Equity Price Dynamics under Shocks: In Distress or Short Squeeze

Abstract

This paper proposes a simple bounded stochastic motion to model equity price dynamics under shocks. The stochastic process has a quasi-bounded boundary which can be breached if the probability leakage condition is met. The quasi-boundedness of the process at the boundary can thus provide an indicator of the possible risk of equities under price shocks or in distress. Empirical calibration of the model parameters of the proposed process for equities can be performed easily due to the availability of an analytically tractable probability density function which generates fat-tailed distributions consistent with empirical observations. The volatility and mean-reversion of the S&P500 dynamics calibrated by the process are positively and negatively co-integrated, respectively, with the VIX index representing the level of market distress. The process captures the high likelihood of Hertz’s default about two months earlier, using only information until that point, and before the firm filed for Chapter 11 bankruptcy in May 2020 as a result of the COVID-19 pandemic. Empirical calibration of the process for GameStop’s stock price shows that the short squeeze in the stock occurred when the condition for breaching the upper boundary was met on 14 January 2021, i.e., about two weeks before major short-sellers closed out their positions with significant losses. The trading volume of the stock was positively co-integrated with the probability leakage ratio.

1. Introduction

Black (1986) argues that an efficient market is one in which price is in the range of a factor of 2, i.e., the price is more than half of the value and less than twice the value as a result of uninform supply and demand flows. He further argues that the factor of 2 is arbitrary, but reasonable, in light of sources of uncertainty about value and the strength of the forces tending to cause prices to return to value. Therefore, prices are within an uncertainty band in which their dynamics are not random, but exhibit trends (see de Bondt and Thaler 1985; DeLong et al. 1990; Daniel et al. 1998; Hong and Stein 1999; Asness et al. 2013; Lempérière et al. 2014; Bouchaud et al. 2019). When prices move towards the boundaries of the band, mean-reversion forces drive prices back to more mean levels. Studies including Fama and French (1988), Day and Huang (1990), Lux and Marchesi (2000), Madan (2017), and Bouchaud et al. (2018) find empirical evidence that in the dynamics of various asset classes, such as stock indexes, commodities, and currencies, mean-reversion represents a self-correcting mechanism. Nartea et al. (2021) investigate the stationarity of the daily stock prices in 12 Asia-Pacific markets during 1991–2020, and find that the stock prices became more mean-reverting and had a faster speed of adjustment towards the mean level after the global financial crisis for all sample markets. For the purposes of pricing derivatives and simulations of future prices, each stochastic process of a given asset class has to be properly modelled to resemble its empirical statistical properties. In view of studies of the existence of uncertainty bands, price trends, and mean-reverting forces, a bounded stochastic process is needed to model the constrained stochastic motion of financial asset prices.

In this paper, we propose a relatively simple stochastic model for equity prices which are subject to either positive or negative shocks. The prices are constrained to lie between two positive bounds representing an uncertainty band. Generally, these two bounds are time-varying. Driven by forecasting with time series decomposition of equity prices, we detrend the time series of the equity prices, and the model simply concentrates on their fluctuations. The lower boundary in the proposed stochastic process is quasi-bounded, implying that the lower boundary can be breached if the probability leakage condition is met. Such a property is studied by Pesz (2002), Silva et al. (2006), and Cardeal et al. (2007). Similarly, completely bounded exchange rate dynamics is studied by Ingersoll (1997) and Larsen and Sørensen (2007).

Given that the well-known Cox–Ingersoll–Ross (CIR) (Cox et al. 1985) process allows the underlying variable to be quasi-bounded at the zero bound and is a mean-reverting process, we select CIR to represent the quasi-bounded process. The CIR model is popular among practitioners and is used for pricing various asset classes, such as equities, fixed income, commodities, foreign exchanges, and their derivatives, as in Carr et al. (2020), and modelling the variance of asset prices, as in Heston (1993). Lo et al. (2015) and Hui et al. (2016) find that the quasi-bounded CIR process is able to describe the exchange rate dynamics in a target zone of the HKD against the USD, and the CHF against the EUR, during September 2011 to January 2015, respectively. Hui et al. (2022) apply the quasi-bounded target-zone model to model yield curve control for Japanese government bonds.

The calibration of model parameters for equity prices can be performed easily, given that an analytically tractable probability density function can be derived from the CIR process. In terms of the calibrated model parameters, assessing the likelihood of future movements of equity prices becomes feasible, in particular, the likelihood of whether or not the stochastic process is still bounded. The quasi-boundedness of the process at the lower boundary can therefore provide an indicator of a possible downside risk of the corresponding equities in distress. Understanding the behaviour of distress equities has proved something of a challenge. The empirical literature documenting the distress anomaly is quite extensive, for example, Elkamhi et al. (2012), Hackbarth et al. (2015), Gao et al. (2017), and Boualam et al. (2020). Several credit risk-based theories have been developed in response to this evidence, including Conrad et al. (2014), Eisdorfer et al. (2019), McQuade (2018), and Opp (2021). The proposed quasi-bounded-process approach to studying distress equity price dynamics is different from the credit risk-based approach.

