Drawdown-based risk indicators for high-frequency financial volumes

Abstract

In stock markets, trading volumes serve as a crucial variable, acting as a measure for a security’s liquidity level. To evaluate liquidity risk exposure, we examine the process of volume drawdown and measures of crash-recovery within fluctuating time frames. These moving time windows shield our financial indicators from being affected by the massive transaction volume, a characteristic of the opening and closing of stock markets. The empirical study is conducted on the high-frequency financial volumes of Tesla, Netflix, and Apple, spanning from April to September 2022. First, we model the financial volume time series for each stock using a semi-Markov model, known as the weighted-indexed semi-Markov chain (WISMC) model. Second, we calculate both real and synthetic drawdown-based risk indicators for comparison purposes. The findings reveal that our risk measures possess statistically different distributions, contingent on the selected time windows. On a global scale, for all assets, financial risk indicators calculated on data derived from the WISMC model closely align with the real ones in terms of Kullback–Leibler divergence.

Introduction

In recent years, there has been a growing interest in drawdown-based risk measures within both academic circles and the financial sector, as evidenced by numerous studies (see e.g., D’Amico et al. (2020), D’Amico et al. (2023), Zhang and Hadjiliadis (2012), Hongzhong (YYYY), Li et al. (2022), Jiang (2022), Cantor (2001)). The primary reason for this increased attention is that these measures consider the temporal progression of financial data. This aspect is overlooked by other inflated risk indicators, such as the value-at-risk and the expected shortfall.

Several risk indicators based on drawdown have been proposed in academic literature. Notably, the expected conditional drawdown measures and the conditional drawdown measures are among the most prevalent (e.g., see Goldberg and Mahmoud (2017) and Chekhlov et al. (2005), respectively). This group of financial indicators also includes the maximum drawdown and average drawdown, which are both straightforward and user-friendly (e.g., see Casati and Tabachnik (2013) and Chekhlov et al. (2005), respectively). Recently, new risk measures based on drawdown, which are closely related to market crashes, have been proposed. Specifically, the drawdown of a fixed level, the time to crash, and the speed of crash were initially introduced in Zhang and Hadjiliadis (2012) and subsequently expanded upon in D’Amico et al. (2020, 2023). These measures provide investors with crucial information about the risk level prior to a crash event occurring. The recovery time and the speed of recovery, which were further developed and examined in D’Amico et al. (2020), examine the phase that transitions an asset from financial collapse to recovery.

To reproduce financial time series, it is necessary to use a stochastic model. There are several models suggested in academic literature, with the most prevalent being econometric models (refer to D’Amico et al. (2023)) or diffusive models (refer to Zhang and Hadjiliadis (2012)). Recently, effective alternatives based on semi-Markovian models have been introduced (refer to D’Amico et al. (2020), Masala and Petroni (2022), Swishchuk and Islam (2011), Swishchuk and Vadori (2017), Vassiliou (2014), Vassiliou (2020), D’Amico and Petroni (2011), D’Amico and Petroni (2012), Limnios and Oprisan (2001), Janssen (2013)).

Semi-Markov processes, which are extensions of Markov processes (refer to Puneet and Dharmaraja (2021), Barbu and Limnios (2009)), describe the time intervals between transitions using any type of distribution. Specifically, a general semi-Markov model, known as the weighted-indexed semi-Markov Chain (WISMC) model, has been used extensively by numerous researchers. It is used in many sectors, including finance, in both its univariate and multivariate versions (see e.g., D’Amico and Petroni (2012), D’Amico and Petroni (2018), De Blasis (2023), Pasricha et al. (2020), D’Amico et al. (2018), D’Amico et al. (2020), Masala and Petroni (2022), D’Amico and Petroni (2021)). The true power of this model, and what distinguishes it from a conventional semi-Markov model, lies in the incorporation of an index process. This process facilitates the aggregation of information derived from the historical trajectory of the financial time series.

This study broadens the scope of all risk measures discussed in D’Amico et al. (2020), including the drawdown of a fixed level, the time to crash, the speed of crash, the recovery time, and the speed of recovery, by considering time-varying windows. Furthermore, it extends their examination to high-frequency financial volumes, given the pivotal role that intraday data play in financial markets, as noted in Zhang et al. (2023). Note that previous studies have consistently applied drawdown-based risk measures to asset returns, not to asset volumes, as seen in D’Amico et al. (2020), D’Amico et al. (2023), Masala and Petroni (2022). Therefore, we posit that incorporating trading volumes into our study represents a novel and original contribution, considering their significant role in stock markets as a key financial quantity for assessing an asset’s liquidity risk, as referenced in Queirós (2005), Martınez et al. (2005), Bank et al. (2011). High volumes indicate a liquid asset that can be sold quickly and easily, while low volumes suggest an illiquid asset that is challenging to trade. This demonstrates that WISMC models are suitable for replicating volume data and can thus be effectively employed to study drawdown-based liquidity risk measures.

In this study, we employ a WISMC process to model the minute-by-minute volumes of Tesla, Netflix, and Apple assets listed on the Nasdaq Stock Exchange. First, we verify the need for a model that considers the correlation between transition times and volumes using a $\chi ^{2}$ test. The results consistently reject the independence hypothesis. Second, we calculate drawdown-based risk indicators on both real and synthetic data, which are generated using the WISMC model, in time-varying windows by setting a starting time s. We then conduct a goodness-of-fit test (Song 2002) on the real data to confirm that the selection of different time-varying windows affects our financial indicators, causing them to vary and thus subjecting the investor to fluctuating risk. Third, for both real and simulated financial indicators, we determine their optimal parametric distribution among the Lognormal, Weibull, Exponential, and Gamma laws using the AIC and BIC criteria, as per (D’Amico et al. 2020). This analysis, performed with Type-1 right censorship, is conducted to compare real and simulated risk indicators. Overall, for each asset, the simulated financial risk indicators closely resemble the real ones in terms of Kullback–Leibler divergence.

The structure of the rest of this paper is as follows. In the section titled “Drawdown-based Risk Measures,” we offer a detailed explanation of the drawdown-based measures we examine. The “Mathematical Model” tion briefly introduces the WISMC model. In the “Application Results” section, we present the findings of our analysis. Finally, in the “Conclusion” section, we share our final thoughts and objectives for future research.

