Modeling the Asymmetric and Time-Dependent Volatility of Bitcoin: An Alternative Approach ^†

Abstract

Volatility as a measure of financial risk is a crucial input for hedging, portfolio diversification, option pricing and the calculation of the value at risk. In this paper, we estimate the asymmetric and time-varying volatility for Bitcoin as the dominant cryptocurrency in the world market. A novel approach that explicitly separates the falling markets from the rising ones is utilized for this purpose. The empirical results have important implications for investors and financial institutions. Our approach provides a position-dependent measure of risk for Bitcoin. This is essential since the source of risk for an investor with a long position is the falling prices, while the source of risk for an investor with a short position is the rising prices. Thus, providing a separate risk measure in each case is expected to increase the efficiency of the underlying risk management in both cases compared to the existing methods in the literature.

1. Introduction

In financial markets, risk is a crucial factor that rational investors assess in when making investment decisions in addition to the rate of return. If the rate of return of any asset is not known in advance, then that asset is considered risky. The simplest measure of financial risk or volatility is the average location of the returns in relation to the center of the distribution of the returns. Basically, the standard deviation of the rate of returns for an asset is a measure of the financial risk for that asset. Since the pioneering work of Mandelbrot [1], it is common knowledge that the volatility of financial assets is usually not constant, but time dependent, and fat tails are common property of the distribution of risky assets. Engle [2] introduced the seminal approach for dealing with the clustering behavior of the risk the characterizes financial and economic variables. This time-dependent measure of risk was called the autoregressive conditional heteroskedasticity (ARCH) model by Engle [2]. A generalization of this approach has been introduced by Bollerslev [3], which is called the GARCH model. This generalized ARCH model is more parsimonious since it allows for moving average lags in addition to autoregressive lags. There have been a significant number of additional extensions of this model. For some interesting reviews of the contributions pertinent to the extensions and the applications of the time-dependent risk models, see, among others, Engle [4,5], Bollerslev et al. [6], Bollerslev et al. (1994) [7], Poon and Granger [8], and Francq and Zakoian [9].

Another important issue within the context of financial risk modeling is the issue of potential asymmetry in the underlying risk measure. There are plenty of contributions in the existing literature that allow for some form of an asymmetric structure in the estimation of the risk measure via thresholds or by using different forms of indicator variables. However, a common factor for all these approaches has been allowing for the asymmetry in the volatility within the same model. However, these models cannot explicitly provide a position-dependent measure of risk. The origin of risk for an investor with a long position in the asset is falling prices, while the origin of risk for an investor with a short position in the asset is rising prices. Hatemi-J [10,11] introduces alternative asymmetric time-dependent measures of financial risk that explicitly provide the relevant risk measure for a long or short position in the investment asset. The aim of the current paper is to model the time-varying risk for Bitcoin using these alternative approaches. Since its introduction by Nakamoto [12], Bitcoin is the cryptocurrency that among all competing ones has the highest market capitalization worldwide. Dwyer [13] and Talib [14] provide interesting insights on cryptocurrencies. For new theoretical developments of cryptocurrency pricing, see El-Khatib and Hatemi-J [15,16]. Bitcoin is also very energy intensive, with severe natural environmental consequences (Sapra et al. [17]).

Volatility as a measure of financial risk is a crucial input for hedging (Johnson [18]), the calculation of the value at risk (Jorion [19,20]; Morgan, [21]; Engle and Manganelli [22]; Abda et. al. [23]; Dyhrberg [23,24]), and portfolio diversification (Markowitz [25]; Hatemi-J and El-Khatib [26]; Hatemi-J et al. [27]), among others. For portfolio diversification potentials of cryptocurrencies, see Liu [28] and Hatemi-J et al. [29], among others. For statistical software components that make the portfolio diversification estimations operational, see Hatemi-J and Mustafa [30,31] in VBA and Python, Hatemi-J and Mustafa [32] in Gauss, and Mustafa and Hatemi-J [33] in Python.

The remainder of the paper consists of the following sections. The methodology is described in Section 2. The tests results and the estimated output of the underlying models are presented in Section 3. The last section provides the final remarks. The graphs of the data for Bitcoin, including the transformed components, are presented in an Appendix A at the end of the paper.

