**1. Introduction**

Forecasting volatility is pivotal for developing accurate and realistic risk management models that perform well in good times and in bad. An accurate volatility forecast depends on the assumptions made by the analyst and selection of proper statistical models that can provide a parsimonious representation of the stylized features of the data. When risk management fails, the blame is squarely placed on risk models. According to Bernanke (2008), "Those institutions faring better during the recent turmoil generally placed relatively more emphasis on validation, independent review, and other controls for models and similar quantitative techniques. They also continually refined their models and applied a healthy dose of skepticism to model output". Hence, a crucial task facing a risk manager is to make sure the models are tested, back-tested, and validated to minimize expected losses.

Academics, practitioners, and regulators have commonly used risk models that were deemed sophisticated in terms of forecasting risk. For instance, JPMorgan and Bank of America use historical simulation to estimate their trading risk. Others rely on volatility forecasting models such as GARCH family models, exponentially moving average, JPMorgan's RiskMetrics, and extreme value theory models. In this respect, academics provided various results of the reality checks of these models and suggested different versions of the GARCH volatility models by alternating between Normal, Student-*t*, and Skewed-*t* distributions in an attempt to better capture tail events and asymmetry of the data generating process (see, for example, Bauwens and Laurent (2005), Danielsson and Morimoto (2000)). Other scholars suggested hybrid models combining, for instance, filtered historical simulation with GARCH models or assuming different error terms in the models. Nevertheless, such models require assumptions about the stochastic processes of the underlying asset prices that are subject to validation failure either because of misspecification or the latent characteristic of the parameters, especially during economic downturns.

On a more macro level, it is now evident that the importance of risk models remains fundamental for capital requirements as imposed by the Basel regulations. Decision-makers rely on these risk models as long as they have passed some validation criteria adopted by financial institutions and regulatory authorities. Three critical model-failures have been noted in the literature—1992 Deutsche Bank loss of \$500 million, the 1998 collapse of Long Term Capital Management (LTCM), and the 2012 "London Whale"1 debacle of JPMorgan Chase & Co. For the Deutsche bank loss, the culprit was the assumption of flat volatility to price options and, in the case of the LTCM debacle, the blame was placed on the model's use of Gaussian copula and the assumption of no contagion (Jorion 2000).2 Finally, the 2012 loss of \$6.2 billion, due to a spreadsheet error in calculating Value-at-Risk (VaR) and operational risk at JPMorgan Chase, highlights why it is important to validate risk models.3

In light of some of these historical data, it is fitting that scholars shifted their approach to stochastic volatility risk models, postulating that volatility is driven by its own stochastic process that accounts for jump dynamics in the returns rather than skewness or excess kurtosis. Such an approach, when pitted against other risk models, outperformed both in and out-of-sample backtesting results (see, for example, Maheu and McCrudy (2005), Su and Hung (2011), and Ze-To (2012)). Their results supported a consensus that jumps are causing extreme value in returns and taking them into consideration provides better VaR forecasts for long and short positions at lower and higher VaR levels. Though such models were successfully validated, they accounted for jumps in the return series and not in volatilities. In addition, many of these risk models were validated in a portfolio context, and little has been done with individual assets with a stochastic model that accounts for both jumps in returns and volatilities (see, for example, Eraker et al. (2003)).

The challenge, therefore, is to identify the best risk model that has passed some validation criteria using risk measures such as VaR and Expected Shortfall (ES), which remain the building-block of market risk regulations. One typical means for identification of the best risk forecast model is by analyzing violation ratios, which is better known as backtesting. Although some scholars argue that risk model choice is the least concern for decision-makers (see, for example, Danielsson et al. (2016)), the scenario takes a different path when dealing with individual financial assets and considering economic events affecting financial markets.