We examine whether the dynamics of the S&P500 Index (S&P) subject to market distress risk can be characterised by the quasi-bounded process. In addition, given that Hertz filed for bankruptcy (Chapter 11) due to the impact of the COVID-19 pandemic on the car-rental industry, we investigate the relationship between the intensity of the pandemic’s spread, which triggered the firm’s financial distress, and its equity price dynamics derived from the model. The Chapter 11 bankruptcy allowed Hertz to stay in business and restructure its obligations. Hertz was under a reorganisation plan in the best interest of the creditors. The corresponding model parameters are related to the level of market distress in terms of the VIX index and the number of COVID-19 cases, respectively, showing the validity of the model. The VIX index is a measure of constant, 30-day expected volatility derived from real-time, mid-quote prices of S&P500 Index call and put options. It is one of the most recognised measures of volatility and also captures the information of market stress conditions embedded in option prices.

Retail trading activity in the US stock market has surged since late 2019, when some brokerages cut commissions to almost zero. Its impact on the market is demonstrated by the price action of GameStop (GME) as a result of co-ordinated buying through forums in January 2021, which put the low-profile GME on the front pages of financial news organisations. Interest in GME emerged from individual investors on social media sites after they discovered the extreme short interest in the stock (139% of free float at the peak). These investors acted as a group to build buying pressure on a short squeeze, leading GME stocks to rise 57% to USD 31.4 on 13 January 2021, followed by another 27% jump the next day. On 27 January, the share price surged to USD 347), with a median target price among analysts of only USD 12.50. On the same day, some of the most prolific hedge funds lost the majority of their GME short positions at a substantial loss (see https://www.wsj.com/articles/melvin-capital-lost-53-in-january-hurt-by-gamestop-and-other-bets-11612103117 (accessed on 31 January 2021) and the timeline at https://www.reuters.com/article/us-gamestop-hot-timeline-idUSKBN29W237 (accessed on 28 January 2021)).

Traders have a limited capacity, meaning that they cannot sustain losses beyond a certain point, as illustrated by Brunnermeier and Pedersen (2005), particularly for funds which have redemption obligations for investors. In addition, their trading limits, measured through value-at-risk, act as constraints on keeping their positions. When stock prices shorted (longed) by traders surge (fall) beyond certain thresholds, the positions must be closed out due to those constraints. Leverage constraints imposed by short-term creditors can also force a financial institution to liquidate long-term investments at fire-sale prices. Brunnermeier and Oehmke (2014) argue that such financial institutions may be vulnerable to predatory short-selling. Brunnermeier et al. (2009) show that the illiquidity in the market suggests more scope for predators to make a profit as those traders need more time to close out their positions with larger movements in the prices. Stein (2009) demonstrates that another destabilising impact is crowding. Boehmer et al. (2008) and Diether et al. (2009) point out that a high short-sale volume of a stock will attract positive-feedback on short-sale trading.

The mean-reversion in the CIR process represents the market actions between long-buyers and short-sellers, which form an error-correction process in the price dynamics. The process allows the equity price to be quasi-bounded at an upper bound, which is the threshold imposed by constraints such as stop-loss limits forcing traders to close out short positions squeezed by co-ordinated buying. Their scramble to buy adds to the upward pressure on the stock’s price. A surge in the equity price breaching the bound triggered by the short squeeze is conditional on the probability leakage ratio of the CIR process.

This paper is organised as follows. We present the model of the quasi-bounded process in the following Section. The calibrations of the equity price dynamics under market distress and short squeezes, respectively, are presented in Section 3. The relationships between the equity dynamics and market conditions are studied empirically and discussed in Section 4. The Section 5 concludes the paper.

2. Equity Price Dynamics

2.1. Modelling Constrained Stochastic Motion

A financial observable (e.g., stock) S is constrained to move within a band ${[S}_L,S_U]$ for $0{\le S}_L<S_U$. Both $S_L$ and $S_U$ are time-varying. The moving average $\overline{S}$ provides a trend component of the time series of S, as shown in Ritschel et al. (2021) and Vinod et al. (2022). The time series of $S-\overline{S}$ is the detrended series assuming the additive decomposition of a time series. Based on qualifying the lower boundary of the financial observable in distress, historical data of the financial observable can be used to set a trading band from which the financial observable is not expected to breach. The historical trend of the financial observable is measured based on the $\overline{S}$ of its current and past values. For a financial observable in distress, the moving average can be scaled by a non-negative parameter, $0<{\eta }_L<1$, such that ${S_L\equiv \eta }_L\overline{S}$ forms a lower boundary for the movement of the financial observable. The parameter ${\eta }_L$, which determines the fraction of $\overline{S}$, suggests how much market participants expect the maximum downside of holding the financial observable. A smaller ${\eta }_L$ indicates that the market expects wider ranges of fluctuations over a given time horizon. Such specification of a lower boundary captures the behaviour of the financial observable over a period of time, instead of its current value.

The lower boundary S_L is defined as the number (Δ) of standard deviations ($\Sigma$) from its mean $\overline{S}$: $S_L=\overline{S}-\Delta \Sigma$. If a financial observable is assumed to be normally distributed and Δ is set equal to 1.5 and 2, respectively, suggesting that falling to the lower boundary means it is in distress, the corresponding cumulative normal probabilities of the financial observable falling below the boundaries are 0.0668 and 0.0227. This approach was used in a study of currency crashes by Jurek (2014), in which the crash level was set at 0.7 and 1.4 standard deviations, respectively, away from the spot exchange rates. This measure is also similar to value-at-risk (VaR), which is defined as the maximum expected loss of assets or portfolios at a pre-defined confidence level (say 95%) over a given time horizon. In other words, a 5% probability of such extreme loss suggests that the asset price is under stress. As long as the level of the boundary is adequately low, the choice of its level does not affect the process of the financial observable dynamics.