Drawdown-based risk measures

Trading volumes denote the aggregate count of security transactions within a given time frame, serving as a crucial financial metric that gauges an asset’s liquidity. High volumes imply a liquid asset, which can be swiftly sold at the preferred price. Conversely, low volumes suggest an illiquid asset, which poses challenges in converting it into cash. In general, financial volumes are at their peak during the opening and closing of the daily stock market, because traders are required to establish a position. These periods are composed of minutes (refer to Graczyk and Duarte Queiros (2016)). Consequently, we incorporate a general starting time s, which may not necessarily align with the initial minute (i.e., $s = 0$) of the trading day, in the definitions of the risk measures. Therefore, time s enables the computation of all the contemplated financial indicators for any sub-daily interval.

Obviously, if time s equals zero we recover definitions in D’Amico et al. (2020).

To qualify the drawdown of an asset, we introduce the discrete time-varying volume process X(t) and its running maximum process $Y^{s}(t)$, defined as

$$\begin{aligned} Y^{s}(t):=\max _{l \in [s,t]} \{X(l) \}. \end{aligned}$$

(1)

The introduction of $s >0$ mitigates the effect of sudden and high initial volume transactions on our risk measures.

Accordingly, the drawdown process $D^{s}(t)$ is determined as the difference between the running maximum process $Y^{s}(t)$ and the volume process X(t), as follows:

$$\begin{aligned} D^{s}(t):= Y^{s}(t)-X(t) \,\, \text {with}\,\, t \ge s. \end{aligned}$$

(2)

This expresses the correction of the security trading volume with respect to a previous relative maximum.

Fig. 1

Volume processes (red lines), running maximum processes (blue lines) and drawdown processes (black lines) for Tesla asset, considering two different starting times ($s=0$ and $s=100$)

To gain a solid understanding of the aforementioned definitions and the function of time s, we provide a visual representation in Fig. 1 for both $s=0$ and $s=100$. The red, blue, and black lines depict the volume processes, the ongoing maximum processes, and the drawdown processes, respectively, of the Tesla asset over the course of a trading day. The x-axis denotes the minutes, which serve as the standard unit of measurement for the trading day. Specifically, the stock market day consists of 391 min when $s=0$, and 291 min when $s=100$. It is observable that Tesla’s volume processes exhibit a characteristic U-shape form (refer to Queirós (2016)), and for $s=0$, the initial fluctuations are significantly more pronounced than for $s=100$. Note that substantial and abrupt initial volume changes have significant effects on the calculation of risk measures based on drawdown. Indeed, the running maximum process, and consequently the drawdown process, assume extremely high values from the first few minutes of the trading day.

To examine the influence of these fluctuations on our risk indicators, we calculate them using various starting times s, which represent the point from which the risk measures need to be computed. We introduce drawdown-based financial indicators that are closely related to market crashes, including the drawdown of a fixed level, the time to crash, the speed of crash, the recovery time, and the speed of recovery. These crash-recovery measures are typically applied to financial returns (refer to D’Amico et al. (2020), D’Amico et al. (2023) for examples). However, applying them to trading volumes alters their financial interpretation. Specifically, when volumes are low, these risk measures imply a lack of interest in the asset. On the other hand, high volumes suggest that the security is highly attractive.

The drawdown of fixed level is the first time that the drawdown process achieves or overcomes a certain threshold M.

Rigorously, this is defined as

$$\begin{aligned} \tau ^{s}(M):=\min \{t\ge s\mid D^{s}(t)\ge M \}\,\,\,\,\, \text {with}\,\,\,\,\, M\ge 0. \end{aligned}$$

(3)

To identify the time to crash, we need to introduce the last visit time of the maximum before the stopping time $\tau ^{s}(M)$; formally, this is defined as

$$\begin{aligned} \rho ^{s}(M):= \max \{t\in [s,\tau ^{s}(M)]\mid X(t)=Y^{s}(t)\}. \end{aligned}$$

(4)

Exploiting the definition of $\tau ^{s}(M)$ and $\rho ^{s}(M)$, we qualify the time to crash as

$$\begin{aligned} T_c^{s}(M):=\tau ^{s}(M)-\rho ^{s}(M). \end{aligned}$$

(5)

This is the time necessary to have the first M-variation in the drawdown process.

Consequently, the speed of crash, that is, the velocity at which the first M-change occurs, is expressed as

$$\begin{aligned} S_c^{s}(M):= \frac{M}{T_c^{s}(M) }. \end{aligned}$$

(6)

These indicators offer insights into the level of risk prior to a crash event. Equally, examining the subsequent phase that transitions the financial asset from the crash to recovery is intriguing. In order to do this, we define the quantity $\gamma ^{s}(M,M')$, as follows:

$$\begin{aligned} \gamma ^{s}(M,M'):= \min \{ t>\tau ^{s}(M) \mid D^{s}(t)\le M'\} \,\,\,\,\, \text {with}\,\,\,\,\, M >M'. \end{aligned}$$

(7)

This describes the first moment in which the drawdown process falls below the threshold $M'$, just after having crossed the threshold M for the first time.

Using the definitions of $\gamma ^{s}(M,M')$ and $\tau ^{s}(M)$, the recovery time is determined as

$$\begin{aligned} R_{t}^{s}(M,M'):= \gamma ^{s}(M,M')-\tau ^{s}(M). \end{aligned}$$

(8)

Hence, the speed of recovery, that is, the velocity at which we attain the $M'$ threshold after exceeding the threshold M for the first time, is defined as

$$\begin{aligned} S_{r}^{s}(M,M'):= \frac{M-M'}{R_{t}^{s}(M,M')}. \end{aligned}$$

(9)

Fig. 2

Drawdown processes of Tesla asset during a trading day, for $s=100$. The orange and blue lines stand for the thresholds M and $M'$

Figure 2 illustrates graphically all the financial indicators just described, considering Tesla and fixing $s=100$.