Risk validation in any financial asset that trades on organized platforms is critical for national and international regulatory bodies that are entrusted with providing a safe and sound financial environment for financial transactions. To this extent, investor safety is paramount for an assessment of risks of cryptocurrencies so that proper regulatory controls, if needed, can be designed and implemented. The popular media have declared the cryptocurrencies as some of the most volatile assets in the financial market worldwide. Such assertions must be validated using appropriate econometric risk models that incorporate stylized features of the market to understand the evolution of risk and the factors that are responsible for it. Most importantly, the structure of the market, transaction costs, market microstructure, price formation, and the volatility should be studied within an appropriate risk model. For the emerging cryptocurrencies market where governmental oversight and regulatory structure is still evolving, model risk due to wrong assumptions can lead to wrong conclusions and incorrect policy implementation.

Overall, cryptocurrencies have taken place in the financial markets and in portfolio management. They may be useful in risk management and ideal for risk-averse investors in anticipation of negative shocks to the market. They are also considered as investment assets useful for portfolio diversification and hedging against movements in other financial assets such as commodities. To sum up, for an

<sup>1</sup> The term "London Whale" was based on the enormous size of the bet on credit default swaps made by the London office of the bank's risk management division.

<sup>2</sup> In addition, the LTCM model made several critical mistakes, including assuming that returns were normally distributed, and the time period to establish the risk parameters was rather short. See Jorion (2000) for more.

<sup>3</sup> Interestingly, JPM CEO Jaime Dimon had initially described the problem as "a tempest in a teapot".

investor trying to manage tail risk in cryptocurrencies, choosing an appropriate model is critical for forecasting volatility.

This paper aims at exclusively identifying a risk model that is valid for the cryptocurrency markets. It also attempts to build up on the consensus that cryptocurrencies exhibit extreme volatility that needs to be properly quantified for risk management purposes. The existing literature suggests that both stochastic volatility and jumps in returns in the equity market are important components of the returns. Hence, we consider theoretical and applied return models that require the specification of a stochastic volatility component. The model that we select accommodates the persistence in volatility, and volatility of a jump to address the unpredictable and large movements in the price process. In essence, our objective is to examine if jumps in returns and volatility can help us predict tail risk and expected shortfall more accurately. Furthermore, it also is important to determine if jumps in returns and volatility can help us accurately predict and manage expected losses from investing in cryptocurrencies. This particular focus on the volatility structure of the cryptocurrency market is incomplete in the literature.

Our risk model validation approach starts with a nonparametric test to detect jumps in the dynamics of the price process in the cryptocurrency market. Next, we introduce the price dynamics as inputs in a stochastic model that allows for jumps in both returns and volatility, as well as their correlation. We call this the Stochastic Volatility with Co-Jumps (SVCJ) model. We further study how such a model could be appropriate for risk measurement and compare its Value-at-Risk and Expected Shortfall predictions with competing models that are frequently applied to financial time series. Backtesting criteria are implemented to test the statistical accuracy of the models, followed by an examination of the statistical significance of the differences between the models.

Our results suggest that no one model universally fits all cryptocurrencies. We find that there are jumps in the returns and volatility of returns in the cryptocurrency market, though jump probability estimates vary across currencies. We find evidence of the leverage effect where volatility has an asymmetric response to good news and bad news. Both the SVCJ and TGARCH models produce accurate forecasts of tail risk and Expected Shortfall (ES) better than the popular RiskMetrics model. Finally, the strongest result in the paper is that the proposed SVCJ model produces lower economic losses than the TGARCH and RiskMetrics models. This implies real savings for an investor for dealing with capital losses for investing in the cryptocurrency market.

The paper proceeds as follows. In Section 2, we discuss the proposed stochastic volatility model with jumps and leverage. In Section 3, we offer empirical results. The final section concludes the paper.