The upper boundary can be defined similarly as ${S_U\equiv \eta }_U\overline{S}$ for a parameter ${\eta }_U>1$. The logarithm normalised fractional deviation x of S from $\overline{S}$ is given by

$$x=-\ln \left[\frac{{\eta }_U\overline{S}-S}{\left({\eta }_U-{\eta }_L\right)\overline{S}}\right],$$

(1)

where $x\in \left[0,\infty \right]$, and x = 0, when ${S=S}_L$. Both ${\eta }_U$ and ${\eta }_L$ are adjustable parameters for the upper and lower boundaries of a band, respectively. It is not necessary for the two time-varying boundaries to be symmetric about the moving average $\overline{S}$. Accordingly, it is clear that the financial observable is confined to a drifting band with time-varying width. Similarly, regarding an upside shock, we can put the origin $x^{\prime }=0$ at an upper boundary by defining

$$x^{\prime }=-\ln \left[\frac{{S-\eta }_L\overline{S}}{\left({\eta }_U-{\eta }_L\right)\overline{S}}\right].$$

(2)

The stochastic process of a financial asset is commonly modelled by the CIR process (see, for example, Heston 1993). The CIR process allows the underlying variable to be quasi-bounded at the zero bound and is a mean-reverting process. Following this approach, we propose that the normalised financial observable x in Equation (1) ($x^{\prime }$ in Equation (2)) follows the CIR process:

$$dx=\kappa \left(\theta -x\right)dt+{\sigma }_x\sqrt{x}dZ,$$

(3)

where dZ is a Wiener process with E[dZ] = 0 and E[dZ²] = dt. Here, ${\sigma }_x\sqrt{x}$ in the second stochastic term depends on the level of x, and k determines the speed of the mean-reverting drift towards the long-term mean $\theta$. According to Feller’s classification of boundary points, the one at the origin is a boundary of no probability leakage under the condition ${(\sigma }_x^2/4\kappa \theta )<1$ (see Karlin and Taylor 1981). In other words, it is a quasi-bounded process, and x could breach the boundary if the no-leakage condition is not met. The infinity is a non-attractive natural (or inaccessible) boundary.

The normalised financial observable x, which is a fractional deviation of S from $\overline{S}$, follows the CIR process by using the normalisation of Equation (1) or (2). The implication is that the normalised financial observable dynamics contain information on the random fluctuation of S. A mean-reverting force in the financial observable dynamics will pull the random variable x away from the lower (upper) boundary towards the mean level. Probability leakage through the quasi-bounded boundary occurs only in rare situations when fluctuations accompanied by a shock in the financial observable surge drastically or the mean-reverting force diminishes sharply. This condition is consistent with the observations in Bates (2012), in which volatility goes up or an error-correction process drops during economic recessions.

2.2. Probability Density Function

The probability density function (PDF) of x under the CIR process is given by:

$$\begin{array}{c}G\left(x,t;x^{\prime },t^{\prime }\right)=\frac{2}{{\sigma }_x^2{C}_1\left(t-t^{\prime }\right)}{\left(\frac{x}{x^{\prime }}\right)}^{\omega /2}\exp \left[-\frac{\omega +2}{2}{C}_2\left(t-t^{\prime }\right)\right]\times \\ \exp \left\{-\frac{2x^{\prime }+2x\exp \left[-C_2\left(t-t^{\prime }\right)\right]}{{\sigma }_x^2{C}_1\left(t-t^{\prime }\right)}\right\}\times \\ I_{\omega }\left\{\frac{4x^{1/2}{x}^{\prime 1/2}\exp \left[-C_2\left(t-t^{\prime }\right)/2\right]}{{\sigma }_x^2{C}_1\left(t-t^{\prime }\right)}\right\}\end{array},$$

(4)

where ${\omega =2\kappa \theta /\sigma }_x^2-1$, $C_1\left(\tau \right)=\left[\mathrm{e}\mathrm{x}\mathrm{p}\left(\kappa \tau \right)-1\right]/\kappa$, and $C_2\left(\tau \right)=-\kappa \tau$, and $I_{\omega }$ is the modified Bessel function of the first kind of order ω. The associated asymptotic PDF will eventually approach the steady-state distribution, which is:

$$K\left(x,t\to \infty ,x^{\prime },t^{\prime }\right)=\frac{2x^{\omega }}{\mathsf{\Gamma }\left(\omega +1\right)}{\left(\frac{2\kappa }{{\sigma }_x^2}\right)}^{\omega +1}\exp (-\frac{2\kappa }{{\sigma }_x^2}x),$$

(5)

where $\mathsf{\Gamma }$ is the gamma function. Given the PDF in Equation (4), the parameters of the CIR process are calibrated by using market data in Section 3.

