neq 0\\\ 1-e^{-x/\sigma}&\textrm{if }\xi=0\end{cases}$

for $x\in[0,\infty)$ if $\xi\geq 0$ and $x\in[0,-\sigma/\xi]$ if $\xi<0$.
Here, $\sigma$ is the scale parameter and $\xi$ is the shape parameter. To
determine the nature of a distribution’s tail, the sign of $\xi$ is crucial.
In particular, we have

  * •

If $\xi>0$, then the distribution has a power-law decay, i.e., heavy-tailed.

  * •

If $\xi=0$, then the distribution is exponentially decaying.

  * •

If $\xi<0$, then the distribution has bounded support.

In this study, we consider the top 5% of log-returns data as the tail and
employ the GPD to model them. Figure 5 presents a comparison between the
empirical CDF of the tail data and the corresponding CDF of the GPD of the
TMOBBAS and GMP at all ten depths for AAPL. From Figure 5, it is evident that
the overall fit is excellent, allowing us to proceed with this method in our
study.

(a)

(b)

Figure 5: Comparison between empirical CDF (represented by black solid lines)
of tail and its fitted CDF of GPD of (a) TMOBBAS and (b) GMP at all 10 depths
for AAPL

To estimate $\xi$, a common approach is to utilize maximum likelihood
estimation (MLE) ((Davison, , 1984), (Smith, , 1985), (Smith, , 1987),
(Hosking and Wallis, , 1987), (Grimshaw, , 1993), etc. among others). By
employing the gpfit() function in MATLAB, we can calculate the estimated $\xi$
and its 95% confidence interval (CI) for each depth. Figure 6 presents the
evolution of the estimated $\xi$ and its corresponding 95% CI across different
depths. From Figure 6, we can observe that the estimated $\xi$ are
significantly greater than zero, and their corresponding 95% CIs for each
depth strictly extend to the right of zero. Thus, we should employ a power-law
decay (i.e., heavy-tailed) distribution to fit the entire density of the log-
returns of TMOBBAS and GMP.

(a)

(b)

Figure 6: The evolution of estimated shape parameter $\xi$ (represented by
blue solid line) and its corresponding 95% CI (depicted by two red dashed
lines) for (a) TMOBBAS and (b) GMP over depth for AAPL

###  4.3 The Hill Method

The alternative method we propose is the Hill method. The crucial component in
the Hill method is the Hill estimator, introduced by (Hill, , 1975). In this
method, we will focus on the tail index $\alpha$, the reciprocal of the shape
parameter $\xi$ (i.e., $\alpha=1/\xi$).

Suppose we have an i.i.d. sample of $X$ with a size of $n$,
$X_{1},X_{2},\cdots,X_{n}$, in non-decreasing order, i.e., $X_{(1)}\geq
X_{(2)}\geq\cdots\geq X_{(n)}$. The $k$-th largest value, $X_{(k)}$, is called
the $k$-th order statistic of the sample. The standard form of the Hill
estimator is

$\hat{\alpha}_{k,n}^{(H)}=\left(\frac{1}{k}\sum_{i=1}^{k}\log\frac{X_{(i)}}{X_{(k+1)}}\right)^{-1}=\left[\frac{1}{k}\left(\sum_{i=1}^{k}\log
X_{(i)}\right)-\log X_{(k+1)}\right]^{-1}.$ (6)

From Equation (6), we can see that $\hat{\alpha}_{k,n}^{(H)}$ uses only the
order statistics up to $k+1$ of the sample to estimate $\alpha$. Also, from
(De Haan and Peng, , 1998), we have

$\displaystyle\hat{\alpha}_{k,n}^{(H)}\overset{\mathbb{P}}{\longrightarrow}\alpha,\quad\textrm{if
}n\to\infty,k\to\infty,k/n\to 0.$

This means that the Hill estimator converges in probability to the real tail
index when $k$ is chosen appropriately with growing $n$. From (McNeil et al.,
, 2015), we know that in empirical work, the best choices of $k$ are
relatively small–say 10–50 order statistics in a sample of size 1000, i.e.,
$k/n$ should be about 0.01–0.05.

Besides the Hill estimator itself, we also compute the CI of
$\hat{\alpha}_{k,n}^{(H)}$. The $(1-\theta)\times 100\%$ Wald CI (Haeusler and
Segers, , 2007) can be expressed as

$I_{n}^{\rm
Wald}(\theta,k)=\left[\left(1+\frac{z_{\theta/2}}{\sqrt{k}}\right)^{-1}\hat{\alpha}_{k,n}^{(H)},\left(1-\frac{z_{\theta/2}}{\sqrt{k}}\right)^{-1}\hat{\alpha}_{k,n}^{(H)}\right]$

where $z_{p}$ means the $p$-th quantile of the standard Gaussian distribution.

In this study, we let $k/n\approx 0.05$ and $\theta=0.05$. From Figure 7, we
can see that the estimated tail indices are significantly greater than zero
and the 95% CIs strictly extend to the right of zero, indicating that the
tails are heavy.

(a)

(b)

Figure 7: The dependence of estimated tail index $\alpha$ (represented by blue
solid line) and its corresponding 95% CI (depicted by two red dashed lines)
for (a) TMOBBAS and (b) GMP on the number of order statistics at all 10 depths
for AAPL

##  5 Dynamics of Log-Returns

In the preceding section, we applied the GPD to characterize the right tails
of the logarithmic returns for both TMOBBAS and GMP. Specifically, we
identified the top 5% of log-return data as the tail and utilized the GPD to
model it. Our analysis included estimating the parameters $\xi$ and the tail
index $\alpha$ along with their corresponding 95% confidence intervals for
each depth. This revealed that the distributions of the log-returns for both
TMOBBAS and GMP exhibit heavy-tailed behavior, indicative of significant
deviations from normality.

The methodology employed in the previous sections relied on a historical or
static approach, which involved sequentially sampling the returns data from
fixed historical periods. This provides a snapshot of market activity and
global events within a finite timeframe. Within this framework, we treated the
dataset as a series of historical log returns, assuming they were
independently and identically distributed. This simplification is widely
adopted by practitioners, including traders and portfolio managers, for its
simplicity and accessibility. However, it is imperative to acknowledge the
common caveat found in fund prospectuses that past performance does not
guarantee future returns.