#### **2. Methodology**

An understanding of the volatility process of financial assets is necessary for investors to manage risks of investing in financial markets. Equally important is that regulators have a more informed view of the underlying volatility structure of these assets so that appropriate regulatory policies can be designed to attract investors and potential new issuers. To this extent, it is important to examine if assets have time varying volatility, jumps, autocorrelation, extreme risk, and how the volatility process responds to good news and bad news in the markets. These issues have been investigated in the literature individually in a disparate manner when they should be addressed simultaneously in an integrated model to allow interaction among these volatility parameters (see Ardia et al. (2019), Barivera et al. (2017), and Segnon and Bekiros (2019), and references therein). Hence, we adopt a model that can capture quick and persistent movements of the conditional volatility of returns as in Eraker et al. (2003), which was an implementation of the model with jumps in both returns and volatility by Duffie et al. (2000). Such models showed that, with jumps in returns and jumps in stochastic volatility, the performance is better than competing models with different specification of the volatility process. A number of papers have examined equity price models with jumps in returns and stochastic volatility (see, for example, Bakshi et al. (1997), Andersen et al. (2002), and Pan (2002)) and made it clear that both stochastic volatility and jumps in returns are important components of the time series properties of financial assets.

Let us begin by defining *logPt* as the logarithmic price process with *Vt* as the stochastic variance. Both processes are assumed to have a continuous path or happen to be discontinuous with the occurrence of at least one jump:

$$\begin{array}{rcl}d\log P\_t &=& \mu dt + \sqrt{\nabla\_t} dW\_t^X + f^X dZ\_t^X\\dV\_t &=& \kappa(\theta - V\_t)dt + \sigma\_V \sqrt{\nabla\_t} dW\_t^V + f\_t^V dZ\_t^V,\end{array} \tag{1}$$

where the stochastic volatility *Vt* has parameters *κ* and *θ* that are the mean reversion rate and mean reversion level, respectively. *W<sup>X</sup>* and *W<sup>V</sup>* are correlated standard Brownian motions with Cov(*dW<sup>X</sup> <sup>t</sup>* , *dW<sup>V</sup> <sup>t</sup>* ) = *ρdt*. *Z<sup>X</sup> <sup>t</sup>* = *Z<sup>V</sup> <sup>t</sup>* are contemporaneous jump arrivals in both prices and volatility and are assumed to follow a Poisson process with constant intensity *λ*. *σ<sup>V</sup>* represents the volatility of volatility and measures the variance responsiveness to diffusive volatility shocks.

Because data are observed in discrete time, it is common to use an Euler discretization of the continuous time process in Equation (1). Assuming a time discretization of one day (*dt* = 1) and *Xt* = *logPt* − *logPt*−1, the discrete model, labeled SVCJ, becomes:

$$\begin{array}{rcl} X\_t &=& \mu + \sqrt{V\_{t-1}} \mathfrak{e}\_t^X + \mathfrak{f}\_t^X Z\_t^X\\ V\_t &=& \kappa (\theta - V\_{t-1}) + \sigma \sqrt{V\_{t-1}} \mathfrak{e}\_t^V + \mathfrak{f}\_t^V Z\_t^V \end{array} \tag{2}$$

where *J<sup>X</sup> <sup>t</sup>* and *J<sup>V</sup> <sup>t</sup>* are the correlated jump sizes with *J<sup>V</sup> <sup>t</sup>* ∼ *exp*(*μV*) and *<sup>J</sup><sup>X</sup> <sup>t</sup>* |*J<sup>V</sup> <sup>t</sup>* ∼ *<sup>N</sup>*(*μ<sup>X</sup>* + *<sup>ρ</sup><sup>J</sup> <sup>J</sup>V*, *<sup>σ</sup>*<sup>2</sup> *X*), and  *<sup>X</sup> <sup>t</sup>* and  *<sup>V</sup> <sup>t</sup>* are standard normal random variables with correlation *ρ*. We note that, when *ρ<sup>J</sup>* = 0 and *μ<sup>V</sup>* = 0, the model turns to a stochastic volatility with jumps of Bates (1996), and, when *ρ<sup>J</sup>* = 0, *μ<sup>V</sup>* = 0, *λ* = 0, *μ<sup>X</sup>* = 0, and *σ<sup>X</sup>* = 0, the model is a stochastic volatility of Heston (1993).