Based on the calibration of the model parameters using the S&P in Section 3, and setting the lower boundary at ${\eta }_L{S}_{At}=S_L=1900$ and upper boundary at ${\eta }_U{S}_{At}=S_U=3320$, the steady-state distributions in S with three values of the long-term mean θ of 0.3, 1.0, and 1.5 are shown in Figure 1. The smaller θ is closer to the lower boundary with S = 2268, 2797, and 3003, respectively. We use the model parameters for ${\sigma }_x=0.05$ (Panel A), and 0.15 (Panel B), and k = 0.05. The distributions peak at θ = 1.0 and 1.5 on the right, indicating a left-fat-tails effect with the probability of outlier negative returns. However, using θ = 0.3, the distributions peak on the left with a fat tail on the right. The different skewness of the distributions is consistent with both left- and right-skewed distributions obtained from empirical observations of equity returns, depending on their sample periods. We performed a comparison between Panels A and B, where σ_x increases from 0.05 to 0.15, and the distributions have fatter tails and become hump-shaped. The distributions generated from the quasi-bounded process show that the probability leakage condition is consistent with the empirical left-skewed distributions for equities in distress.

3. Model Calibration

3.1. Calibrations of S&P500 Index in Distress

We examine whether the dynamics of the S&P in distress can be characterised by the quasi-bounded process. The model parameters of the process specified in Equation (3) are calibrated based on the PDF of Equation (4) and the logarithm-normalised x in Equation (1). The calibration employs maximum likelihood estimation (MLE) of the time series of the daily spot index of the S&P data from 1 January 1990 to 20 June 2020. Figure 2 (Panel A) shows the S&P in S and the associated moving lower and upper boundaries with the parameters of ${\eta }_L=0.7$ and ${\eta }_U=1.2$ with the 50-day moving average, and the normalised index in x. The estimation is based on a 1-year rolling window. The market data used for the estimations are from Bloomberg.

Figure 3 reports the estimates of drift k (Panel A) at a 5% significance level with its z-statistic above 1.96 when k is above 0.02. k was in the range from 0.02 to 0.12 for most of the time and dropped sharply under shocks or crashes in the market. As the S&P dropped towards its lower boundary when the market crashed, the mean-reverting force weakened with the dropped k. The estimation of k became insignificant (not significantly different from zero) in very short periods of time after the crashes. Subsequently, the estimation rebounded to the 0.04 level and was significant. Panel B shows a significant steady mean θ with values ranging between 0.7 and 1.2. The estimation of θ became insignificant in very short periods of time after the crashes, similar to the changes in k. The mean-reversion represented by k and θ was found to be present in the estimations.

The volatility σ_x in Panel C is estimated to be significant between 0.02 and 0.14. The results suggest that the volatility part of the quasi-bounded dynamics is robust. The volatility increased sharply during 2008–2009 when the global financial crisis emerged, and during the COVID-19 outbreak in March 2020.

Panel D displays the leakage condition of $({\sigma }_x^2/4\kappa \theta )$ to identify periods when this measure is greater than 1 to portray the crash risk of the S&P at the lower boundary. The measure was generally below 0.01, suggesting that the crash risk was immaterial when the S&P was well bounded above the lower boundary. In recent history, the measure rose sharply and breached 1.0 on 23 July 2002 (the dotcom bubble crash), 9 October 2008 (the subprime mortgage crisis), 8 August 2009 (the global financial crisis), and 9 March 2020 (the COVID-19 pandemic), with the existence of the leakage condition. The diminishing mean-reverting force in the S&P dynamics and the leakage condition reflect the crash risk being built up during those periods of financial distress.

We use the Breusch–Godfrey serial correlation Lagrange-multiplier test to study autocorrelation in the errors of the model estimations. The test uses the residuals from the model, and the corresponding test statistic is derived from these. The null hypothesis is that there is no serial correlation of any order up to p. Panel E shows that despite some rejections of null hypothesis (i.e., low p-values), the majority of the tests support the hypothesis that the residuals are not serially correlated up to 5 days, indicating that the estimations are adequate.

3.2. Calibrations of Hertz Equity Price in Distress

To investigate how financial distress affects equity price dynamics, we calibrate the dynamics of the Hertz equity (HTZ) price according to the quasi-bounded process. Hertz filed voluntary petitions for re-organisation under Chapter 11 on 22 May 2020 as the COVID-19 pandemic crushed the car-rental industry. The sample period covers the daily price of HTZ from 18 August 2014 to 31 August 2020. Panel B of Figure 2 shows the price of HTZ in S and the associated moving lower and upper boundaries with the parameters of ${\eta }_L=0.4$ and ${\eta }_U=1.6$ with the 50-day moving average, and the normalised price in x. The lower and upper boundaries correspond to about 2.4 standard deviations, respectively. The estimations are based on the 2-year rolling window.

Figure 4 reports the estimated drift term $\kappa$ (Panel A) at the 5% significance level when $\kappa$ is above 0.02. $\kappa$ was between 0.02 and 0.1 during most of the time. It dropped sharply and was not significantly different from zero on 16 March 2020. As the HTZ price dropped towards its lower boundary during March 2020, the mean-reverting force in its dynamics weakened substantially. Panel B shows a significant steady estimated mean $\theta$ ranging between 0.5 and 0.8, which became insignificant on 16 March 2020. The volatility ${\sigma }_x$ displayed in Panel C is estimated to be highly significant, with a value between 0.06 and 0.13

Panel D shows the measure of the probability leakage condition, which was below 0.2 before March 2020, suggesting that the default risk was not significant as the HTZ price was bounded above the lower boundary. The measure rose sharply and breached 1.0 on 16 March 2020 with the existence of the leakage condition when the HTZ price fell sharply. While the HTZ price was bounded above the lower boundary for much of the time, as indicated by its dynamics, the condition for breaching the boundary was met on 16 March 2020 using only information until that point, about two months before the company filed for bankruptcy. The existence of the leakage condition suggests that the default risk of the firm occurs and its default probability accumulates.