In contrast to the historical approach, dynamic methods investigate more
deeply the informational content in the historical data. These methods operate
under the assumption that historical returns arise from a dynamic univariate
distribution, where characteristics such as dependency may vary over time.
Seeking to uncover the nature of this distribution, dynamic methods generate
extensive predictive samples of asset returns, with a particular focus on
extreme events or tail behavior. The rationale behind using dynamic methods
lies in their ability to adapt to changing market conditions and account for
evolving risk factors that may not be captured by static models. By
continuously updating the model parameters based on new information, dynamic
methods offer improved accuracy in predicting future returns and mitigating
potential risks. While historical methods remain crucial, especially for
modeling new high-frequency spread indices and providing insights into risk
management frameworks like Basel I and II, dynamic approaches offer heightened
sensitivity to potential significant shifts in market performance. By
incorporating the time-varying nature of market dynamics, dynamic methods
provide more robust and reliable predictions, making them invaluable tools for
investors and risk managers alike.

In this section, we adopt a time-series approach designed to identify the best
fitting models for time series data, considering both in-sample and out-of-
sample performance. By leveraging Monte Carlo simulations, our goal is to
generate a true sample of i.i.d. log-returns for subsequent periods. This
methodological shift distinguishes our approach from historical returns, which
inherently lack the i.i.d. property.

###  5.1 ARMA(1,1)–GARCH(1,1) with Normal Inverse Gaussian Distribution

Our dynamic framework incorporates several key components. We utilize a
versatile ARMA(1,1)-GARCH(1,1) model to fit the returns data, incorporating a
Normal Inverse Gaussian (NIG) distribution to model the innovations within the
ARMA-GARCH framework, accommodating heavier tails than a normal distribution.
Subsequently, we generate a large sample of asset innovation values from the
NIG distribution. By applying inverse ARMA-GARCH to this sample set, we derive
a comprehensive set of values of the returns. These values are then
incorporated into our analysis to model spread indices across various depths
of the order book, providing deeper insights into market dynamics and risk
management strategies. Additionally, we aim to employ the same dynamic
framework for analyzing the GMP data alongside TMOBBAS to enhance the
comprehensiveness of our analysis.

If a time-series of returns, $r_{t}$, is stationary, a useful general model
for describing it is the synthesis of the autoregressive moving-average (ARMA)
model and the generalized autoregressive conditional heteroscedasticity
(GARCH) model. The ARMA (Eagle, , 1982) component explicitly models the
behavior of the returns, whereas the GARCH (Bollerslev, , 1986) component
explicitly models its variance. Both models contain theoretically infinite
parameters; the variations in the models are denoted by the finite number of
parameters employed. The ARMA($p,q$) model (Tsay, , 2005) is

$r_{t}=\phi_{0}+\sum_{i=1}^{p}\phi_{i}r_{t-i}+a_{t}+\sum_{j=1}^{q}\theta_{j}a_{t-j},$

where each shock, $a_{t}$, is a zero-mean random variable. The first two terms
in Equation (5.1) describe the autoregressive dependence of $r_{t}$ on
previous returns; the second two terms add the influence of a weighted
(moving) average of shocks, $a_{t}$. The GARCH($m,s$) model relates $a_{t}$
to, and provides a model for, the variance $\sigma_{t}^{2}$ of the series:

$a_{t}=\sigma_{t}\epsilon_{t},\quad\sigma_{t}^{2}=\alpha_{0}+\sum_{i=1}^{m}\alpha_{i}a_{t-i}^{2}+\sum_{j=1}^{s}\beta_{j}\sigma_{t-j}^{2}.$
(7)

Here, the so-called innovations, $\epsilon_{t}$, are zero-mean, unit-variance,
independent, identically distributed random variables. The GARCH model is
clearly autoregressive in both $\sigma_{t}^{2}$ and $a_{t}^{2}$.

Identifying the daily variance as the volatility of the time series, Equation
(7) captures the property of conditional heteroscedasticity, that is, the
property that the volatility is not constant relative to that of prior days.
With six parameters, the ARMA(1,1)–GARCH(1,1) model

$\displaystyle r_{t}$
$\displaystyle=\phi_{0}+\phi_{1}r_{t-1}+a_{t}+\theta_{1}a_{t-1},$
$\displaystyle a_{t}$ $\displaystyle=\sigma_{t}\epsilon_{t},$
$\displaystyle\sigma_{t}^{2}$
$\displaystyle=\alpha_{0}+\alpha_{1}a_{t-1}^{2}+\beta_{1}\sigma_{t-1}^{2},$

provides enough generality to model many time series of returns. However,
providing a fit to a particular time series requires the specification of the
distribution governing the random variables of the innovations. In the dynamic
optimization method, we assume that the innovations, $\epsilon_{t}$, are
governed by the NIG distribution given by

$f(x|\alpha,\beta,\mu,\delta)=\frac{\delta}{2\alpha
K_{1}(\delta)}\exp\left(-\frac{\delta}{\alpha}\left(\sqrt{\beta^{2}+(x-\mu)^{2}}-\sqrt{\beta^{2}+\mu^{2}}\right)\right),$

where $K_{1}(\cdot)$ is the modified Bessel function of the second kind, and
$\alpha$, $\beta$, $\mu$, and $\delta$ are shape and scale parameters.

###  5.2 The TMOBBAS Hill Index Estimations

The dynamic module seeks to provide a statistically accurate larger sampling
of returns for computing the tail index. This is achieved by fitting an
ARMA(1,1)–GARCH(1,1) with an NIG-distribution model to the time series of the
TMOBBAS index of depths $s=1,2,...,10$, generating the model parameters,
computing the shock series $a_{t}$ and the variances $\sigma_{t}^{2}$
predicted from the fitted GARCH(1,1) model, generating $10,000$ of the
innovation series $\epsilon_{t}$, and performing the inverse transformations
$\epsilon_{t}$ in the ARMA(1,1)–GARCH(1,1) model to generate a dynamic
ensemble of predicted returns for the TMOBBAS index for each depth. The
ensemble of returns $\\{r_{t}^{(s)}\\}_{s=1}^{10}$ represents the output from
the dynamic module, which is then incorporated into our analysis to model
spread indices across various depths of the order book, providing deeper
insights into market dynamics and risk management strategies.

Figure 8 illustrates the evolution of the estimated $\xi$ and its
corresponding 95% CI across different depths.