We use a likelihood-based framework for estimating multivariate jump-diffusion models using the Markov Chain Monte Carlo (MCMC) method. This method is based on Bayesian modeling that requires using a likelihood, a priori distribution, and a posteriori distribution. Prior distributions are required for the initial volatility state, *V*0, and for all parameters governing the dynamics of the volatilities. Moreover, the prior contains information about both the parameters and the structure of the latent processes: the stochastic specifications of the jump sizes, and jump times. As in Eraker et al. (2003), the priors are always consistent with the intuition that jumps are "large" and infrequent. More specifically, we choose a prior that places low probability on the jump sizes being small, say less than one percent, and a prior that places low probability on the daily jump probability being greater than 10 percent. In this paper, we generate results with priors.

Next, the forecastability of the SVCJ model is compared to commonly adopted alternative volatility models within the popular GARCH family. For this and to be in line with the stylized facts that financial time series have leptokurtosis, heavy tail, and autocorrelation, we impose volatility dynamics within the universe of GARCH specifications. We choose the TGARCH specification of Glosten et al. (1993) is due to its ability to capture the so-called leverage effect, the tendency of volatility to increase more with negative news rather than positive news. Brownlees and Engle (2012) argued that this volatility model has superior forecasting performance than other known volatility models4. The model takes into consideration any presence of autocorrelation of order *p* and is presented as follows:

<sup>4</sup> Other volatility forecasting models would include ARCH, GARCH, I-GARCH, GARCH-M, GJR-GARCH, and TARCH, for example. However, it is very tough to generalize the statement because results from the above models may vary due to differences in assets, data, and time period under study. See, for example, Ali (2013).

$$\begin{aligned} X\_t &= \quad a\_0 + \sum\_{j=1}^p a\_j X\_{t-1} + u\_t \\ \sigma\_t^2 &= \quad \omega + \alpha u\_{t-1}^2 + \gamma u\_{t-1}^2 I\_{t-1}^- + \beta \sigma\_{t-1}^2 \end{aligned} \tag{3}$$

with *ut* ∼ *<sup>D</sup>*(0, *<sup>σ</sup>*<sup>2</sup> *<sup>t</sup>* ) representing independent and identically distributed shocks with zero mean and time-varying variance, and *I* − *<sup>t</sup>*−<sup>1</sup> <sup>=</sup> 1 if *ut* <sup>&</sup>lt; 0, and zero, otherwise. In this model, the parameters *<sup>α</sup>* and *β* are respectively the ARCH and GARCH coefficients, and the parameter *γ* captures the leverage effect of the returns. In line with the stylized facts observed in the cryptocurrency market (see for example Chan et al. (2017), Caporale and Zeokokh (2019), and Ardia et al. (2019)) and, because there is a large departure of the cryptocurrencies returns from normality, we allow for the distribution *D* of shocks to follow a Student-*t* or skewed Student-*t* with *ν* degrees of freedom.

We explore whether the forecasts generated from the two models are able to provide an investor with a valid tool to hedge risk. Therefore, we derive VaR and ES using the simulated volatility series when fixing the parameter estimates produced by the models. An *n*-day *τ*% VaR is defined as

$$\text{VaR}\_t^\pi(X) = \inf\{\mathbf{x} \mid \Pr(X\_t < -\mathbf{x}) \le \tau\},\tag{4}$$

and, once *X* is below VaR*τ*, we define

$$\mathrm{ES}^{\tau}(X) = \frac{1}{\pi} \int\_{0}^{\tau} \mathrm{VaR}\_{\mathsf{u}}(X) d\mathsf{u}.\tag{5}$$

To concentrate on a specific return bracket, we adopt a non-parametric technique based on Filtered Historical Simulation of Barone-Adesi et al. (1999) to simulate 5000 returns' paths from both the SVCJ and the AR(2)-TGARCH (1,1)∼ *t* models. For the latter, we first standardize returns by quantiles and volatility estimates and then generate returns' paths serving as the basis for calculating VaR and ES.