As indicated by the definitions provided, all risk measures depend on a threshold. The risk ’appetite of investors dictates the levels of M and $M'$. Investors who are more adventurous tend to favor higher thresholds, thereby accepting a higher risk. On the other hand, investors who are more risk-averse opt for lower thresholds, which signifies less risk. However, the selection of these thresholds must be contextualized within the trend of the drawdown process. If the drawdown process exhibits high values and an investor chooses a threshold that is too low, it will be surpassed immediately, placing the investor in a perceived high-risk area. Conversely, if an investor sets a threshold that is too high, there is a possibility that the drawdown will never exceed it, resulting in the stock being perceived as less risky than the investor’s maximum ’risk tolerance.

All of these risk indicators provide investors with useful financial tools to evaluate the liquidity risk of a particular investment or investment portfolio. Specifically, $\tau ^{s}$ Provides information about the hazard of an asset and depends on the selected M-value. Small M values indicate low-risk events, that is, a typical market condition in which the stock can be sold easily (i.e., a liquid asset). Conversely, high M values denote very risky events, signifying very uncertain and unstable market conditions for the asset and, thus, a general difficulty in trading it (i.e., an illiquid asset).

$T_c^{s}(M)$ and $S_c^{s}(M)$ quantify how long it takes and how quickly such risky events occur and, consequently, the duration and speed, respectively, at which a stock becomes illiquid.

Unlike previous metrics, $R_{t}^{s}(M, M')$ and $S_{r}^{s}(M, M')$ examine the behavior of a stock after reaching a specific M-level in its drawdown, thereby indicating a dependency on both the M and $M'$ thresholds. Specifically, the recovery time and the speed of recovery indicate the duration of the drawdown process to reach the $M'$ threshold after already hitting the M threshold, and the rate at which this happens. In simpler terms, these metrics measure the number of minutes an asset requires to shift from an illiquid state to a more liquid one, and the speed of this transition.

Mathematical model

Semi-Markov chain models extend the capabilities of Markov chain models by allowing the time intervals between transitions to be shaped by any given distribution, as opposed to a predetermined one. As such, Markovian processes can be viewed as a specific instance of semi-Markovian processes. This is evident when the waiting time distributions in the states of the system are memoryless distributions, such as the geometric distribution in discrete time or the exponential distribution in continuous time.

Consequently, semi-Markovian models consider the present state of the system and its duration in that state, but they disregard the preceding history. This is a significant limitation for these models, especially because past events are important in financial time series. A potential solution to this problem is the application of a more comprehensive semi-Markov chain model, namely the WISMC model (e.g., D’Amico and Petroni (2012), D’Amico and Petroni (2018), D’Amico and Petroni (2021)).

For this reason, we apply a WISMC process to model the high-frequency financial volumes.

Fundamentally, the WISMC model is described by three stochastic processes: $\{J_n\}_{n\in {\mathbb {N}}}$, $\{T_n\}_{n\in {\mathbb {N}}}$, and $\{U_n^{\lambda } \}_{n\in {\mathbb {N}}}$, which in our context are the trading volume process, the corresponding jump time process, and the index process, respectively.

Assuming that X(t) is the discrete time-varying volume process of an asset, we transform it into a series of discrete volumes, denoted by $X_d(t)$, following the map described in D’Amico and Petroni (2012). Then, the sequence of discrete volumes $\{X_d(t)\}_{t\in {\mathbb {N}}}$ is converted into a series of volumes $\{J_n\}_{n\in {\mathbb {N}}}$, denoting the value of the volumes process at the nth change, and into a series of corresponding jump times $\{T_n\}_{n\in {\mathbb {N}}}$, signifying the time at which the nth change in the volumes process occurred. In order to do this, we set $T_0=0$ and $J_0=X_d(0)$, and for $n\ge 1$,

$$\begin{aligned} T_n= & {} \inf \{t \in {\mathbb {N}}, t>T_{n-1}: X_d(t)\ne X_d(T_{n-1})\}, \end{aligned}$$

(10)

$$\begin{aligned} J_n= & {} X_d(T_n). \end{aligned}$$

(11)

The addition of the stochastic process $\{U_n^{\lambda }\}_{n \in {\mathbb {N}}}$, with values in ${\mathbb {R}}$, is the major extension to the traditional semi-Markov model. The random variable $U_n^{\lambda }$ describes the value of the index process at the nth transition, that is, it synthesizes the information contained in the past trajectory of the volume process up to the nth transition. It is defined as

$$\begin{aligned} U_n^{\lambda }=\sum _{k=0}^{n-1} \sum _{a=T_{n-1-k}}^{T_{n-k}-1} f^{\lambda }(J_{n-1-k},T_n,a)+f^{\lambda }(J_{n},T_n,T_n). \end{aligned}$$

(12)

To better understand equation (12), we include a numerical example considering only 3 transitions for the volume process (i.e., $n=3$). In Table 1, we report the volume values ($\{J_0, J_1, J_2, J_3\}$) and the corresponding jump time values ($\{T_0, T_1, T_2, T_3\}$) required in our example.

Note that the volume values in Table 1 are completely unrealistic. We selected them to simplify the notation and thus, construct the example in a functional manner.

By fixing $n=3$, equation (12) becomes

$$\begin{aligned} U_3^{\lambda }=\sum _{k=0}^{2} \sum _{a=T_{2-k}}^{T_{3-k}-1} f^{\lambda }(J_{2-k},T_3,a)+f^{\lambda }(J_{3},T_3,T_3). \end{aligned}$$

Using data in Table 1, we get

$$\begin{aligned} U_3^{\lambda }=\sum _{k=0}^{2} \sum _{a=T_{2-k}}^{T_{3-k}-1} f^{\lambda }(J_{2-k},50,a)+f^{\lambda }(4,50,50). \end{aligned}$$

Next, we perform the first summation (i.e., the summation with index k), and we obtain

$$\begin{aligned} U_3^{\lambda }=\sum _{a=T_{2}}^{T_{3}-1} f^{\lambda }(J_2,50,a)+\sum _{a=T_{1}}^{T_{2}-1} f^{\lambda }(J_1,50,a)+\sum _{a=T_{0}}^{T_{1}-1}f^{\lambda }(J_0,50,a) +f^{\lambda }(4,50,50). \end{aligned}$$