Figure 8: The evolution of estimated shape parameter $\xi$ (represented by
blue solid line) and its corresponding 95% CI (depicted by two red dashed
lines) over depth for AAPL

From Figure 8, we observe that the estimated values of $\xi$ are significantly
greater than zero, and their corresponding 95% CIs for each depth strictly
extend to the right of zero. Thus, we should consider a power-law decay (i.e.,
heavy-tailed) distribution to fit the entire density of the log-returns of the
TMOBBAS and confirm our findings with the historical methodology.

For estimating the Hill index for each depth, consistent with our historical
methodology, we set $k/n\approx 0.05$ and $\theta=0.05$. From Figure 9, it is
evident that the estimated tail indices are significantly greater than zero,
with the 95% confidence intervals extending strictly to the right of zero.
This observation indicates that the tails of the distribution should be
characterized as heavy-tailed, corroborating our findings from the historical
methodology.

Furthermore, in Figure 9, we observe a decreasing trend in the evolution of
the estimated tail index $\alpha$ and its associated confidence intervals as a
function of the number of order statistics. As we include more extreme values
(higher order statistics), the variability in the estimates of the tail index
decreases, leading to a more conservative estimate. In other words, a larger
number of order statistics results in a more robust estimation, yielding a
stable and lower estimate of the tail index.

Figure 9: The dependence of estimated tail index $\alpha$ (represented by
black solid lines) and its corresponding 95% CI (depicted by two red dashed
lines) on the number of order statistics at depths 2, 3, $\cdots$ 10 for AAPL

Furthermore, we examined the tail behavior of an Equally Weighted Portfolio
(EWP) comprising returns from all ten depths. Adopting an EWP approach
facilitated the aggregation of returns across multiple depths, offering a
holistic view of market behavior while minimizing the influence of individual
depth-specific factors. As illustrated in Figure 10, our analysis reaffirms
the heavy-tailed nature of the distribution, indicating significant deviations
from normality. This observation underscores the robustness of our methodology
in capturing heavy-tailed behavior across the TMOBBAS index, further enhancing
our understanding of tail risk dynamics within the market.

Figure 10: The dependence of estimated tail index $\alpha$ (represented by
black solid line) and its corresponding 95% CI (depicted by two red dashed
lines) on the number of order statistics for EWP for AAPL

These findings from our analysis provide valuable insights into the tail
behavior of the log-returns for the TMOBBAS index, particularly when comparing
results obtained from historical methodologies with dynamic approaches. Our
examination of the Hill index for each depth, consistent with our historical
methodology, reaffirms the heavy-tailed nature of the distribution.

###  5.3 The GMP Hill Index Estimations

In this subsection, we extend our analysis to the GMP, applying the same
methodology previously employed for the analysis of the TMOBBAS. By leveraging
the dynamic framework described earlier, we aim to provide insights into the
tail behavior and dynamics of GMP returns across various depths of the order
book.

The dynamic module utilized an ARMA(1,1)–GARCH(1,1) model a NIG distribution
to fit the time series of the GMP index for depths $s=1,2,...,10$. This
generated a statistically accurate ensemble of predicted returns for GMP at
each depth, facilitating the modeling of spread indices and offering deeper
insights into market dynamics and risk management strategies.

We estimated the Hill index for each depth of the GMP, consistent with our
methodology applied to TMOBBAS. Figure 11 demonstrates the evolution of the
estimated tail index $\alpha$ and its corresponding 95% confidence intervals
on the number of order statistics at each depth for GMP.

Figure 11: The dependence of estimated tail index $\alpha$ (represented by
black solid lines) for GMP and its corresponding 95% CI (depicted by two red
dashed lines) on the number of order statistics at depth 2,3,$\cdots$ 10 for
AAPL

Additionally, we examined the tail behavior of an EWP comprising returns from
all ten depths of the GMP. This approach allowed us to gain a comprehensive
understanding of market behavior while mitigating the influence of individual
depth-specific factors. Figure 12 presents the evolution of the estimated tail
index $\alpha$ and its corresponding $95\%$ confidence intervals on the number
of order statistics for the EWP of GMP.

Figure 12: The dependence of estimated tail index $\alpha$ (represented by
black solid line) of GMP and its corresponding 95% CI (depicted by two red
dashed lines) on the number of order statistics for EWP for AAPL

By conducting a thorough analysis of the GMP using the established
methodology, we aim to enhance our understanding of market dynamics and tail
risk within the context of the order book. This analysis complements our
investigation of TMOBBAS and GMP, providing valuable insights into the
behavior of both indices and informing effective risk management strategies in
financial markets.

Furthermore, by comparing the analyses of the tail behavior of GMP and
TMOBBAS, we observe that while there are similar trends in the shape of their
tail indices, there are also notable differences, suggesting distinct risk
profiles for each index. It is worth noting that the Hill index estimated for
all depths of GMP shows very similar values, with minimal variation. This
observation contrasts with the behavior observed for TMOBBAS, where the Hill
index varies significantly between depths, ranging from 0 to 3. The consistent
behavior of the Hill index as a function of depth for GMP suggests a more
stable tail behavior compared to TMOBBAS.

This phenomenon may be due to the nature of the market and the characteristics
of the order book. The GMP may exhibit a more uniform distribution of returns
across depths, leading to consistent tail behavior. On the other hand, the
TMOBBAS may experience more variability in trading activity and liquidity
across depths, resulting in fluctuating tail behavior.

This analysis complements our investigation of TMOBBAS and GMP, providing
valuable insights into the behavior of both indices and informing effective
risk management strategies in financial markets.

##  6 Option Pricing and Implied Volatility

The primary objective of our paper is to introduce and analyze the new high-
frequency spread indices, TMOBBAS and GMP, which capture the dynamics of
variations in the spread across different depths of the order book. Our
findings from previous sections show that the log-returns and spreads at
different depths exhibit heavy-tailed characteristics, highlighting the
potential challenges faced by market participants, particularly high-frequency
traders, who are sensitive to fluctuations in liquidity.

As we investigated the analysis of both TMOBBAS and GMP, it became evident
that global spreads and mid-prices at various depths can significantly vary,
thereby impacting liquidity. This observation underscores the potential
challenges faced by market participants, particularly high-frequency traders,
who are sensitive to fluctuations in liquidity. In response to this concern,
there arises an opportunity to explore innovative risk management strategies.
One such approach involves the adoption of insurance instruments akin to
portfolio insurance, aimed at hedging against the risk of low liquidity.

Thus, there is a compelling rationale for the development of an option pricing
model tailored specifically for spreads at different levels. By constructing
an equally weighted index comprising these spreads and offering derivative
pricing on this index, we can furnish market participants with a robust
mechanism for hedging against the adverse effects of low liquidity. This
initiative not only addresses the practical needs of traders but also
contributes to the ongoing discourse on risk management in financial markets.

