Pricing Cryptocurrency Options

Cryptocurrencies, especially Bitcoin (BTC), which comprise a new digital asset class, have drawn extraordinary worldwide attention. The characteristics of the cryptocurrency/BTC include a high level of speculation, extreme volatility and price discontinuity. We propose a pricing mechanism based on a stochastic volatility with a correlated jump (SVCJ) model and compare it to a flexible co-jump model by Bandi and Ren\`o (2016). The estimation results of both models confirm the impact of jumps and co-jumps on options obtained via simulation and an analysis of the implied volatility curve. We show that a sizeable proportion of price jumps are significantly and contemporaneously anti-correlated with jumps in volatility. Our study comprises pioneering research on pricing BTC options. We show how the proposed pricing mechanism underlines the importance of jumps in cryptocurrency markets.


Introduction
Bitcoin (BTC), the network-based decentralized digital currency and payment system, has garnered worldwide attention and interest since it was first introduced in 2009. The rapidly growing research related to BTC shows a prominent role of this new digital asset class in contemporary financial markets. 1 Several studies have suggested econometric methods to model the dynamics of BTC prices, including cross-sectional regression models involving the major traded cryptocurrencies and also multivariate time-series models. 2 Scaillet et al. (2019) show that jumps are much more frequent in the BTC market than, for example, in the US equity market (see e.g., Bajgrowicz et al. (2015), Eraker (2004), Bandi and Renò (2016) and among others).These earlier studies suggest that jumps should be considered when modeling BTC prices. 1 see e.g., Becker et al. (2013), Segendorf (2014), Dwyer (2015), also studies on economics (Kroll et al., 2013), alternative monetary systems (Rogojam (2014) and Weber (2016)) and financial stability (Ali, 2014;Badev, 2014;ECB, 2015). An analysis of the legal issues involved in using Bitcoin can be found in Elwell et al. (2013). 2 For example, Hayes (2017) performs a regression using a cross-section dataset consisting of 66 traded digital currencies to understand the price driver of cryptocurrencies. Kristoufek (2013) proposes a bivariate Vector-AutoRegression (VAR) model for the weekly log returns of Bitcoin prices. Bouoiyour (2019) investigates the long and short-run relationships between BTC prices and other related variables using an autoregressive distributed lag model. However, research on the BTC derivative markets is still limited despite the rapidly growing availability of BTC futures and options traded on an unregulated exchange platform (i.e., Deribit). Especially, the CME (Chicago Mercantile Exchange) Group, the world's leading derivatives marketplace, launched BTC futures based on the CME CF Bitcoin Reference Rate (BRR) on 18 December 2017. The limited research on pricing and hedging Bitcoin derivatives is partly attributed to the fact that pricing BTC derivatives (e.g., options) encounters econometric challenges from the extraordinary occurrence of jumps as this market is unregulated, lacks of central settlement and is highly speculation-driven. This calls for a more flexible model to capture the sudden jumps appearing in both the returns and variance processes.
In this paper, we contribute to the existing literature by exploring the stochastic and econometric properties of BTC dynamics and then pricing the BTC options based on these properties. The investigation is carried out by using the most advanced stochastic volatility models, i.e., the stochastic volatility with the correlated jump (SVCJ) model of Duffie et al. (2000) and the stochastic volatility with the possible non-linearity structure of Bandi and Renò (2016) (BR hereafter). The employed SVCJ model incorporates jumps in both returns and the stochastic volatility process, while the BR model captures the possible non-linearity of return and variance processes and characterizes a non-affine structure. We aim to provide a theoretical foundation for the future development of derivative markets on cryptocurrencies.
Numerous empirical studies have applied the SVCJ model in different markets. For example, Eraker et al. (2003) and Eraker (2004) use the SVCJ model to describe equity market returns and estimate equity option pricing. They find strong evidence of jumps in returns and volatility in the US equity market. We further compare the SVCJ estimates to the simplified versions such as Bates (2000) (SVJ hereafter) and the stochastic volatility (SV) model.
For a purpose of robustness check, we compare our results with those from the BR model. Bandi and Renò (2016) propose a price and variance co-jump model that generalizes the SVCJ model to capture the possible nonlinearity in the parameters of the returns and variance processes.
The BR model characterizes independent and correlated jumps and allows for a nonparamet-ric parameter structure, and estimates the parameters by using high-frequency data. We also apply this model to the dynamics of BTC. We base our option pricing on an experimental simulation where the parameters used to execute a simulation are from the SVCJ and BR model, respectively.
We summarize our main empirical findings as follows. First, as in the existing literature, the results from the SVCJ and BR models indicate that jumps are present in the returns and variance processes and adding jumps to the returns and volatility improves the goodness of fit. Second, in contrast to existing studies that commonly report a negative leverage effect, we find that the correlation between the return and volatility is significantly positive in the SVCJ model. However, we cannot find significant negative relations between risk and return in the BR model. This implies that a rise of price is not associated with a decrease in volatility, which is consistent with the "inverse leverage effect" found in the commodity markets (Schwartz and Trolled, 2009).
Third, we find that the jump size in the return and variance of BTC is anti-correlated. The parameter estimates of the jump size (ρ j ) from both the SVCJ and BR models are negative (though the SVCJ estimate is insignificant). It is worth noting that the correlation between the price jump size and the volatility jump size turns out to be significant with a negative coefficient with high-frequency data, while tending to be insignificant for the SVCJ fitting using daily prices. This finding is in line with existing studies of the stock market from Eraker (2004), Duffie et al. (2000) and Bandi and Renò (2016), among others. For example, Bandi and Renò (2016) report an anti-correlation with the nonaffine structure. Eraker (2004) finds a negative correlation between jump size only when augmenting return data with options data, and the negative correlation between co-jump size being identified in the implied volatility smirk. Using high-frequency data, Jocod and Todorov (2009) and Todorov and Tauchen (2010) also report that the large jump size of prices and volatility are strongly anti-correlated.
Finally, we observe that the option price level is prominently dominated by the level of volatility and therefore overwhelmingly affected by jumps in the volatility processes. The results from the plots of implied volatility (IV) indicate that adding jumps in the return increases the slope of the IV curves. The greater steepness of the IV curve can be strengthened by the presence of jumps in volatility. The presence of co-jumps enlarges the IV smile further. As evidenced from the IVs curve, options with a short time to maturity are more sensitive to jumps and co-jumps. To fulfill a hedge or speculation need from institutional investors, we replicate the entire analysis for the CRyptocurrency IndeX (CRIX), a market portfolio comprising leading cryptocurrencies (see more detail in www.thecrix.de). A recent volatility index, VCRIX, created by Kim et al. (2019) also shows the evidence of jumps in CRIX.
To summarize our contributions, this study is the first paper to extensively investigate the stochastic and econometric properties of BTC and incorporate these properties in the BTC options pricing. Our results have practical relevance in terms of model selection for characterizing the BTC dynamics. We document the necessity of incorporating jumps in the returns and volatility processes of BTC, and we find that jumps play a critical role in the option prices.
Our approach is readily applicable to pricing BTC options in reality. Our results are also important for policymakers to design appropriate regulations for trading BTC derivatives and for institutional investors to launch effective risk management and efficient portfolio strategies.
The paper is organized as follows. Section 2 briefly introduces the BTC market. Section 3 studies the BTC return and variance dynamics with the SV, SVJ and SVCJ models. Fitting of the BR model is investigated in Section 4. Section 5 implements the option-pricing exercises.
Section 6 documents an examination of the CRIX, while Section 7 concludes the study. A few preliminary econometrics analysis and estimation results for the CRIX are in the Appendix.
The codes for this research can be found in www.quantlet.de.