3.3. Calibration of GameStop Stock Price Dynamics under Short Squeezes

The calibration was conducted by applying the MLE and the logarithm normalised in Equation (2) to the daily GME price data from 1 January 2017 to 21 January 2021. Figure 5 (Panel A) shows the GME price in S and the moving upper boundary with ${\eta }_U=2.5$ with the 50-day moving average, and the equity price in x. The upper boundary is about 2.5 standard deviations, covering the probability of 99.38%.

Using the 2-year rolling window, Figure 6 reports the estimates of the drift term k (Panel A), which are statistically significant at the 5% significance level. When the GME price surged sharply in January 2021, k fell and was not significantly different from zero on 14 January 2021. Panel B shows the estimated mean θ, which is significant, with the corresponding z-statistic above the 1.96 level until the price spiked in January 2021. The results show that the mean-reverting force in terms of k and θ was present in the GME price dynamics during the estimation period and diminished under the short squeeze in January 2021. The volatility σ_x of the quasi-bounded process, which is displayed in Panel C, is estimated to be highly significant and to have an increasing trend with a jump in January 2021.

As the probability leakage condition of ${(\sigma }_x^2/4\kappa \theta )$ portrays the likelihood of the GME price breaching the upper boundary under the short squeeze, Panel D displays this measure to identify when the leakage condition is greater than 1. The measure was, in general, below 0.05, suggesting that the short squeeze was not relevant to the GME price bounded below the upper boundary. The measure rose sharply and breached 1.0 on 14 January 2021 with a leakage condition greater than 1 when the GME price escalated. The diminishing mean-reverting force in the GME price dynamics and the existence of the leakage condition indicate that the short squeeze built up through co-ordinated buying.

4. Discussion

4.1. The Dynamic Relationship of S&P500 Index Dynamics with Market Stress Condition

To test the validity of incorporating crash/distress risk into the model, a co-integration analysis is used in this paper to test the relationship between the dynamics of the S&P and the VIX. The VIX measures the level of risk aversion anticipated by market participants to put capital into the market. This is noted as the model parameters based on the MLE are estimated using the S&P data only, without any information from the option market.

We use the following dynamical error-correction representation to investigate any long-run equilibrium relationship between the model parameters and VIX:

$$∆y_t=a_{10}+{\alpha }_y\left(y_{t-1}-{\beta }_1{X}_{t-1}\right)+{\sum }_k{b}_{1k}∆y_{t-k}+{\sum }_k{c}_{1k}∆X_{t-k}+{\epsilon }_{yt},$$

(6)

where $y_t$ is either k, $S_{\theta }$, or ${\sigma }_x$ at time t; ${\alpha }_y$ is negative; and $X_{t-1}$ is the logarithm of the VIX at time t − 1. $y_t$ represents the logarithm of model parameters ($\kappa ,{\sigma }_x$), and the mean level $S_{\theta }$ will respond to stochastic shocks represented by ${\epsilon }_{yt}$ and the long-run equilibrium deviation in the previous period ($y_{t-1}-{\beta }_1{X}_{t-1}$). The mean level $S_{\theta }$ in the original price measure of S is used instead of θ, because risk sentiment in the market is based on the level of the price in S, not in x normalised by the time-varying upper and lower boundaries. The co-integration relationship specification of error-correction requires that the estimated speed of adjustment (i.e., ${\alpha }_y$) should be negative and nonzero. A larger ${\alpha }_y$ in absolute terms indicates higher sensitivity of $y_t$ to the long-run equilibrium deviation.

As there was a structural break during the global financial crisis in 2009, the estimation is conducted at a daily frequency for the pre-crisis period from 1 January 2001 to 31 December 2009, and the post-crisis period from 1 January 2011 to 30 June 2020. The summary statistics, correlation coefficient, and respective Augmented Dickey–Fuller (ADF) test results for the variables, both in the levels and the first differences, for the pre- and post-crisis periods, respectively, are reported in

Table 1. Descriptive statistics of ln(κ), ln(S_θ), and ln(σ_x) for S&P500 Index and ln(VIX).