This section focuses on pricing options on both the TMOBBAS and GMP indices
and deriving their implied volatilities. Measures of implied volatility play a
crucial role in this context by encapsulating the current perspective of the
market on the future risk, as implied by the observed transaction prices of
option contracts. Given the dynamic nature of the high-frequency market and
the presence of temporal dependence, as manifested in volatility clustering,
traditional measures of historical volatility may be inadequate for capturing
the prevailing risk environment. Implied volatility offers a forward-looking
outlook, integrating market expectations and sentiments, thereby furnishing
valuable insights for traders and investors.

Our approach involves employing a double-subordinated process to model both
the TMOBBAS and GMP indices. This method is chosen to enhance the accuracy and
reliability of the pricing, particularly in capturing the dynamic nature of
the high-frequency market and addressing the challenges associated with the
heavy-tailed distributions of asset returns. This choice is informed by
empirical evidence suggesting that the returns of speculative assets, such as
the TMOBBAS and GMP indices, are heavy-tailed distributions and asymmetric,
challenging the assumption of normality underlying traditional option pricing
models like the Black–Scholes–Merton (BSM) formula. A double-subordinated
process enables the variance of the normal distribution to vary over time,
thereby effectively capturing heavy-tailed phenomena. Specifically, one of the
subordinated processes models intrinsic time, enhancing our ability to capture
the intricate dynamics of both the TMOBBAS and GMP indices and provide more
accurate pricing for options based on them.333For a further discussion of the
use of subordinators in financial modeling, see (Sato, , 1999), (Schoutens, ,
2003), and (Shirvani et al., 2021a, ).

###  6.1 Double Subordination Model for TMOBBAS and GMP

In this subsection, we define the double-subordinated process for modeling
both the TMOBBAS and GMP indices and pricing options based on their dynamics.
By outlining the model parameters and processes, we lay the groundwork for our
analysis, aiming to capture the nuanced behavior of these indices and provide
accurate pricing for options, contributing to risk management and investment
strategies in the high-frequency trading domain. Through this exploration, we
aim to uncover valuable insights into the dynamics of both the TMOBBAS and GMP
indices and their implications for market participants.444For more details on
Lévy subordinator processes, refer to (Duffie, , 2010).

The double-subordination framework involves a Lévy subordinator process,
denoted by $\mathbb{X}=(X(t),t\geq 0)$, defined on a stochastic basis
$(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$. A process $\mathbb{X}$ is
considered a Lévy process if $X(0)=0$ almost surely under $\mathbb{P}$, it has
independent increments, has stationary increments, and is continuous in
probability. 555For more details on Lévy subordinator processes, refer to
(Sato, , 1999) and (Schoutens, , 2003). A Lévy process $\mathbb{T}=(T(t),\
t\geq 0,\ T(0)=0)$ with non-decreasing trajectories (i.e., non-decreasing
sample paths) is called a Lévy subordinator. Since $T\left(0\right)=0$, the
trajectories of $\mathbb{T}$ take only non-negative values. In the BSM option
pricing model, the price dynamics of the underlying asset are given by

$S_{t}^{\textrm{(BSM)}}=e^{X_{t}^{\textrm{(BSM)}}},\quad t\in[0,\tau],$

where the log-process is

$X_{t}^{\textrm{(BSM)}}=X_{0}+\mu_{1}t+\sigma_{1}B_{t},\quad\mu_{1}\in\mathbb{R},\
\sigma_{1}>0,\ X_{0}=\ln(S_{0}),\ S_{0}>0,$

and $\mathbb{B}=(B_{t},\ t\geq 0)$ is a standard Brownian motion. To
accommodate the non-normality of asset returns, (Mandelbrot and Taylor, ,
1967) and (Clark, , 1973) proposed the use of a subordinated Brownian motion,
where the price process $S_{t}^{\textrm{(ss)}}$ and the log-price process are
defined by

$S_{t}^{\textrm{(ss)}}=e^{X_{t}^{\textrm{(ss)}}},\quad t\in[0,\tau],$
$X_{t}^{\textrm{(ss)}}=X_{0}+\mu_{2}t+\sigma_{2}B_{T(t)},\quad\mu_{2}\in\mathbb{R},\
\sigma_{2}>0,$ (8)

where $\mathbb{T}=(T(t),\ t\geq 0,\ T(0)=0)$ is a Lévy subordinator.

Various studies have demonstrated that single-subordinated log-price models
commonly fail to capture the heavy-tailedness observed in financial
returns.666 For example, see (Lundtofte and Wilhelmsson, , 2013) and (Shirvani
et al., 2021b, ). (Shirvani et al., 2021a, ) defined and investigated the
properties of various multiple-subordinated log-return processes designed to
model leptokurtic asset returns. They showed that multiple-subordinated log-
return processes can have heavier tails than single-subordinated models and
that they are capable of capturing skewness and kurtosis. Therefore, a double
subordination framework may be a more appropriate candidate for modeling the
rather extreme behavior of both the TMOBBAS and GMP indices.

To apply double subordination to modeling the TMOBBAS and GMP index price
processes, let $S_{t}$ denote the price process with the dynamics

$\displaystyle S_{t}=e^{X_{t}},\quad t\in[0,\tau],$ $\displaystyle
X_{t}=X_{0}+\mu_{3}t+\gamma U(t)+\rho T(U(t))+\sigma_{3}B_{T(U(t))},\quad
t\geq 0,$ $\displaystyle\mu_{3}\in\mathbb{R},\ \sigma_{3}>0,\
X_{0}=\ln(S_{0}),\ S_{0}>0.$ (9)

where the components of the triple ${B_{s},T(s),U(s)},s\geq 0$ are independent
processes generating the stochastic basis $(\Omega,\mathcal{F},\
\mathbb{F}=\left({\mathcal{F}}_{t},t\geq 0\right),\mathbb{P})$, which
represents the real world.