2. Methodology

In this section, the econometric methodology for modeling the asymmetric and time-varying risk of Bitcoin is presented, which explicitly provides a position-dependent (i.e., long position or short selling) financial risk measure for this most valuable cryptocurrency. This approach is expected to enhance the precision of risk management of the specific investor since the source of risk for normal trading is falling prices, while rising prices represent the source of risk for a short seller. According to Hatemi-J [10,11], the position dependent measure of risk can be obtained by transforming the data into positive and negative components. Let B represent the price of Bitcoin in terms of the US dollar that is generated by the following integrated autoregressive model with deterministic trend parts:

$$B_t=c+dt+B_{t-1}+w_t$$

(1)

where c and d are parametric constants that can be estimated by the least squares method, and t represents the time trend covering the period t = 1, …, T. The error term w_t is assumed to be an identical independently distributed (IDD) random process. By substituting recursively, the following solution to Equation (1) is obtained:

$$B_t=ct+\frac{t\left(t+1\right)}{2}d+B_0+\sum _{i=1}^t{w}_i$$

(2)

where B₀ is the initial value of Bitcoin. The data are transformed into positive and negative components by applying the following definition: $w_i^{+}:=\mathit{max}\left(w_i,0\right)$ and $w_i^{-}:=\mathit{min}\left(w_i,0\right)$. By using these values, the following expression is obtained:

$$B_t=c+dt+B_{t-1}+w_t=ct+\frac{t\left(t+1\right)}{2}d+B_0+\sum _{i=1}^t{w}_i^{+}+\sum _{i=1}^t{w}_i^{-}$$

(3)

Consequently, the positive and negative partial cumulative sums for Bitcoin are defined as the following:

$$B_t^{+}:=\frac{ct+\left[\frac{t\left(t+1\right)}{2}\right]d+B_0}{2}+\sum _{i=1}^t{w}_i^{+}$$

(4)

Plus

$$B_t^{-}:=\frac{ct+\left[\frac{t\left(t+1\right)}{2}\right]d+B_0}{2}+{\sum }_{i=1}^t{w}_i^{-}$$

(5)

It should be noted that these partial cumulative sums ensure that the necessary theoretical restriction for a correct transformation, i.e., $B_t=B_t^{+}+B_t^{-}$, is satisfied. Also note that $B_t^{+}$ represents the quantified good news for a long position in Bitcoin; however, it represents the bad news for the short seller. Likewise, $B_t^{-}$ is the quantified good news for a short seller; nevertheless, it is the bad news for the owner of the asset. The source of risk is naturally the quantity that represents the bad news in each case. The standard methods in the literature do not make this crucial distinction.

The next step is to test for the ARCH effects of a certain degree in each case. If the null hypothesis is rejected at the conventional significance levels, then each model should be estimated subject to the time-varying risk model via the maximum likelihood method. Thus, there is need for a model to separately model the time-varying risk for each case if the null hypothesis is rejected. For example, the following regression needs to be estimated to test for the ARCH effects in the positive component.

$$e_t^{2+}=g_0^{+}+\sum _{i=1}^{L+}{g}_i^{+}{e}_{t-i}^{2+}+v_t^{+}$$

(6)

Here, $v_t^{+}$ is an IDD error term, L⁺ is the lag length in the autoregressive model that also represents the ARCH order, and $e_t^{+}$ is the estimated value for error term in the following regression model:

$$B_t^{+}=a^{+}+\sum _{i=1}^{L+}{\gamma }_i^{+}{B}_{t-i}^{+}+e_t^{+}$$

(7)

The random variable $e_t^{+}$ is presumed to follow a normal distribution as $e_t^{+}\text{~}\mathrm{N}\left(0,{\sigma }_t^{2+}\right)$. The default hypothesis of no ARCH of order, L⁺ in this case, is defined as the following for testing:

$$H_0:g_1^{+}=g_2^{+}=\cdots =g_{L+}^{+}=0$$

(8)

This null hypothesis is empirically tested by the following test statistic:

$$L{M}^{+}=T\times R_{+}^2$$

(9)

which is estimated by the Lagrange multiplier (LM) approach. T signifies the sample period and $R_{+}^2$ represents the measure of goodness of fit in the unrestricted model defined by model (6). The distribution of the test statistics (9) is chi-squared asymptotically, and the degrees of freedom is equal to L⁺, which also represents the ARCH order of interest being tested.