1 January 2001–31 December 2009
	ln(κ)			ln(Sθ)			ln(σ)			ln(vix)
	Level	Change		Level	Change		Level	Change		Level	Change
Mean	−3.02	0.000		6.94	0.000		−2.87	0.000		3.00	−0.002
Median	−3.01	0.000		6.96	0.000		−2.81	0.000		3.00	−0.006
Maximum	−2.10	2.134		7.29	0.239		−1.97	0.088		4.39	0.496
Minimum	−5.86	−2.167		6.59	−0.251		−3.45	−0.038		2.29	−0.300
Std. Dev.	0.27	0.081		0.16	0.010		0.43	0.006		0.41	0.059
Skewness	−1.01	−0.490		−0.23	−9.265		0.39	2.627		0.50	0.548
Kurtosis	12.24	469.397		2.09	604.122		2.17	39.756		2.92	7.483
ADF test statistics	0.09	−24.35	***	−1.21	−19.46	***	−0.13	−4.583	***	−1.56	−50.95	***
Phillips–Perron test statistics	0.10	−59.40	***	−1.90	−57.26	***	−0.48	−55.546	***	−2.94	−53.32	***
Correlation with ln(vix)	−0.303	-		−0.413	-		0.802	-		-	-
Observations	2236	2140		2262	2179		2263	2181		2263	2181
1 January 2011–30 June 2020
	ln(κ)			ln(Sθ)			ln(σ)			ln(vix)
	Level	Change		Level	Change		Level	Change		Level	Change
Mean	−2.94	0.000		7.48	0.000		−3.09	0.000		2.78	−0.001
Median	−2.87	0.001		7.52	0.001		−3.12	0.000		2.71	−0.007
Maximum	−2.26	0.372		8.27	0.297		−2.26	0.130		4.42	0.768
Minimum	−13.82	−0.321		6.96	−0.301		−3.77	−0.080		2.21	−0.314
Std. Dev.	0.43	0.048		0.29	0.014		0.28	0.008		0.33	0.080
Skewness	−8.09	−0.234		−0.20	2.802		0.29	4.180		1.35	1.231
Kurtosis	190.56	11.297		1.94	435.728		3.61	64.434		5.52	9.968
ADF test statistics	−0.86	−49.86	***	−1.49	−49.08	***	−0.91	−13.17	***	−1.03	−50.90	***
Phillips–Perron test statistics	−0.86	−49.77	***	−2.02	−75.27	***	−0.61	−44.89	***	−0.97	−54.12	***
Correlation with ln(vix)	−0.357	-		−0.162	-		0.622	-		-	-
Observations	2312	2211		2386	2296		2389	2301		2389	2302

Notes: *** indicates significance at a level of 1%. Both tests check the null hypothesis of unit root existence in the time series.

. The ADF test results show that ln$\left(\kappa \right)$, ln($S_{\theta })$, ln$\left({\sigma }_x\right)$, and the logarithm value of the VIX [ln$\left(VIX\right)]$ have a unit root in level. However, these variables are co-integrated with the same order I(1) given the lack of presence of a unit root at the 1% level for their first differences.

We adopt the single-equation test in Engle and Granger (1987) to test the co-integration relationship between [ln$\left(\kappa \right)$, ln($S_{\theta })$, ln$\left({\sigma }_x\right)$] and ln$\left(VIX\right)$. It tests the stationarity of the residuals of the linear combinations between i. ln$\left(\kappa \right)$ and ln$\left(VIX\right)$; ii. ln($S_{\theta })$ and ln$\left(VIX\right)$; and iii. ln$\left(\sigma \right)$ and ln$\left(VIX\right)$. The co-integration tests between ln(VIX) and [ln$\left(\kappa \right)$, ln($S_{\theta })$, ln$\left({\sigma }_x\right)]$ are reported in

Table 2. Tests for co-integration of ln(κ), ln(S_θ), and ln(σ_x) for S&P500 Index and ln(VIX).

1 January 2001–31 December 2009
Engle–Granger Single-Equation Test	ln(κ)		ln(Sθ)		ln(σ)
Dependent Variable:
ADF test statistic	−5.01	***	−1.95	**	−3.66	***
Phillips–Perron test statistic	−7.09	***	−2.59	***	−3.62	***
1 January 2011–30 June 2020
Engle–Granger Single-Equation Test	ln(κ)		ln(Sθ)		ln(σ)
Dependent Variable:
ADF test statistic	−3.55	***	−6.46	***	−4.47	***
Phillips–Perron test statistic	−3.19	***	−4.54	***	−3.59	***

Notes: *** and ** indicate significance at levels of 1% and 5%, respectively. The co-integration test uses the Augmented Dickey–Fuller and Phillips–Perron tests to check the null hypothesis that the residuals of the regression of ln(VIX) and the parameters from the MLE calibration with the 3-year rolling window are non-stationary, assuming zero mean in the test equation. The critical value of the test was obtained from MacKinnon (1996).

for the pre- and post-crisis periods, respectively. On the whole, the results show that there is at least one co-integrating vector of ln$\left(VIX\right)$ and [ln$\left(\kappa \right)$, ln($S_{\theta })$, ln$\left({\sigma }_x\right)]$, respectively, given the statistical significance at the 1% or 5% level.

Table 3. Estimates of long-run coefficient (β) ln(κ), ln(S_θ), and ln(σ_x) for S&P500 Index and ln(VIX).

1 January 2001–31 December 2009
	ln(κ)		ln(Sθ)		ln(σ)
Dependent Variable:
ln(vix)	−0.20	***	−0.14	***	0.83	***
Constant	−2.41	***	7.38	***	−5.36	***
1 January 2011–30 June 2020
	ln(κ)		ln(Sθ)		ln(σ)
Dependent Variable:
ln(vix)	−0.52	***	−0.14	***	0.56	***
Constant	−1.51	***	7.89	***	−4.65	***

Notes: *** indicates significance at a level of 1%.

shows that the co-integrating vectors expressed by β between ln$\left(VIX\right)$ and [ln$\left(\kappa \right)$, ln($S_{\theta })$, ln$\left({\sigma }_x\right)]$ are estimated to be [−0.2, −0.14, 0.83] and [−0.52, −0.14 and 0.56] in the pre-crisis and post-crisis periods, respectively, at the 1% significance level. The estimated negative coefficients indicate that a higher VIX would decrease $\kappa$ and $S_{\theta }$. The negative relationship suggests that when the VIX increases, the crash risk of the S&P rises, with a weakened mean-reverting force in the S&P dynamics. On the other hand, the estimated positive coefficient for ${\sigma }_x$ shows that a higher VIX would increase ${\sigma }_x$ as expected.