Consider the case where the subordinators $T(t)$ and $U(t)$ are inverse
Gaussian (IG) Lévy processes; In this case, (Shirvani et al., 2021a, )
referred to $T(U(t))$ as the double inverse Gaussian subordinator and to
$X_{t}$ as a normal double inverse Gaussian (NDIG) log-price process, with the
characteristic function (CF) of $X_{1}$ given by

$\varphi_{X_{1}}(v)=\\\
\exp\left\\{iv\mu_{3}+\frac{\lambda_{U}}{\mu_{U}}\left[1-\sqrt{1-\frac{2\mu_{U}^{2}}{\lambda_{U}}\left(\frac{\lambda_{T}}{\mu_{T}}\left(1-\sqrt{1-\frac{\mu_{T}^{2}}{\lambda_{T}}(2iv\rho-\sigma_{3}^{2}v^{2})}\right)+iv\lambda\right)}\
\right]\right\\},$ (10)

with $v\in\mathbb{R}$.

To price European contingent claims of both TMOBBAS and GMP indices, we follow
(Shirvani et al., , 2023) and assume that the log-price process $X_{t}$
follows a NDIG model. An equivalent martingale measure $\mathbb{Q}$ is derived
from the risk-neutral measure $\mathbb{P}$, ensuring that the discounted price
process $e^{-rt}S_{t}$ is a martingale (Duffie, , 2010, Chapter 6). Using the
martingale-corrected moment matching (MCMM) approach, the parameters of the
process are estimated, and an appropriate drift term is added to ensure the
martingale property.777(Yao et al., , 2011) constructed a martingale measure
using the MCMM approach for the geometric Lévy process model and showed that
this measure is an equivalent martingale measure if there is a continuous
Gaussian part in the Lévy process. If $X_{t}$ is a pure-jump Lévy process,
they pointed out that this measure cannot be equivalent to a physical
probability. However, pricing European options under this measure is still
arbitrage-free.

The price of a European call option $\mathcal{C}$ with an underlying asset
$\mathcal{S}$ is given by

$C(S_{0},r,K,\tau)=e^{-r\tau}\
\mathbb{E}_{\mathbb{Q}}\left[\mathrm{max}(S^{(\mathbb{Q})}_{\tau}-K,0)\right],$

where $\tau$ denotes the maturity, $K$ is the strike price, and
$S_{t}^{(\mathbb{Q})}$ is the price dynamics of $\mathcal{S}$ under
$\mathbb{Q}$. (Shirvani et al., , 2023) dervied the price dynamics of
$\mathcal{S}$ under $\mathbb{Q}$ and the CF of the log-price $\ln
S^{(\mathbb{Q})}_{t}$, respectively, given by

$\displaystyle S^{(\mathbb{Q})}_{t}$
$\displaystyle=\frac{b_{t}S_{t}}{M_{X_{t}}(1)}=S_{0}e^{\left(r-K_{X_{1}}(1)\right)t+X_{t}},\quad
t\in[0,\tau],$ (11) $\displaystyle\varphi_{\ln S^{(\mathbb{Q})}_{t}}(v)$
$\displaystyle=S^{iv}_{0}\exp\left\\{\left[iv(r-K_{X_{1}}(1))+\psi_{X_{1}}(v)\right]t\right\\},\quad
t\in[0,\tau].$ (12)

The efficient computation of option prices relies on (Carr and Madan, ,
1998)’s option pricing formula, utilizing the fast Fourier transform (FFT) and
necessitating access to the CF of the log-price, represented by

$C(S_{0},r,k,\tau)=\frac{e^{-r\tau-
ak}}{\pi}\int_{0}^{\infty}e^{-ivk}\frac{{\varphi}_{\ln
S^{(\mathbb{Q})}_{\tau}}(v-i(a+1))}{a^{2}+a-v^{2}+i(2a+2)v}dv,$ (13)

where $a>0$ and $E(S^{(\mathbb{Q})}_{t})^{a}<\infty$. 888Limits for $a$
ensuring stable option prices are determined through numerical experiments,
see (Shirvani et al., , 2023).

For pricing options on both the TMOBBAS and GMP indices and deriving their
implied volatility, we employ a double-subordinated process to model these
indices. The method of moments and the empirical CF are utilized to estimate
the parameters of the model, following (Paulson et al., , 1975) and (Yu, ,
2003).999Exploiting the fact that the probability density function (PDF) is
the Fourier transform of the CF, the objective is to minimize the squared
differences of the first four moments and the empirical and theoretical CFs.
This involves estimating model parameters through optimization subject to
constraints based on the statistical properties of the data. See (Shirvani et
al., , 2023).

###  6.2 Option Pricing and Implied Volatility of TMOBBAS

To explore the realm of option pricing and implied volatility for the TMOBBAS
index, we employ the NDIG Lévy model. This model facilitates the pricing of
plain vanilla European options on the TMOBBAS index, providing valuable
insights into market dynamics and risk management strategies. Let’s explore
the details.

We apply the NDIG Lévy model to the pricing of plain vanilla European options
on the TMOBBAS index. Let $\mathcal{C}$ be a European call option, where the
underlying risky asset $\mathcal{S}$ follows the log-price process given in
Equation (9). The dynamics of $\mathcal{S}$ on $\mathbb{Q}$ are given by
Equation (11), and the CF of the log-price is given by Equation (12). We
evaluate the integral Equation (13) using the FFT for a range of strike levels
and maturity horizons.

Figure 13 depicts the resulting prices for call and put options plotted
against the time to maturity $\tau$ and the strike price $K$ for depths 5 and
10. Since the call and put surfaces for different depths of the TMOBBAS index
are similar, we only plot the TMOBBAS index with depths 5 and 10. It’s worth
noting that for close-to-the-money values ($K\approx S$), where the strike
price is close to the current asset price, the call and put option prices
increase with $\tau$, reflecting increased future uncertainty. This behavior
is consistent with market expectations, as higher uncertainty about the future
value of the underlying asset leads to higher option prices. This phenomenon
is often observed in financial markets, where options with longer maturities
tend to command higher premiums due to the additional time value and increased
uncertainty associated with longer time horizons.

(a)

(b)

(c)

(d)

Figure 13: NDIG-based call price surfaces for TMOBBAS index of AAPL (a) for
depth 5, (b) for depth 5, (c) for depth 10, and (d) for depth 10 as functions
of the time to maturity and strike price.

Figures 14 and 15 also plot the implied volatility surface computed from call
price options for the TMOBBAS index for depths 1 to 10 and EWP as a function
of $T$ and moneyness, $K/S_{0}$. As is typically observed, at constant values
of $T$ the implied volatility (future uncertainty) increases as strike prices
move away from the value $K/S_{0}$ (the volatility “smile”). At constant
values of $K/S_{0}$, the implied volatility decreases as time to maturity
increases.