Finally, employing numerical values in Table 1, we get

$$\begin{aligned} U_3^{\lambda }=\sum _{a=0}^{2} f^{\lambda }(2,50,a)+\sum _{a=3}^{9} f^{\lambda }(5,50,a)+\sum _{a=10}^{49}f^{\lambda }(3,50,a) +f^{\lambda }(4,50,50). \end{aligned}$$

Table 1 Volumes values $J_n$ and corresponding jump times values $T_n$ considering $n=3$

Based on previous applications to high-frequency financial data (e.g., D’Amico et al. (2020), D’Amico and Petroni (2018), D’Amico and Petroni (2021)), we employ an exponentially weighted moving average of the squared J as a function f. Formally, we assume the following functional form:

$$\begin{aligned} f^{\lambda }(J_{n-1-k},T_n,a)=\frac{\lambda ^{T_{n}-a} J_{n-1-k}^{2}}{\sum _{k=0}^{n-1}\sum _{b=T_{n-1-k}}^{T_{n-k}-1}\lambda ^{T_{n}-b}} =\frac{\lambda ^{T_{n-a}}J_{n-1-k}^2}{\sum _{b=1}^{T_n}\lambda ^{b}}. \end{aligned}$$

(13)

This depends on the past values of volumes $J_{n-1-k}$ that occurred at time a, the current time $T_n$, and the parameter ${\lambda }$, which balances past information.

In the next section, we describe the calibration of the parameter $\lambda$ and determining the optimal number of states for the financial volumes, as well as the index process.

To construct the WISMC model, we explicitly define the dependency structure between the random variables $J_n$, $T_n$, and $U_n^{\lambda }$. To this end, we adopt the following assumption:

$$\begin{aligned} \begin{aligned} {\mathbb {P}}[J_{n+1}&=j,\, T_{n+1}-T_{n}\le t \mid \sigma (J_h,T_h,U_h^{\lambda })_{h=0}^{n},J_n=i, U_n^{\lambda }=u]\\&={\mathbb {P}}[J_{n+1}=j,\, T_{n+1}-T_{n}\le t \mid J_n=i, U_{n}^{\lambda }=u]=:Q_{ij}^{\lambda }(u,t), \end{aligned} \end{aligned}$$

(14)

where $\sigma (J_{h},T_{h},U_{h}^{\lambda })$ is the natural filtration of the three-variate process $\{J_{n},T_{n},U_{n}^{\lambda }\}$. Relation (14) asserts that knowledge of the last volume value ($J_n = i$) and of the value of the index process ($U_n^{\lambda }=u$) suffices to give the conditional distribution of the couple $(J_{n+1}, T_{n+1} - T_{n})$.

In this mathematical model, the matrix of functions ${{\textbf {Q}}}^{\lambda }(u,t)=(Q_{ij}^{\lambda }(u,t))_{i,j\in E}$, known as the weighted-indexed semi-Markov kernel, is a crucial component. If ${{\textbf {Q}}}^{\lambda }(u,t)$ is constant in u, then the weighted-indexed semi-Markov kernel degenerates into a regular semi-Markov kernel.

We denote by ${{\textbf {P}}}^{\lambda }(u)=({p_{ij}^{\lambda }(u)})_{i,j \in E}$ the transition probabilities of the embedded indexed Markov chain $J_n$, where

$$\begin{aligned} p_{ij}^{\lambda }(u)={\mathbb {P}}[J_{n+1}=j\, \mid J_n=i, U_n^{\lambda }=u]. \end{aligned}$$

We assume that ${{\textbf {G}}}^{\lambda }(u,t)=(G_{ij}^{\lambda }(u,t))_{ij\in E}$ are the conditional waiting time distribution functions, where

$$\begin{aligned} G_{ij}^{\lambda }(u,t)={\mathbb {P}}[T_{n+1}-T_{n} \le t \mid J_n=i, J_{n+1}=j, U_n^{\lambda }=u]. \end{aligned}$$

Then, equation (14) is equivalent to

$$\begin{aligned} \begin{aligned} {Q}_{ij}^{\lambda }(u,t)={\mathbb {P}}[J_{n+1}&=j\, \mid J_n=i, U_n^{\lambda }=u]\cdot {\mathbb {P}}[T_{n+1}-T_{n} \le t \mid J_n=i, J_{n+1}=j, U_n^{\lambda }=u]\\&=p_{ij}^{\lambda }(u) \cdot G_{ij}^{\lambda }(u,t). \end{aligned} \end{aligned}$$

Accordingly, the evolution of the system can be summarized as follows: given a certain state i and a value u of the index process, the transition in the state j is determined using the probability $p_{ij}^{\lambda }(u)$, and the permanence of the system in the state i before moving to state j is determined by using the waiting time distribution function $G_{ij}^{\lambda }(u,t)$.

The triple of processes {$J_n,T_n,U_n^\lambda$} describes the system’s behavior for the transition time $T_n$. To characterize the behavior of our model at any time t, which can be a transition time or a waiting time, we must specify additional stochastic processes.

Let $N(t)=\sup \{ n \in {\mathbb {N}}: T_n \le t\}$ be the number of transitions up to time t, and let $Z(t)=J_{N(t)}$ be the state of the system at time t.

We refer to Z(t) as a weighted indexed semi-Markov process.

For a thorough explanation of the estimation procedures for each quantity included in this model, see D’Amico and Petroni (2018).

Application results

The content of this section is divided into two sub-sections. The first sub-section provides a summary of the key statistical characteristics of the financial data set used in our study, followed by details about the application of the WISMC model. The second sub-section discusses the analysis of risk measures, developed using both actual and simulated data.