Next, we evaluate the accuracy of each model through backtesting the estimated VaR and ES. The backtesting relies on comparing the risk measures estimated by the models under analysis with the actual trading results. The cases in which the actual loss exceeds the VaR estimate are called exceptions. According to Christoffersen (1998), the exception sequence is defined as:

$$I\_t^\tau = \begin{cases} 1, \text{if } X\_t < -\text{VaR}\_t^\tau & \text{violation occurs} \\ 0, & \text{otherwise} \end{cases} \tag{6}$$

for *t* = *T* + 1, ... , *T* + *n*, where *T* is the number of return observations used to estimate the VaR of the day *T* + 1, and *n* is the number of one-step-ahead estimates of that risk measure included in the test. Consequently, Christoffersen's conditional coverage test (*LRcc*) for VaR backtesting consists of determining whether the probability of occurrence of an exception, *p* = *Pr*[*Xt* < VaR*<sup>τ</sup> <sup>t</sup>* ] is significantly different from the defined *τ* (unconditional coverage test *LRuc*) and whether the exception sequence is serially independent (independence test *LRind*) 5. The likelihood ratio statistics for the test of correct conditional coverage is defined as:

$$LR\_{cc} = 2\ln\left[ (1 - \pi\_{01})^{n\_{00}} \pi\_{01}^{n\_{01}} (1 - \pi\_{11})^{n\_{10}} \pi\_{11}^{n\_{11}} \right] - 2\ln\left[ (1 - \pi)^{n\_0} \pi^{n\_1} \right] \tag{7}$$

where *n*<sup>0</sup> and *n*<sup>1</sup> are respectively the number of 0s and 1's in the indicator series, *nij* is the number of observations with value *i* followed by value *j* in the *I<sup>τ</sup> <sup>t</sup>* series. The value *i*, *j* = 0 denotes no violation, while *i*, *j* = 1, denotes the opposite. The series *I<sup>τ</sup> <sup>t</sup>* are assumed to be a first-order Markov process

<sup>5</sup> The probability of an exception does not depend on the previous day's outcome.

with transition probabilities *<sup>π</sup>ij* <sup>=</sup> *nij* ∑*<sup>j</sup> nij* 6. The likelihood function *LRcc* follows a *χ*<sup>2</sup> (2) and tests the independence of exceedance (loss) across time periods. If the sequence of losses is independent, then *π*<sup>01</sup> = *π*<sup>11</sup> = *p*. Hence, this test can reject a model that generates too many or too few violations.

Given that VaR passes this test, we then proceed with backtesting the excess loss component, *<sup>L</sup>* <sup>=</sup> ES*<sup>τ</sup>* <sup>−</sup> VaR*τ*, using the McNeil et al. (2005) 'zero mean test' and the bootstrap method of Efron and Tibshirani (1994), which requires no assumption on the distribution of *<sup>S</sup>* = (*<sup>L</sup>* <sup>−</sup> ES*τ*)1*L*>VaR*<sup>τ</sup>* .

Lastly, we test the superiority of a model vis-à-vis a competing model with respect to the loss function of Angelidis et al. (2004) and using Sarma et al. (2003) 'zero median test'. The loss function is defined as:

$$\mathbf{C}\_{t} = \begin{cases} \left(\mathbf{X}\_{t} - \left(-\text{VaR}\_{t}^{\tau}\right)\right)^{2}, & \text{if } \text{violation occurs} \\ \left(q\_{\tau}[\mathbf{X}\_{t}]\_{T+1}^{T+n} - \left(-\text{VaR}\_{t}^{\tau}\right)\right)^{2}, & \text{otherwise} \end{cases} \tag{8}$$

where *qτ*[*Xt*] *<sup>T</sup>*+*<sup>n</sup> <sup>T</sup>*+<sup>1</sup> is the quantile of the out-of-sample returns used for backtesting. At each time *<sup>t</sup>*, *Ct* increases either by excess loss, if a violation occurs, or by the difference between VaR*<sup>τ</sup> <sup>t</sup>* forecast and the future quantile. It follows that choosing the best accurate model *i* over model *j*, which will minimize the total loss ∑*<sup>T</sup> <sup>t</sup>*=<sup>1</sup> *Ct*, can be decided by testing the hypothesis that the median of the distribution *Bt* = *Cit* − *Cjt* is equal to 0. Here, *Bt* is known as the loss differential between model *i* and model *j* at time *t*, and a negative value indicates the superiority of model *i* over *j*. This loss function is of practical interest to investors seeking to reduce market risk and avoiding allocating more money than needed.