Figure 14: Implied volatility surface for depths $1,2,...,10$ for AAPL Figure
15: Implied volatility surface for EWP

We also observe a decrease in implied volatility as depth increases, as
illustrated in Figure 14, which presents an intriguing trend in the context of
option pricing dynamics for the TMOBBAS index. Market depth, representing the
level of liquidity and trading activity, plays a crucial role in determining
option prices and their associated implied volatilities.

The decrease in implied volatility with increasing depth suggests a
relationship between market depth and option pricing stability. Deeper markets
typically exhibit higher liquidity, characterized by a larger volume of buy
and sell orders across various price levels. This increased liquidity
contributes to a more stable and efficient pricing environment, reducing the
uncertainty surrounding future price movements. As a result, options priced in
deeper markets tend to reflect lower implied volatility levels, as market
participants perceive less risk and uncertainty in price fluctuations.

Moreover, deeper markets often feature narrower bid–ask spreads, indicating
tighter pricing and improved market efficiency. The reduced spread suggests
that buyers and sellers are more closely aligned in their pricing
expectations, further contributing to the lower implied volatility observed in
options priced based on the TMOBBAS index.

Additionally, our analysis reveals a striking similarity between the implied
volatility surface of the EWP encompassing all depths of the TMOBBAS index and
the implied volatility of the TMOBBAS index with depth $10$. This observation
suggests that the implied volatility of the EWP, which aggregates returns from
all depths, closely mirrors the implied volatility behavior of the TMOBBAS
index with the greatest market depth. Essentially, the EWP serves as a
holistic representation of market sentiment and volatility by amalgamating
information from diverse depths of the TMOBBAS index. Consequently, the
implied volatility surface of the EWP offers a comprehensive perspective on
market volatility, integrating insights from various depths.

Moreover, the notable resemblance between the implied volatility of the EWP
and the TMOBBAS index with depth $10$ indicates the predominant influence of
deeper market levels on overall market volatility. This underscores the
critical role of deep liquidity levels in shaping market sentiment and option
pricing dynamics. The convergence of implied volatility behavior between the
EWP and the TMOBBAS index with the greatest depth underscores the importance
of considering depth-specific factors in understanding and predicting market
volatility.

To summarize, the inverse relationship between implied volatility and market
depth underscores the importance of liquidity and stability in option pricing.
The decreasing trend in implied volatility with increasing depth highlights
the role of deeper markets in providing a more stable and predictable pricing
environment for options, benefiting both investors and market participants
alike. Furthermore, the remarkable resemblance between the implied volatility
of the EWP encompassing all depths of the TMOBBAS index and the TMOBBAS index
with depth $10$ suggests that deeper market levels have a dominant influence
on overall market volatility, emphasizing the significance of high levels of
liquidity in shaping market sentiment and option pricing dynamics.

###  6.3 Option Pricing and Implied Volatility of GMP

In this subsection, we present the results of analyzing option pricing and
implied volatility for the GMP index, leveraging the NDIG Lévy model. Our
objective is to gain insights into the pricing dynamics of plain vanilla
European options on the GMP index and assess implied volatility trends.

(a)

(b)

(c)

(d)

Figure 16: NDIG-based call price surfaces for TMOBBAS index of AAPL (a) for
depth 5, (b) for depth 5, (c) for depth 10, and (d) for depth 10 as functions
of the time to maturity and strike price.

Figure 16 showcases the pricing surfaces for call and put options plotted
against the time to maturity $\tau$ and the strike price $K$, focusing on
depths 5 and 10 of the GMP index. Notably, these surfaces offer valuable
insights into the option pricing behavior specific to the GMP index.

Observing the patterns in the pricing surfaces reveals intriguing dynamics.
Particularly, for near-the-money options ($K\approx S$), where the strike
price closely aligns with the current asset price, both call and put option
prices demonstrate an increasing trend with $\tau$. This trend reflects the
market’s anticipation of heightened future uncertainty, leading to elevated
option premiums for longer-maturity contracts. Such observations align with
established market phenomena, where longer-maturity options command higher
premiums due to increased time value and augmented uncertainty associated with
extended time horizons.

Figures 17 and 18 depict the implied volatility surface computed from call
price options for the GMP index across depths 1 to 10 and EWP as a function of
$T$ and moneyness, $K/S_{0}$. Notably, the implied volatility exhibits a
distinct “smile” pattern, characterized by a sharp increase in volatility as
the moneyness approaches zero.

Figure 17: GMP Implied volatility surfaces for depths $1,2,...,10$ for AAPL
Figure 18: GMP Implied volatility surface for EWP

Moreover, there is a notable minimum volatility observed close to zero
moneyness, particularly evident as moneyness approaches 1 from both in-the-
money and out-of-the-money options. This phenomenon suggests that market
participants perceive lower risk and uncertainty for options with moneyness
near 1, reflecting a more stable pricing environment.

Interestingly, the implied volatility surface for GMP exhibits a consistent
shape across different depths, with minimal variation in implied volatility as
depth increases. Unlike the TMOBBAS index, where implied volatility decreases
with increasing depth, the implied volatility of GMP remains relatively stable
across all depths. This uniformity in implied volatility suggests a robust and
stable pricing environment for options on the GMP index, regardless of the
market depth.

Comparing the implied volatility of the GMP with that of the TMOBBAS across
all depths reveals that the implied volatility of the GMP is consistently
lower. This disparity in implied volatility underscores the differences in
market dynamics and risk perceptions between the two indices, with the GMP
exhibiting lower perceived risk and volatility than the TMOBBAS.

Regarding the implied volatility surface of the EWP for the GMP, our analysis
indicates that it has a slightly higher volatility than the implied volatility
of each depth of GMP. This suggests that while the overall level of volatility
may vary, the underlying volatility dynamics captured by the implied
volatility surface exhibits uniform characteristics across different depths of
the GMP. Also, despite this difference in overall volatility levels, we
observe a remarkable similarity in the shapes of the implied volatility
surfaces.

Similar to the individual depths of GMP, the implied volatility surface of the
EWP also displays the characteristic volatility “smile,” with a sharp increase
in volatility as the moneyness approaches zero. Additionally, the minimum
volatility is observed close to zero moneyness, as the options move both in-
the-money and out-of-the-money.