Data analysis and preliminary model results

Table 2 Mean, median, standard deviation, skewness, and kurtosis of Tesla, Netflix, and Apple volumes

The study was conducted using high-frequency trading volumes of three Nasdaq-listed stocks, specifically, Tesla, Netflix, and Apple, which were recorded on a minute-by-minute basis. These corporations were chosen because they represent different industrial sectors. For instance, Tesla is primarily involved in the Automotive and Renewable sector, while Netflix and Apple are part of the Communication Services, Technology & Entertainment, and Information Technology sectors, respectively. The data were sourced from Thomson Reuters EikonTM (https://eikon.thomsonreuters.com/index.html) and covers the period April 2022–September 2022. For each corporation, 49,266 min-by-minute trading volumes were analyzed, corresponding to 126 trading days, with each day comprising 391 min. Table 2 provides the main descriptive statistics of the trading volumes for Tesla, Netflix, and Apple. The results in the table show that Netflix stock seems to have the smallest mean, median, and standard deviation values, but also the largest values of kurtosis and skewness.

Table 3 Jarque-Bera test results performed on Tesla, Netflix, and Apple log-volumes considering both $\alpha =0.01$ and $\alpha =0.05$

In Table 3, using a Jarque–Bera test, we rule out the Gaussian hypothesis for the volumes X(t) at the $1\%$ and $5\%$ significance levels for every asset. The results reject the Gaussian hypothesis for all the considered assets.

Table 4 Contingency table to test the dependence between X(t) and T for Tesla

Table 5 Contingency table to test the dependence between X(t) and T for Netflix

Table 6 Contingency table to test the dependence between X(t) and T for Apple

Finally, in Tables 4, 5, and 6, we create contingency tables by grouping volumes in five intervals, each containing about $20\%$ of observations. Then, we apply the $\chi ^{2}$ test to check whether the volumes and the waiting times rely on one another. The results show that the independence hypothesis is uniformly rejected by all of the stocks. Specifically, the statistical test scores for Tesla, Netflix, and Apple are 1.4892e+03, 1.1630e+03, and 1.3283e+03, respectively, and 20.0902 is the critical value of the $\chi ^{2}$ statistics with eight degrees of freedom and $\alpha =0.01$. This is reflected in the low p-values (****).

These arguments highlight the necessity for all assets to adopt a model where the intervals between transitions and volumes are interdependent, and not based on the Gaussian distribution. This necessity prompted us to replicate volumes using a WISMC process.

To properly realize the investigation, we judged real data above the 95th percentile to be outliers, and replaced them with the semi-sum of the nearest data.

Table 7 Estimated coefficient values of the fourth-degree polynomial and relative 95% Confidence intervals (95% CI) for Tesla, Netflix, and Apple assets

In addition, all our high-frequency volumes display a daily pattern, increasing at the opening of the market, decreasing throughout the trading day, and then increasing again at market close. By taking the average for each trading day on a minute-by-minute basis, we establish the daily trend values. To accurately simulate financial volumes using a WISMC model, we removed all trends from the financial time series by modeling these trends with a fourth-degree polynomial. Table 7 shows the estimated coefficient values of the fourth-degree polynomial, along with their respective 95% confidence intervals, for all stocks. The Adjusted-$R^2$ values of the polynomial regressions are 0.9801, 0.9580, and 0.9677 for Tesla, Netflix, and Apple, respectively.

The WISMC estimations are conducted following the methodology outlined in D’Amico and Petroni (2018). Specifically, to shape volumes using a WISMC process, it is necessary to establish the number of states for the volume process and the parameter $\lambda$ for each asset. These two quantities are calibrated by minimizing the mean percentage error between actual and synthetic autocorrelation functions, in line with the algorithm detailed in D’Amico and Petroni (2018). In essence, the algorithm aims to create a trajectory by arbitrarily initiating a set of states and a $\lambda$ value, estimate the weighted-indexed semi-Markov kernel using the approximate nonparametric maximum likelihood estimator derived in D’Amico et al. (2013), and subsequently conduct a Monte Carlo simulation to generate a synthetic series. The autocorrelation functions for both actual and simulated data are then calculated, and the mean percentage error values are compared. This process is repeated with varying state values and $\lambda$. Ultimately, the number of states and the $\lambda$ value that most accurately represent the data are determined by minimizing the mean percentage error.

Using this optimization procedure, we find that for all stocks, the ideal number of states is five and the optimal value of $\lambda$ is 0.93. However, note that in previous financial applications, based on the financial returns of other assets, even higher $\lambda$ values have been observed (see D’Amico and Petroni (2021)).

Specifically, we discretize financial volumes in five states not symmetrical to each other. Denoting as L the detrended financial volume series, we discretize them using the following grid:

$$\begin{aligned} Mean + Std \cdot [-1, -0.5, 0.5, 1], \end{aligned}$$

where Mean and Std are the mean and standard deviation values, respectively, of the L financial time series. The grid detects the following five intervals: $(-\infty , Mean-Std]$, $(Mean-Std, Mean-0.5\cdot Std]$, $(Mean-0.5\cdot Std, Mean+0.5\cdot Std]$, $(Mean+0.5\cdot Std, Mean+Std]$, and $(Mean+Std, +\infty )$. The mean and standard deviation values for the detrended volume series for Tesla, Netflix, and Apple are shown in Table 8.

Table 8 Mean and standard deviation values for the detrended series of Tesla, Netflix, and Apple

We also discretized the index process into five states: low, medium-low, medium, medium-high, and high volume levels (Fig. 3).

In Figs. 4, 5, and 6, we show the probability plots for both the detrended volumes states and index states for Tesla, Netflix, and Apple, respectively.

Results on drawdown-based risk measures

First, we compute the drawdown-based risk measures using real data. Following this, we examine the hypothesis that different distributions exist as s changes on real risk measures. We do this by employing the goodness-of-fit test suggested in Song (2002). This test, which is grounded in relative entropy or Kullback–Leibler divergence (see Kullback and Leibler (1951)), ascertains whether the distributions of two data sets are statistically identical or not. Formally, the test assumes the following form:

$$\begin{aligned} \left\{ \begin{array}{rl} H_0: D(P, Q)=0,\\ H_1: D(P, Q)>0. \end{array} \right. \end{aligned}$$

(15)

Here, P and Q are two probability distributions, and D(P, Q) indicates their Kullback–Leibler divergence. We define the Kullback–Leibler divergence in its continuous version, denoted also by $D(P \Vert Q)$, as follows:

$$\begin{aligned} D(P \Vert Q)= \int _{-\infty }^{+\infty } p(x) log_{2} \bigg (\frac{p(x)}{q(x)} \bigg )dx, \end{aligned}$$

(16)

where p and q are the probability densities of P and Q. This expresses the measure of the information lost when the distribution Q is employed to approximate the distribution P.