If the default hypothesis presented in Equation (8) is not accepted empirically, subsequently, the GARCH(K⁺, Q⁺) model defined below needs to be estimated via the maximum likelihood.

$$\begin{array}{c}{B}_t^{+}=a^{+}+{\displaystyle \sum _{i=1}^{L+}}{\gamma }_i^{+}{B}_{t-i}^{+}+e_t^{+}\\ {\sigma }_t^{2+}=c^{+}+{\displaystyle \sum _{i=1}^{K+}}{\beta }_j^{+}{e}_{t-i}^{2+}+{\displaystyle \sum _{j=1}^{Q+}}{\lambda }_j^{+}{\sigma }_{t-j}^{2+}\end{array}$$

(10)

The variable ${\sigma }_t^{2+}$ represents a measure of time-dependent conditional variance for $e_t^{+}$. The unconditional variance of $B_t^{+}$ via GARCH(K⁺, Q⁺) is the following:

$$Var\left(B_t^{+}\right)=E\left(e_t^{+}\right)=E\left({\sigma }_t^{+}\right)=\frac{c^{+}}{1-{\sum }_{i=1}^{K+}{\beta }_j^{+}-{\sum }_{j=1}^{Q+}{\lambda }_j^{+}}$$

(11)

The measure defined by Equation (11), or the squared root of it, is an important input in, among others, portfolio diversification, hedging, and asset pricing, such as determining the fair premium for call and put options.

Similarly, a test for ARCH effects in $B_t^{-}$ can be implemented. If the null hypothesis is rejected, then a separate GARCH(K⁻, Q⁻) model needs to be estimated for the negative component also. The data can be transformed by making use of a software component produced by Hatemi-J and Mustafa [34] in the Visual Basics for Applications (VBA), which is available online.

An important issue that needs attention when dynamic models are used is the determining the optimal lag order, since the entire empirical inference is a direct function of the chosen lag order. The selection of an optimal lag length can be fulfilled in each case by making use of a lag selection criterion, such as the univariate version of the information criterion advocated by Hatemi-J [35,36]. A VBA module created for this purpose is described by Mustafa and Hatemi-J [37]. For additional references on software development, see Hatemi-J and Mustafa [38,39].

3. Empirical Estimation Results

The empirical investigation is conducted by using weekly data for the exchange rate of Bitcoin versus the US dollar. Weekly data are used in order to avoid the weekend effect that can characterize the daily data. The sample period covers the last week of December 2017 until the first week of March 2024. The source of the data is Yahoo Finance. The data are transformed into positive and negative components via the solutions presented in Equations (4) and (5). It should be mentioned that only a drift was needed for the transformation without the time trend based on the visual inspection of the time path, and the exchange rate for Bitcoin presented in Figure 1.

The estimation results for the ARCH tests are presented in

Table 1. Test results of arch impacts.

The Null Hypothesis	Estimated Test Value	Probability Value
No ARCH(4) Effects on the Positive Element of Bitcoin	52.15584	<0.0001
No ARCH(4) Effects on the Negative Element of Bitcoin	23.53037	0.0001

Note: An ARCH model of order four was considered for testing in each case to capture the effect of one month of trading days.

. The probability of the test for each case is much lower than any conventional significance level. That means the null hypothesis of no ARCH of degree four is strongly rejected for both negative and positive components of Bitcoin. The implication of these results is that volatility, as a measure of risk, is strongly time dependent in each case. Thus, the linear regression model for the rising as well as the falling Bitcoin prices needs to be estimated separately by using the maximum likelihood method that accounts for time-dependent volatility. For a long position in Bitcoin, the basis of risk is falling prices. For the short seller of Bitcoin, the origin of risk is the rising prices. Therefore, the time-dependent volatility for the positive component of Bitcoin is useful to a trader with a short position in Bitcoin, while the time-based volatility for the negative component of Bitcoin is worthwhile to a trader with a long position in Bitcoin. This position-dependent volatility measure that this paper offers is expected to enhance the accuracy of investment decisions as well as of the financial risk management in each case.