In conclusion, our analysis of the option pricing and implied volatility
surfaces for both the TMOBBAS and GMP indices reveals intriguing insights into
the dynamics of the market and the behavior of the volatility. While the
TMOBBAS index exhibits distinct characteristics in implied volatility across
different depths, with a noticeable decrease in volatility as the depth
increases, the GMP index has a more uniform implied volatility across
different depths. Despite differences in the overall volatility levels, both
indices display similar shapes in their implied volatility surfaces,
characterized by the volatility “smile.” Moreover, the EWP for both indices
serves as a comprehensive representation of market sentiment and volatility,
aggregating information from diverse depths. Overall, these findings
underscore the importance of considering the market depth and composition of
the index in understanding and predicting option pricing dynamics and implied
volatility in high-frequency trading environments.

##  7 Evaluation of Risk–Return Performance

In this section, we will focus on measuring the risk–return performance for
TMOBBAS and GMP. To do this, we will employ the Rachev ratio (Biglova et al.,
, 2004) as the metric. Unlike reward-to-variability ratios such as the well-
known Sharpe ratio (Sharpe, , 1966) and Sortino ratio (Sortino and Price, ,
1994), the Rachev ratio is a reward-to-risk ratio designed to assess the
potential for extreme positive returns relative to the risk of extreme losses
in a non-Gaussian setting. Intuitively, it reflects the potential for
significant gains compared to the risk of significant losses, at a rarity
frequency $q$ (quantile level) defined by the researcher.

The formula to compute Rachev ratio for the TMOBBAS or GMP is

$\textrm{RR}_{(\beta,\gamma)}(X):=\frac{\textrm{AVaR}_{\beta}(-X)}{\textrm{AVaR}_{\gamma}(X)}$
(14)

where

  * •

$X$ denotes the log-return of TMOBBAS or GMP at a given time interval.

  * •

$\textrm{AVaR}_{\gamma}(X):=\gamma^{-1}\int_{0}^{\gamma}\textrm{VaR}_{u}(X)\textrm{
d}u$ is the average Value-at-Risk (VaR) at the level $\gamma\in(0,1]$.

  * •

$\textrm{VaR}_{u}(X):=\inf\left\\{m\in\mathbb{R}:\mathbb{P}[X+m<0]\leq
u\right\\}$ is the VaR at the level $u$.

The formula to compute $\textrm{AVaR}_{\gamma}(X)$ may seem a bit challenging.
However, according to (Föllmer and Schied, , 2004), we know that

$\textrm{AVaR}_{\beta}(X)=\frac{\mathbb{E}[(q-X)^{+}]}{\beta}-q$ (15)

where

  * •

$q$ is the $\beta$-quantile of $X$

  * •

$(q-X)^{+}:=\max(q-X,0)$.

In this study, we set $\beta=\gamma=0.05$ to explore the relationship between
excess profit and excess loss when “investing” in the TMOBBAS. Next, we merge
Equation (14) and Equation (15), applying them to our dataset for AMZN, AAPL,
and GOOG. Figure 19 illustrates the evolution of the Rachev ratio of TMOBBAS
and GMP for AMZN, AAPL, and GOOG across different depths.

(a)

(b)

Figure 19: The evolution of Rachev ratio of (a) TMOBBAS and (b) GMP for AMZN,
AAPL, and GOOG over depth (AMZN, AAPL, and GOOG represented by red, green, and
blue line, respectively)

From Figure 19(a), we can observe that

  * •

The general trend shows that as the depth increases, the Rachev ratio
initially rises and then declines.

  * •

For AAPL, the Rachev ratio remains consistently above 1 across different
depths, suggesting that excess losses can always be offset by excess profits
when investing in TMOBBAS at varying depths of AAPL. In particular, the
highest Rachev ratio occurs at a depth of 2.

  * •

In the case of AMZN, as the depth increases to 5 or beyond, the Rachev ratio
drops below 1, indicating that excess losses cannot be balanced by excess
profits when investing in TMOBBAS at such depths. Similarly to AAPL, the
Rachev ratio for AMZN peaks at a depth of 2.

  * •

For GOOG, as the depth increases to 6 or beyond, the Rachev ratio falls below
1, suggesting that excess losses cannot be balanced by excess profits when
investing in TMOBBAS at such depths. Notably, the Rachev ratio for GOOG
achieves relatively high values at depths 2 and 3.

From Figure 19(b), we can observe that

  * •

The general trend shows that as the depth increases, the Rachev ratio
initially rises and then declines.

  * •

For AAPL, the Rachev ratio increases first, reaching its peak at a depth of 2,
approximately 1.001. Then, this value remains close to, but less than, 1 for
the next eight depths. This suggests that the excess loss can almost be
balanced by the excess profit when investing in GMP for AAPL at various
depths.

  * •

For AMZN, the Rachev ratio also increases initially, reaching its peak at a
depth of 4, around 1. Afterward, the overall trend of the Rachev ratio is to
decrease until it reaches a value of approximately 0.993 at a depth of 10.
This suggests that the excess loss cannot almost be balanced by the excess
profit when investing GMP for AMZN at various depths.

  * •

For GOOG, the Rachev ratio initially increases, reaching its peak at a depth
of 2, approximately 1.005. It then decreases to its lowest point at a depth of
8, around 0.981, before increasing again to about 0.988 at a depth of 10.
Overall, we can see that the excess loss can be balanced by the excess profit
when investing in GMP for GOOG at depths of 2, 3, and 4. However, for other
depths, the excess loss cannot be balanced by excess profit.

##  8 Summary

This paper has ventured beyond traditional analyses of limit order books
(LOBs) to introduce and scrutinize two pivotal concepts: the total market
order book bid–ask spread (TMOBBAS) and the global mid-price (GMP). Our
exploration, grounded in high-frequency data, achieves nuanced understandings
of market liquidity, volatility, and efficiency through the lens of these
novel metrics. The study highlights the significance of considering the entire
depth of the LOB in capturing the complexity of market dynamics, a perspective
that enriches our comprehension of financial markets in the era of high-
frequency trading.

Our findings underscore the multifaceted nature of the TMOBBAS and GMP,
illustrating their roles in offering strategic insights into the prediction of
spreads and understanding the behavior of the market. By examining the
analysis of the tail behavior, the dynamics of the log-returns, and
risk–return performance, we have provided a comprehensive assessment of the
risks and returns associated with these measures under different market
conditions.

The implications of our study are many, touching upon practical aspects for
traders and risk managers, as well as theoretical contributions to financial
market research. Our analysis not only advances the academic discourse by
introducing innovative concepts in the study of LOBs but also offers valuable
insights for market participants navigating the complexities of modern
financial systems. In doing so, this paper bridges the gap between theoretical
models and practical realities, paving the way for future research and
strategic applications in financial market analysis.