The BTC dynamic
We start by briefly introducing BTC. BTC was the first open source distributed cryptocurrency released in 2009, after it was introduced in a paper "Bitcoin: A Peer-to-Peer Electronic Cash System" by a developer under the pseudonym Satoshi Nakamoto. It is a digital, decentralized, partially anonymous currency, not backed by any government or other legal entity. The system has a pre-programmed money supply that grows at a decreasing rate until reaching a fixed limit.
Since all is based on open source, the design and control is open for all. Traditional currencies are managed by a central bank, while BTCs are not regulated by any authority; instead, they are maintained by a decentralized community. The transactions of bitcoins are recorded in the ledgers (known as the blockchain), which is maintained by a network of computers (called 'miners'). Since bitcoin is not a country-specific currency, international payments can be carried out more economically and efficiently.
Our empirical analyses are carried out based on both daily closing (SVCJ model) prices and five minutes intra-daily (BR model) prices. The data cover the period from 1 August 2014 to 29 September 2017 and are collected from Bloomberg. The dynamics of BTC daily prices (left panel) and BTC returns (right panel) are depicted in Figure 1. It shows that the BTC return is clearly more volatile than the stock return, along with more frequent jumps or the scattered volatility spikes. Bitcoin's price spent most of the year 2015 relatively stable. The BTC price in the first four months of 2016 was in the range of 400-460 USD. It moved upward dramatically after 2016 and increased to almost 5000 USD by the end of our sample period in 2017. At the time of the writing of this paper, the BTC market capitalization is more than USD 7 billions (source: Coinmarketcap 2017).
Both the BTC prices and returns react to big events in the BTC market. A dramatic surge observed after March 2017 was due to the widespread interest in cryptocurrencies (CCs). The subsequent drop in June 2017 was caused by a sequence of political interventions. Several governmental announcements of bans on initial coin offerings (ICOs) have spurred intensive movements on CC markets. For example, the Chinese SEC (Securities and Exchange Commission) denied permission for a bitcoin ETF on March 10, 2017; and Bitcoin crashed down after China banned initial coin offerings on September 4, 2017. The large upward movements in BTC prices caused the returns of BTC displaying extremely high volatility and with scattered spikes/jumps. Several large jumps triggered by a series of big events in the BTC market can be detected from the returns series, see also Kim et al. (2019). We have implemented a number of time series models to the BTC returns and the results are shown in Appendix 8.1 and Appendix 8.2 . We find that the standard set of stationary models, such as ARIMA and GARCH, cannot fit the BTC returns well due to the presence of jumps. Notes: This figure graphs the BTC daily price (left panel) from 01/08/2014 to 29/09/2017 and BTC returns (right panel). The returns (R t ) are calculated as R t = log(P t ) − log(P t−1 ), where P t is the BTC price at time t.