The separation of modeling for positive and negative components can also be useful for capturing the so-called leverage effect. Investing in risky assts in via loans in addition to own capital (i.e., trading on margin) has a magnifying impact on the rate of investment return. During favorable market conditions, trading on margin results in a higher rate of turn compared to trading on own-capital only. The reverse is also true since the loss is bigger for the trader on margin compared to trading on own-cash only when the market conditions are unfavorable.

Based on the test results, there is a need for calculating the autoregressive model dependent on the time-varying volatility. Thus, an AR(1)-GARCH(1, 1) was fitted for each component. The estimation results for the positive component of Bitcoin (i.e., $B_t^{+}$) are the following:

$$\begin{array}{c}{B}_t^{+}=0.066196+0.999723B_{t-1}^{+}\\ {\sigma }_t^{2+}=0.000037+0.053874e_{t-1}^{2+}+0.854373{\sigma }_{t-1}^{2+}\end{array}$$

(12)

Likewise, the estimated results for the negative element of Bitcoin (i.e., $B_t^{-}$) are the following:

$$\begin{array}{c}{B}_t^{-}=-0.069265+0.998649B_{t-1}^{-}\\ {\sigma }_t^{2-}=0.000490+0.005505e_{t-1}^{2-}+0.920827{\sigma }_{t-1}^{2-}\end{array}$$

(13)

The logarithmic values are used for estimating the AR(1)-GARCH(1, 1) model in each case. The coefficient of determination was more than 99% in each case. All estimated parameters were strongly significant statistically, except for the slope regarding $e_{t-1}^{2-}$. We can observe that positive and negative components of Bitcoin are integrated processes. The empirical findings also confirm that volatility as a measure of risk is time dependent in each case. In addition, it can be observed that the unconditional variance is lower for the negative component compared to the positive one (i.e., $0.000037$ for the positive component versus $0.000490$ for the negative one). Thus, there is indeed an asymmetric structure in the volatility of Bitcoin. Since the source of risk for a trader with a short position in Bitcoin is rising prices, then the risk model estimated by system of Equation (12) should be used in this case for hedging purposes via derivative securities, such as futures. Subsequently, as the source of risk for a long position in Bitcoin is falling prices, then the estimated risk model of (13) should be used by this investor for risk management purposes. The daily unconditional variance for the positive component is 0.000403, and it is 0.006651 for the negative component calculated using Equation (11). The daily unconditional volatility as a measure of risk for the long position is 0.081556, while the corresponding value for the short seller is 0.020081. The yearly risk measure that the long position faces is 1.294669 for 252 trading days. The yearly risk measure for the short seller is 0.318773, which is four-times lower compared to the risk for the long position. These values have important repercussions for each case in terms of input for important calculations that involve a measure of risk, such as portfolio diversification, hedging, or the value at risk calculation. Volatility is also a crucial input in option pricing. These calculations would not be accurate if they were based on the same level of risk for both trading positions.

4. Concluding Statements

Cryptocurrencies are increasingly utilized by investors and financial institutions both as an investment asset and as a decentralized payment tool. Despite the benefits that a cryptocurrency can provide, such as no need for intermediation pertinent to the financial services and the independence of any central authority, it also has potential disadvantages. For example, cryptocurrencies commonly suffer from three characteristics—namely (1) exceptionally high-risk levels that cluster, (2) illiquidity, and (3) structural breaks or regime shifts that sometimes alternate.

The aim of this paper is to model the asymmetric and time-dependent volatility of Bitcoin since it is the cryptocurrency that currently has the highest market capitalization among thousands of competing ones. A novel approach is applied for this purpose that provides position-dependent risk measures. This is an important issue, and it is expected to enhance the accuracy of the underlying risk management. This conclusion is based on the fact that the basis of risk for the buyer is the rising price, while the source of risk for the seller is the falling price. Therefore, using a position-dependent risk measure accords better with reality.

The estimation results clearly show that volatility as a measure of risk is asymmetric and time dependent for both positive and negative components of Bitcoin. A separate model is calculated for the positive changes as well as for the negative changes in the exchange rate of Bitcoin against the US dollar. The estimated parameters are strongly significant, statistically speaking, and the explanatory power of each estimated model is exceptionally high.

A potential extension of the current approach in the future might be to make use of the multivariate version of testing and modeling the dynamic interaction that could prevail between the volatility of major cryptocurrencies using the method suggested by Hacker and Hatemi-J [40].