Accepting the null hypothesis means stating that we have identical probability distributions, regardless of the starting time s. Conversely, rejecting the null hypothesis implies that we have different probability distributions, both functions of the selected time s.

Practically, we reject $H_0$ if $Z_0>z_\alpha$, where $Z_0=\sqrt{n} \cdot {\hat{D}}(Q,P) /{{\hat{\sigma }}}$, and $z_\alpha$ is the $1-\alpha$ quantile of the standard normal distribution. For a detailed discussion, see Song (2002); Koukoumis and Karagrigoriou (2021).

Table 9 Song test results for the risk measure $\tau ^{s}$ computed on Tesla real data, considering $M=30\%$, $M=40\%$, $M=80\%$. $Z_0$ is the statistic test value under the null hypothesis

Table 10 Song test results for the risk measure $\tau ^{s}$ computed on Netflix real data, considering $M=30\%$, $M=40\%$, $M=80\%$. $Z_0$ is the statistic test value under the null hypothesis

Table 11 Song test results for the risk measure $\tau ^{s}$ computed on Apple real data, considering $M=30\%$, $M=40\%$, $M=80\%$. $Z_0$ is the statistic test value under the null hypothesis

Table 12 Song test results for the risk measure $T_c^{s}$ computed on Tesla real data, considering $M=30\%$, $M=40\%$, $M=80\%$. $Z_0$ is the statistic test value under the null hypothesis

Table 13 Song test results for the risk measure $T_c^{s}$ computed on Netflix real data, considering $M=30\%$, $M=40\%$, $M=80\%$. $Z_0$ is the statistic test value under the null hypothesis

Table 14 Song test results for the risk measure $T_c^{s}$ computed on Apple real data, considering $M=30\%$, $M=40\%$, $M=80\%$. $Z_0$ is the statistic test value under the null hypothesis

Table 15 Song test results for the risk measure $R_t^{s}$ computed on Tesla real data, fixing $M=80\%$ and considering $M'=30\%$, $M'=40\%$, $M'=50\%$. $Z_0$ is the statistic test value under the null hypothesis

Table 16 Song test results for the risk measure $R_t^{s}$ computed on Netflix real data, fixing $M=80\%$ and considering $M'=30\%$, $M'=40\%$, $M'=50\%$. $Z_0$ is the statistic test value under the null hypothesis

Table 17 Song test results for the risk measure $R_t^{s}$ computed on Apple real data, fixing $M=80\%$ and considering $M'=30\%$, $M'=40\%$, $M'=50\%$. $Z_0$ is the statistic test value under the null hypothesis

Tables 9, 10, 11, 12, 13, 14, 15, 16, 17 show the test results for the measures $\tau ^{s}$, $T_c^{s}$, and $R_t^{s}$, respectively. Based on the results of the statistical tests, the null hypothesis is nearly always rejected for all securities and risk measures. The significance of this rejection intensifies as the selected s value increases. However, the assumption of distributive equality is occasionally accepted for relatively modest s values, such as $s=3$ and $s=5$. Consequently, the chosen time s, representing the point at which an investor takes action, modifies our financial indicators. This causes them to diverge from one another, exposing the investor to varying risks.

Table 18 Descriptive statistics of $\tau ^{s}$ (first quartile, second quartile (median), third quartile, mean, standard deviation, asymmetry index) and related censored unit as a function of the threshold M

Table 19 Descriptive statistics of $\tau ^{s}$ (First Quartile, Second Quartile (Median), Third Quartile, Mean, Standard Deviation, Asymmetry Index) and related censored units as a function of the threshold M. All results refer to Netflix real data

Table 20 Descriptive statistics of $\tau ^{s}$ (First Quartile, Second Quartile (Median), Third Quartile, Mean, Standard Deviation, Asymmetry Index) and related censored units as a function of the threshold M. All results refer to Apple real data

Tables 18, 19, 20 show the main descriptive statistics and the number of censored units for the risk measure $\tau ^{s}$ for real data on Tesla, Netflix, and Apple, respectively. The mean and standard deviation estimates are obtained using the Kaplan–Meier estimator (Kaplan and Meier 1958) in order to treat the censored units. Recall that the risk measures $\tau ^{s}$, $T_c^{s}$, and $R_t^{s}$ are right-censored, owing to the choice of a fixed observation time, namely, the trading day consisting of 391 min. For an extensive description of the Type-1 right censorship issue and the drawdown-based measures used in this study, see D’Amico et al. (2020).

In terms of the mean value, for each s, all stocks display a comparable trend, with only a few deviations, implying that more severe events occur at a slower pace, and hence, it requires more time for an asset to become highly illiquid. For instance, in the case of Tesla, when $s=50$, a drawdown variation of 30% is typically reached just before the second minute. However, to witness a significant shift, such as 80%, we need to wait longer, approximately 14 min, on average.

For all three assets, we simulate daily volumes using the WISMC model, and then compute the analyzed risk indicators for these synthetic series.

Given the real and simulated measures, we can estimate their best parametric distribution among the lognormal, Weibull, exponential, and gamma laws by using the AIC and BIC, according to D’Amico et al. (2020). For the measure $T_c^{s}$, we choose the model with the smallest AIC and BIC values, fixing s and varying M. We repeat the same steps for the measure $R_t^{s}$, setting s and M, while varying $M'$. All estimation procedures involving parametric models are performed in Matlab software, considering the right censoring issue.