## **3. Data and Empirical Results**

In this section, we describe the details of the procedures for the comparison of the previously discussed risk models for the matter of validation, and, for a better understanding of our results, we divide this section in three parts. In the first part, we describe the stylized facts of the sample and conduct preliminary diagnostics. The second part presents the details of the in-sample estimation of the risk models, namely SVCJ, TGARCH, and RM. In the third part, we evaluate the out-of-sample forecasting ability of the models in terms of VaR and ES, and then perform backtesting for validation purposes.

#### *3.1. Data*

Over the last few years, the most important aspect of cryptocurrencies which has gained prominence in the media is the realized market volatility. To be fair, the media's infatuation with cryptocurrencies is manifested in the actual market data. Between 26 April 2013 and 16 May 2019, the daily average return from the largest cryptocurrency Bitcoin (BTC) was 0.3% with 4.34% standard deviation. There were 174 days with daily returns falling by more than 5%, and 178 days with daily returns increasing by more than 5%. The maximum daily return during this period was 43.58% (19 November 2013) and the largest one-day change was –23.43% (12 December 2013). On 18 December 2017, the market cap for BTC was \$320 billion and the price soared to \$19,783 (17 December 2017). One year later, the market cap for the currency declined to \$63 billion (28 December 2018). As of this writing (23 May 2019), BTC had a market cap of \$138.5 billion. Such large, unprecedented swings in the market value can be terrifying for some investors, while others see opportunities. In more recent days, however, there is a lot more emphasis on avoiding volatility and promoting the stability of the cryptocurrencies to bring some sense of calm in the market. For example, companies like Google, IBM, and Facebook<sup>7</sup> have announced their plans to introduce newer coins and each one is claiming that their currency will be a more stable asset than the others (Forbes, 16 April 2019).

<sup>6</sup> *π*<sup>01</sup> = *Pr*[*I<sup>τ</sup> <sup>t</sup>* = <sup>1</sup> | *<sup>I</sup><sup>τ</sup> <sup>t</sup>*+<sup>1</sup> = 0), and *<sup>π</sup>*<sup>11</sup> = *Pr*[*I<sup>τ</sup> <sup>t</sup>* = <sup>1</sup> | *<sup>I</sup><sup>τ</sup>*

*<sup>t</sup>*+<sup>1</sup> <sup>=</sup> <sup>1</sup>]. <sup>7</sup> In fact, Facebook is planning to introduce a cryptocurrency, appropriately named as 'Stablecoin' for its "WhatsApp" platform.

We use daily prices of seven successful8 cryptocurrencies: Bitcoin (BTC), Ripple (XRP), Litecoin (LTC), Stellar (XLM), Monero (XMR), Dash (DASH), and Bytecoin (BCN), all collected from cryptocompare.com9. The data span the period 5 August 2014 to 24 March 2019, with a total of 1693 daily observations. Table 1 reports the summary statistics including the mean, standard deviation, minimum, maximum, skewness, kurtosis, and the *p*-values of the Ljung–Box test for first-order autocorrelation for all cryptocurrencies. Ripple has the highest mean of 0.24% and Bytecoin has the highest standard deviation of 11.44%. All cryptocurrencies display excess kurtosis and the Ljung–Box test shows that data exhibit first and second-order autocorrelation except for Stellar at the 5% confidence level.


**Table 1.** Descriptive statistics of daily log-returns of cryptocurrencies.

Data spans from 5 August 2014 until 24 March 2019. AR1 and AR2 display the *p*-values of the Ljung–Box for autocorrelation of first and second order. *p*-values below the 1% significance level indicate rejection of the null hypothesis of no autocorrelation.