Table 4. Estimation results of the short-run dynamics for ln(κ), ln(S_θ), and ln(σ_x) for S&P500 Index and ln(VIX).

1 January 2001–31 December 2009
	ln(κ)		ln(Sθ)		ln(σ)
Dependent Variable:
ln(vix)
Speed of adjustment	−0.04	***	−0.005	*	−0.04	***
Lag length	2		2		2
1 January 2011–30 June 2020
	ln(κ)		ln(Sθ)		ln(σ)
Dependent Variable:
ln(vix)
Speed of adjustment	−0.0100	***	−0.001	*	−0.0045	***
Lag length	5		26		3

Notes: *** and * indicate significance at levels of 1% and 10%, respectively.

reports the estimates of the speed of adjustment (α_y) for [ln$\left(\kappa \right)$, ln($S_{\theta })$ and ln$\left({\sigma }_x\right)],$ which are negative, with [−0.04, −0.005, −0.04] and [−0.01, −0.001, −0.0045], in the pre-crisis and post-crisis periods, respectively, suggesting that the model parameters will adjust to recover the long-run equilibrium. The error-correction specification is thus valid, with the presence of a self-restoring force, such that the gaps between the model parameters and VIX will be closed.

4.2. The Relationship between Hertz Equity Price Dynamics and COVID-19

Given that Hertz filed for Chapter 11 due to the impact of the COVID-19 pandemic on the car-rental industry, we investigate the relationship between the intensity of the pandemic’s spread, which triggered the firm’s financial distress, and its equity price dynamics derived from the model. We estimate the simple linear regression as follows:

$$ln\left(y_t\right)=a+bln\left(COVID19\right),$$

(7)

where $y_t$ is the model parameters {$\kappa$, $\theta$, ${\sigma }_x$}, $COVID19$ is the number of new confirmed COVID-19 cases in the US, and a and b are the coefficients to be estimated.

As the first confirmed COVID-19 case in the US was on 22 January 2020 and the quasi-bounded process for HTZ was no longer bounded after 16 March 2020, the estimation is conducted for the period between these two dates. Figure 7, Figure 8 and Figure 9 show the scatter plots of the model parameters and the number of daily new COVID-19 cases in the US in logarithm. The data of COVID-19 cases were compiled by the Johns Hopkins University Center for Systems Science and Engineering (JHU CCSE) from various sources and can be found at https://data.humdata.org/dataset/novel-coronavirus-2019-ncov-cases (accessed on 15 September 2020). The coefficients $[a$, $b]$ are estimated to be [−2.37, −0.16] for $\kappa$, [−0.32, −0.01] for $\theta$, and [−2.55, 0.019] for ${\sigma }_x$, respectively. All the estimates of coefficients are statistically significant at the 1% level. The $R^2$ of 0.777, 0.463, and 0.737 for $\kappa$, $\theta$, and ${\sigma }_x$, respectively, are reasonably high.

The negative coefficients $b_{\kappa }$ and $b_{\theta }$ show a negative relationship between the mean-reversion (represented by $\kappa$ and $\theta$) and the number of COVID-19 cases in the US. Conversely, the relationship in terms of $b_{{\sigma }_x}$ between the volatility ${\sigma }_x$ and the number of confirmed cases is positive. An increase in the number of confirmed cases weakened the mean-reverting force, but enhanced the volatility in the HTZ price dynamics, suggesting an increased default risk for the firm due to the COVID-19 pandemic in the US. These results are consistent with those of the model calibration results and the measure of the probability leakage condition at the lower boundary for the HTZ price in Section 3.2.

4.3. The Relationship between the GameStop Stock Price Dynamics and Trading Volume

A co-integration analysis is used to test the relationship between the mean-reverting force in the GME price dynamics and the daily trading volume using Equation (7), where $y_t$ is k or θ at time t, and $X_{t-1}$ is the logarithm of the trading volume at time t−1. The estimation is conducted at a daily frequency from 1 January 2019 to 21 January 2021.

Table 5. Descriptive statistics of $\mathrm{l}\mathrm{n}\left(volume\right)$, $\kappa$, and $\theta$ of GameStop stock price.

	κ			θ			ln(volume)
	Level	Change		Level	Change		Level	Change
Mean	0.04	0.000		0.91	−0.002		15.37	0.006
Median	0.04	0.000		0.94	0.002		15.26	−0.031
Maximum	0.08	0.017		1.47	0.428		18.79	3.140
Minimum	0.00	−0.024		0.01	−0.426		14.06	−1.676
Std. Dev.	0.02	0.002		0.25	0.064		0.73	0.561
Skewness	0.87	−1.118		−0.65	−0.271		1.25	1.043
Kurtosis	2.81	44.585		3.42	13.705		5.41	7.352
ADF test statistics	−0.64	−27.62	***	−1.94	−22.938	***	0.32	−16.79	***
Phillips–Perron test statistics	−0.63	−27.86	***	−2.01	−22.944	***	−9.76	−67.73	***
Correlation with ln(vix)	−0.34	-		−0.44	-		-	-
Observations	518	517		518	517		518	517

Notes: *** indicates significance at a level of 1%. Both tests check the null hypothesis of unit root existence in the time series.

reports the summary statistics, correlation coefficient, and respective ADF test results for the variables both in levels and first differences, suggesting that these three variables are co-integrated in the same order I(1) at a 1% significance level.