Table 21 Selection of the best parametric model as a function of s and M for the measure $T_c^{s}$ computed on Tesla real data

Table 22 Selection of the best parametric model as a function of s and M for the measure $T_c^{s}$ computed on Netflix real data

Table 23 Selection of the best parametric model as a function of s and M for the measure $T_c^{s}$ computed on Apple real data

Table 24 Selection of the best parametric model as a function of s and M for the measure $T_c^{s}$ computed on Tesla simulated data

Table 25 Selection of the best parametric model as a function of s and M for the measure $T_c^{s}$ computed on Netflix simulated data

Table 26 Selection of the best parametric model as a function of s and M for the measure $T_c^{s}$ computed on Apple simulated data

Table 27 Selection of the best parametric model as a function of s, M and $M'$ for the measure $R_t^{s}$ computed on Tesla real data

Table 28 Selection of the best parametric model as a function of s, M and $M'$ for the measure $R_t^{s}$ computed on Netflix real data

Table 29 Selection of the best parametric model as a function of s, M and $M'$ for the measure $R_t^{s}$ computed on Apple real data

Table 30 Selection of the best parametric model as a function of s, M and $M'$ for the measure $R_t^{s}$ computed on Tesla simulated data

Table 31 Selection of the best parametric model as a function of s, M and $M'$ for the measure $R_t^{s}$ computed on Netflix simulated data

Table 32 Selection of the best parametric model as a function of s, M and $M'$ for the measure $R_t^{s}$ computed on APPLE simulated data

Tables 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32 show the parametric model selections for the measures $T_c^{s}$ and $R_t^{s}$, respectively. The smallest AIC and BIC values are shown in bold. For both real and simulated data, the lognormal law is invariably the best statistical parametric model. This result underlines the fact that the WISMC model exhibits steady outcomes, consistent with real data, and in line with the findings of D’Amico et al. (2020) in terms of financial returns.

Table 33 Parameters of the best parametric models for $T_c^{s}$ computed on real and simulated data for Tesla

Table 34 Parameters of the best parametric models for $T_c^{s}$ computed on real and simulated data for Netflix

Table 35 Parameters of the best parametric models for $T_c^{s}$ computed on real and simulated data for Apple

Table 36 Parameters of the best models for $R_t^{s}$ computed on real and simulated data for Tesla

Table 37 Parameters of the best models for $R_t^{s}$ computed on real and simulated data for Netflix

Table 38 Parameters of the best models for $R_t^{s}$ computed on real and simulated data for Apple

Accordingly, Tables 33, 34, 35, 36, 37, 38 show the parameter point estimates of the best-selected models for the time to crash $T_c^{s}$ and the recovery time $R_t^{s}$, respectively.

We can obtain the best parametric distributions of $S_c^{s}$ and $S_r^{s}$ from the best $T_c^s$ and $R_t^s$ parametric distributions, respectively, because $S_c^{s}$ and $S_r^{s}$ are their nonlinear transformations. All mathematical steps used to derive the two speeds’ best parametric distributions from the best $T_c^s$ and $R_t^s$ are reported in D’Amico et al. (2020).

Table 39 Descriptive statistics of $T_c^{s}$ (first quartile, second quartile (median), third quartile, mean, standard deviation, asymmetry index) and related censored units as a function of M and s

Table 40 Descriptive statistics of $T_c^{s}$ (first quartile, second quartile (median), third quartile, mean, standard deviation, asymmetry index) and related censored units as a function of M and s

Table 41 Descriptive statistics of $T_c^{s}$ (first quartile, second quartile (median), third quartile, mean, standard deviation, asymmetry index) and related censored units as a function of M and s

In Tables 39, 40, 41, we present both the main descriptive statistics and the number of censored units for the time to crash $T_c^{s}$ for Tesla, Netflix, and Apple stocks. The computation is carried out on real data, using the best statistical parametric model as a function of the threshold M. For every s, we observe that the average values and the standard deviation values increase with M. For Tesla and Apple, there are no censored units, but Netflix shows slight censorship for every s. For context, considering Tesla at $s=50$, a 30% change in its drawdown, occurs in less than two minutes, on average, while an 80% variation is reached in about three minutes, on average. In other words, it takes Tesla much longer to achieve a more illiquid position, that is, an $80\%$-drawdown variation, than to achieve a more liquid one, that is, a $30\%$-drawdown variation.

Table 42 Descriptive statistics of $S_c^{s}$ (First Quartile, Second Quartile (Median), Third Quartile, Mean, Standard Deviation, Asymmetry Index) and related censored units as a function of M and s

Table 43 Descriptive statistics of $S_c^{s}$ (First Quartile, Second Quartile (Median), Third Quartile, Mean, Standard Deviation, Asymmetry Index) and related censored units as a function of M and s

Table 44 Descriptive statistics of $S_c^{s}$ (First Quartile, Second Quartile (Median), Third Quartile, Mean, Standard Deviation, Asymmetry Index) and related censored units as a function of M and s

Given that we know the cumulative distribution function and the probability distribution function for the time to crash in the most effective statistical parametric models, we can convert these results to the speed of crash, which is a nonlinear transformation of the original data. The mathematical calculations needed to derive the cumulative distribution function and probability distribution function for the speed of crash from the time to crash data are reported in D’Amico et al. (2020). Tables 42, 43, 44 show the main descriptive statistics and the censoring rate for $S_c^{s}$. Note that when M rises, the average speed drops. This means that all stocks approach higher thresholds more slowly than they do lower ones, gradually transitioning from a liquid to an illiquid state. Using Tesla as an example, when $s=50$, a 30% change in its drawdown is reached with an average velocity of 0.1232 $min^{-1}$, while a bigger variation, such as 80%, is attained more slowly, with an average speed of 0.2883 $min^{-1}$.

Table 45 Descriptive statistics of $R_t^{s}$ (First Quartile, Second Quartile (Median), Third Quartile, Mean, Standard Deviation, Asymmetry Index) and related censored units as a function of s, M and $M'$

Table 46 Descriptive statistics of $R_t^{s}$ (First Quartile, Second Quartile (Median), Third Quartile, Mean, Standard Deviation, Asymmetry Index) and related censored units as a function of s, M and $M'$

Table 47 Descriptive statistics of $R_t^{s}$ (First Quartile, Second Quartile (Median), Third Quartile, Mean, Standard Deviation, Asymmetry Index) and related censored units as a function of s, M and $M'$

We display the main descriptive statistics and the censoring rate for the measure $R_t^{s}$ in Tables 45, 46, 47. We observe that fixing $M'$, for each s, if M increases, then the mean value, the standard deviation value, and the censoring rate decrease. In particular, the behavior of the average values of $R_t^s$ emphasizes that after experiencing an $80\%$-liquidity drop (i.e., $M=80\%$), all the considered assets need fewer minutes, on average, to recover a more liquid position (i.e., high $M'$ values, and thus small $M-M'$) than they for an illiquid position (i.e., small $M'$ values, and thus high $M-M'$).