## Data Availability Statement

Publicly available datasets were analyzed in this study. This data can be
found here: LOBSTER (https://lobsterdata.com/).

## Author Contributions

YH: Data curation, Formal analysis, Investigation, Methodology, Software,
Validation, Visualization, Writing - original draft, Writing - review &
editing; AS: Data curation, Formal analysis, Methodology, Software,
Validation, Writing - original draft, Writing - review & editing; BS: Writing
- review & editing; SR: Conceptualization, Investigation, Methodology, Project
administration, Resources, Supervision, Visualization, Writing - review &
editing; FF: Resources, Writing - review & editing

## Disclosure Statement

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be constructed as a potential
conflict of interest.

## Funding

The authors declare that no financial support was received for the research,
authorship, and publication of this article.

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##  Appendix A Figures for AMZN

###  A.1 Static Method

(a)

(b)

Figure 20: The evolution of (a) TMOBBAS, (b) GMP and their log-returns over
time at all 10 different depths for AMZN (Each graph contains two $y$-axes.
The left $y$-axis is for TMOBBAS (or GMP), and the right $y$-axis is for the
log-return. The evolution of TMOBBAS (or GMP) is represented by the blue line
while the evolution of the log-returns is represented by the orange line.)

(a)

(b)

Figure 21: Comparison of the kernel density (represented by black solid lines)
of log-returns of (a) TMOBBAS, (b) GMP with the corresponding Gaussian
distribution with the same sample mean and standard deviation (depicted by the
red dashed lines) at all 10 depths for AMZN

(a)

(b)

Figure 22: Comparison of empirical CDF (represented by black solid lines) of
tail and its fitted CDF of GPD of (a) TMOBBAS and (b) GMP at all 10 depths for
AMZN

(a)

(b)

Figure 23: The evolution of estimated shape parameter $\xi$ (represented by
blue solid line) and its corresponding 95% CI (depicted by two red dashed
lines) for (a) TMOBBAS and (b) GMP over depth for AMZN

(a)

(b)

Figure 24: The dependence of estimated tail index $\alpha$ (represented by
blue solid line) and its corresponding 95% CI (depicted by two red dashed
lines) for (a) TMOBBAS and (b) GMP on the number of order statistics at all 10
depths for AMZN

###  A.2 Dynamic Method

####  A.2.1 Figures for TMOBBAS

Figure 25: The dependence of estimated tail index $\alpha$ (represented by
black solid lines) and its corresponding 95% CI (depicted by two red dashed
lines) on the number of order statistics at depths 2, 3, $\cdots$ 10 for AMZN
Figure 26: The dependence of estimated tail index $\alpha$ (represented by
black solid line) and its corresponding 95% CI (depicted by two red dashed
lines) on the number of order statistics for EWP for AMZN

Figure 27: TMOBBAS Implied volatility surfaces for depths $1,2,...,10$ for
AMZN Figure 28: TMOBBAS Implied volatility surface for EWP

####  A.2.2 Figures for GMP

Figure 29: The dependence of estimated tail index $\alpha$ (represented by
black solid lines) for GMP and its corresponding 95% CI (depicted by two red
dashed lines) on the number of order statistics at depths 2, 3, $\cdots$ 10
for AMZN Figure 30: The dependence of estimated tail index $\alpha$
(represented by black solid line) of GMP and its corresponding 95% CI
(depicted by two red dashed lines) on the number of order statistics for EWP
for AMZN

Figure 31: GMP Implied volatility surfaces for depths $1,2,...,10$ for AMZN
Figure 32: GMP Implied volatility surface for EWP

##  Appendix B Figures for GOOG

###  B.1 Static Method

(a)

(b)

Figure 33: The evolution of (a) TMOBBAS, (b) GMP and their log-returns over
time at all 10 different depths for GOOG (Each graph contains two $y$-axes.
The left $y$-axis is for the TMOBBAS (or GMP), and the right $y$-axis is for
the log-returns. The evolution of TMOBBAS (or GMP) is represented by a blue
line while the evolution of the log-returns is represented by an orange line.)

(a)

(b)

Figure 34: Comparison of the kernel density (represented by black solid lines)
of log-returns of (a) TMOBBAS, (b) GMP and the corresponding Gaussian
distribution with the same sample mean and standard deviation (depicted by the
red dashed lines) at all 10 depths for GOOG

(a)

(b)

Figure 35: Comparison of empirical CDF (represented by black solid lines) of
tail and its fitted CDF of GPD of (a) TMOBBAS and (b) GMP at all 10 depths for
GOOG

(a)

(b)

Figure 36: The evolution of estimated shape parameter $\xi$ (represented by
blue solid line) and its corresponding 95% CI (depicted by two red dashed
lines) for (a) TMOBBAS and (b) GMP over depth for GOOG

(a)

(b)

Figure 37: The dependence of estimated tail index $\alpha$ (represented by
blue solid line) and its corresponding 95% CI (depicted by two red dashed
lines) for (a) TMOBBAS and (b) GMP on the number of order statistics at all 10
depths for GOOG

###  B.2 Dynamic Method

####  B.2.1 Figures for TMOBBAS

Figure 38: The dependence of estimated tail index $\alpha$ (represented by
black solid lines) and its corresponding 95% CI (depicted by two red dashed
lines) on the number of order statistics at depths 2, 3, $\cdots$ 10 for GOOG
Figure 39: The dependence of estimated tail index $\alpha$ (represented by
black solid line) and its corresponding 95% CI (depicted by two red dashed
lines) on the number of order statistics for EWP for GOOG

Figure 40: TMOBBAS Implied volatility surfaces for depths $1,2,...,10$ for
GOOG Figure 41: TMOBBAS Implied volatility surface for EWP

####  B.2.2 Figures for GMP

Figure 42: The dependence of estimated tail index $\alpha$ (represented by
black solid lines) for GMP and its corresponding 95% CI (depicted by two red
dashed lines) on the number of order statistics at depths 2, 3, $\cdots$ 10
for GOOG Figure 43: The dependence of estimated tail index $\alpha$
(represented by black solid line) of GMP and its corresponding 95% CI
(depicted by two red dashed lines) on the number of order statistics for EWP
for GOOG

Figure 44: GMP Implied volatility surfaces for depths $1,2,...,10$ for GOOG
Figure 45: GMP Implied volatility surface for EWP