Table 6. Tests for co-integration of $\mathrm{l}\mathrm{n}\left(volume\right)$, $\kappa$, and $\theta$ of GameStop stock price.

Engle–Granger Single-Equation Test	κ		θ
Dependent Variable:
ADF test statistic	−2.51	**	−3.69	***
Phillips–Perron test statistic	−3.84	**	−4.00	***

Notes: *** and ** indicate significance at levels of 1% and 5%, respectively. The cointegration test uses the Augmented Dickey–Fuller and Phillips–Perron tests to check the null hypothesis that the residuals of the regression of ln(volume) and the parameters from the MLE calibration with a 2-year rolling window are non-stationary. The critical value of the test was obtained from MacKinnon (1996).

reports the co-integration tests between ln$\left(volume\right)$ and ($\kappa$, $\theta )$, showing the presence of at least one co-integrating vector between ln$\left(volume\right)$ and $(\kappa$, $\theta )$, respectively, at 5% or 1% significance levels. The co-integrating vectors between ln$\left(volume\right)$ and ($\kappa$, $\theta )$ expressed by β, as shown in

Table 7. Estimates of long-run coefficient (β) for $\mathrm{l}\mathrm{n}\left(volume\right)$, $\kappa$, and $\theta$ of GameStop stock price.

	κ		θ
Dependent Variable:
ln(volume)	−0.003	***	−0.15	***
Constant	0.094	***	3.23	***

Note: 1. *** indicates significance at a level of 1%.

, are estimated to be −0.003 and −0.15 at the 1% significance level, respectively. The estimated negative coefficients indicate that a higher trading volume reduces $\kappa$ and $\theta$, suggesting that the restoring force in the GME price dynamics weakens with a higher trading volume.

Table 8. Estimation results of the short-run dynamics for $\mathrm{l}\mathrm{n}\left(volume\right)$, $\kappa$, and $\theta$ of GameStop stock price.

	κ		θ
Dependent Variable:
ln(volume)
Speed of adjustment	−0.011	**	−0.03	**
Lag length	1		1

Note: ** indicates significance at a level of 5%.

illustrates that the speeds of adjustment (α_y) for $\kappa$ and $\theta$ are negative, showing that the gap between the model parameters and the trading volume will be closed subsequently.

5. Conclusions

A simple stochastic approach has been presented for modelling equity prices under shocks which are constrained to lie within a band. The model is able to capture most stylised facts of equity prices. While the proposed stochastic process has a quasi-bounded boundary which can be breached if the probability leakage condition is met, the other boundary of the band is an inaccessible boundary. The quasi-boundedness at the boundary can be considered a measure of possible downside or upside risk of the corresponding equity prices.

The empirical results show that the S&P500 dynamics can be calibrated according to the quasi-bounded process, in which the volatility and mean-reversion are positively and negatively co-integrated, respectively, with the VIX index, representing the market distress risk implied in the option market. The process also captures the default risk incorporated in stock prices. Hertz filed for Chapter 11 bankruptcy in May 2020 as a result of the COVID-19 pandemic. The volatility and mean-reversion of the calibrated Hertz stock price dynamics are positively and negatively related, respectively, to the number of COVID-19 cases. The process indicates a high likelihood of default in terms of the probability leakage condition at the lower boundary in March 2020 using only information until that point.

The proposed stochastic approach is also able to model equity price dynamics under a short squeeze. A quasi-bounded moving upper bound is the threshold imposed by constraints, such as stop-loss limits forcing traders to close out their short-selling positions squeezed by co-ordinated long-buying. Calibration of the process for the GameStop stock price shows that the quasi-bounded property can adequately describe the price dynamics. The model captures the short squeeze of the stock, as indicated by its price dynamics, when the probability leakage condition for breaching the upper bound was met on 14 January 2021 using only information until that point, i.e., about two weeks before major short-sellers closed out their positions with a significant loss. The trading volume of the stock is negatively co-integrated with the mean reversion of the dynamics, suggesting overshooting prices with increased trading volume during co-ordinated short squeezes.

Given the availability of an analytically tractable probability density function, empirical calibration of the model parameters of the proposed process for different financial observables in general can also be easily performed. Hence, making predictions of future movements of these observables becomes feasible. The leakage condition at the quasi-bounded boundary can thus provide us with a forward-looking indicator of potential crash risk or short squeeze of the corresponding financial observables, including equites for risk management purposes. While only a limited number of firms’ equity prices are calibrated in this sample, we leave the investigation of a larger sample of companies in distress or experiencing short squeeze for future research. In addition, one of the limitations of this study is that fundamental factors such as a firm’s balance-sheet information is not incorporated into the model. Future theoretical research on equity price dynamics could incorporate such factors.