Table 48 Descriptive statistics of $S_r^{s}$ (First Quartile, Second Quartile (Median), Third Quartile, Mean, Standard Deviation, Asymmetry Index) and related censored units as a function of s, M and $M'$

Table 49 Descriptive statistics of $S_r^{s}$ (First Quartile, Second Quartile (Median), Third Quartile, Mean, Standard Deviation, Asymmetry Index) and related censored units as a function of s, M and $M'$

Table 50 Descriptive statistics of $S_r^{s}$ (First Quartile, Second Quartile (Median), Third Quartile, Mean, Standard Deviation, Asymmetry Index) and related censored units as a function of s, M and $M'$

Once the speed of recovery $S_r^{s}$ is a nonlinear transformation of the recovery time $R_t^{s}$, we derive its cumulative distribution function and probability distribution function from those of $R_t^{s}$, following (D’Amico et al. 2020). Tables 48, 49, 50 report the descriptive statistics and the censoring rate for the measure $S_r^{s}$, which denotes the velocity at which a stock goes from an illiquid state to a more liquid state. Note that for Netflix, when fixing s and $M'$ and increasing M, the mean value, the standard deviation, and the censorship rate decrease. Apple shows the same behavior, except for its mean values when $s=5$. Finally, Tesla always follows this pattern for the standard deviation values, but not for its average values when $s=5$ and $s=50$.

To measure the distance between real and simulated measures, we compute the Kullback–Leibler divergence (Kullback and Leibler 1951) for $T_c^{s}$ and $R_t^{s}$.

Table 51 Kullback–Leibler divergence computed on the risk measure $T_c^{s}$ as a function of M and s for Tesla, Netflix and Apple

Table 52 Kullback–Leibler divergence computed on the risk measure $R_t^{s}$ as a function of M, $M'$ and s for Tesla, Netflix and Apple

Table 51 shows the Kullback–Leibler divergences for the time to crash $T_c^{s}$ while, in Table 52, we present finding values of the recovery time $R_t^{s}$. The shortest distances for each s are shown in bold.

Because the Kullback–Leibler divergence is invariant with regard to parameter transformations, its computation is not needed for $S_c^{s}$ and $S_r^{s}$ (see D’Amico et al. (2020)).

For $T_c^{s}$, note that between the smallest distances highlighted in bold, the greatest distance always occurs for $s=0$ for each security. Conversely, for $R_t^{s}$, the greatest distance between the smaller ones varies for $s=5$ for both Tesla and Apple, but for Netflix, this occurs when $s=0$.

To apply our findings, we conduct a simple practical example, using Tesla stock. First, we set $s=5$, followed by $s=50$, and then assess our risk measures using their parametric distributions. Concerning the time to crash $T_c^{s}$, in the first scenario, when $s=5$, there is approximately an 81

Now, we extend the example to the recovery time $R_t^{s}$ and speed of recovery $S_r^{s}$, always using $s=5$ and $s=50$, and setting $M=80\%$ and $M'=30\%$. For $R _t^{s}$, at $s=5$, the probability of having a drawdown change of M intensity and, immediately after, a new variation equal to $M'$ within the first five minutes of the trading day is 43%. Furthermore, the probability that these drawdown variations occur with a speed $S_r^{s}$ less than 0.5 $min^{-1}$ is equal to 89.6%. For the second scenario, when $s=50$, the likelihood of first experiencing a M drawdown variation and then a $M'$ change during the first five minutes is equal to 34%. In addition, these changes will happen with a speed less than 0.5 min$^{-1}$, with a probability equal to 93.4%.

Fig. 3

Cumulative distribution functions (CDFs) on the first 20 min of the trading day for $T_c^s$, $S_c^s$, $R_t^s$, and $S_r^s$. All the blue lines represent the CDFs for scenario $s=0$ while, the red lines are the CDFs for scenario $s=100$

Fig. 4

Probability plots for both the detrended volume states and index states for Tesla asset

Fig. 5

Probability plots for both the detrended volume states and index states for Netflix asset

Fig. 6

Probability plots for both the detrended volume states and index states for Apple asset

Figure 3 shows the results of the example just stated for $s=5$ and $s=50$. The first part of the graph shows the cumulative probability distributions for the first 20 min of the trading day for the time to crash and the speed of crash, considering the $M=30\%$ threshold. Considering this, the cumulative probability distributions of the time to crash and the speed of crash provide information on the likelihood of crossing the $30\%$ threshold, and how rapidly this occurs, minute by minute. Specifically, the second portion of the graph shows the cumulative probability distributions for both the recovery time and the speed of recovery during the initial 20 min of the trading day, with $M=80\%$ and $M'=30\%$. Consequently, these cumulative probability distributions provide insights into the likelihood of surpassing the $30\%$ threshold after surpassing the $80\%$ threshold, as well as the rate at which this happens per minute.

Conclusion

In this study, we examine various risk indicators related to market crises for high-frequency financial volumes of assets listed on the Nasdaq Stock Exchange, such as Tesla, Netflix, and Apple. We introduce a commencement time s to calculate the drawdown-based risk indicators in daily intervals, thereby preventing our risk indicators from being affected by the initial large volume of transactions. We generate artificial time series of volumes using the WISMC model, a variant of the conventional semi-Markov chain model. Next, we compute all drawdown-based risk measures on both actual and simulated data, and investigate them using parametric models. The estimation procedures for these models consider the right censorship. Lastly, we measure the distance between real and simulated risk measures using the Kullback–Leibler divergence. Overall, our findings are financially significant, because they offer insight into an asset”s liquidity risk, and therefore, can be used for financial investments.

Our long-term objective is to jointly examine volumes and returns using the multivariate WISMC model described in D’Amico and Petroni (2021). In order to confirm the efficacy of our model, we also wish to compare the outcomes of the WISMC model with those of inflated econometric models, such as the autoregressive moving average and generalized autoregressive conditional heteroskedasticity models.