SVCJ: affine specification
In this section, we estimate the SVCJ model using BTC prices. We begin with a simple SVCJ jump specification, and switch to the BR model in Section 4. We focus the analysis on BTC and then introduce CRIX in Section 6.

Models
In order to estimate the BTC dynamics with the SV and SVCJ models regarding returns and volatility, we employ the continuous time model of Duffie et al. (2000) that encompasses the standard jump diffusion and the SV with jumps in returns only (SVJ) model of Bates (1996).
More precisely, let {S t } be the price process, {d log S t } the log returns and {V t } be the volatility process. The SVCJ dynamics are as follows: Like in the Cox-Ingersoll-Ross model, κ and θ are the mean reversion rate and mean reversion level, respectively. W (S) and W (V ) are two correlated standard Brownian motions with correlation denoted as ρ. N t is a pure jump process with a constant mean jump-arrival rate λ.
The random jump sizes are Z y t and Z v t . Since the jump-driving Poisson process is the same in both (1), (2), the jump sizes can be correlated. The random jump size Z y t conditional on Z v t , is assumed to have a Gaussian distribution with a mean of µ y + ρ j Z v t and standard deviation set to σ y . The jump in volatility Z v t is assumed to follow an exponential distribution with mean µ v : The correlation ρ between the diffusion terms is introduced to capture the possible leverage effects between returns and volatility. The jumps may be correlated as well. The correlation term ρ j takes care of that. The SV process √ V t is modelled as a square root process. With no jumps in the volatility, the parameter θ is the long-run mean of V t , and the process reverts to this level at a speed governed by the parameter κ. The parameter σ V is referred to as the volatility of volatility, and it measures the variance responsiveness to diffusive volatility shocks. In the absence of jumps, the parameter µ measures the expected log-return.
SVCJ is a rich model since it encompasses the SV and SVJ approaches. If we set Z v t = 0 in (5), then jumps are only present in prices, we obtain the SVJ model of Bates (1996). Taking λ = 0 such that jumps are not present, the model reduces to the pure SV model originally proposed by Heston (1993). If we set κ = θ = σ V = 0 and define Z v t = 0, the model reduces to the pure jump diffusion introduced in Merton (1976).

Estimation: Markov Chain Monte Carlo (MCMC)
There are plenty of different methods to estimate the diffusion process to real data. The generality of simulation-based methods offers obvious advantages over the method of simulated moments of Duffie and Singleton (1993), the indirect inference methods of Gourieroux et al. (1993) and the efficient method of moment (EMM) method of Gallant and Tauchen (1996). For example, Jacquier et al. (1994) show that MCMC is particularly well suited to deal with SV models. Eraker et al. (2003) and Eraker (2004) identify several advantages of using the MCMC approach over other estimation models because MCMC methods are computationally efficient and the estimating is more flexible when using simulations. The MCMC method also provides more accurate estimates of latent volatility, jump sizes, jump times, etc. A general discussion and review of the MCMC estimation of continuous-time models can be found in Johannes and Polson (2009).
For the reasons discussed above, we estimate the SVCJ model using the MCMC method. Doing this allows for a wide class of numerical fitting procedures that can be steered by a variation of the priors. Given that there are no BTC options yet, the MCMC method is more flexible in estimating the stochastic variance jumps and thus able to reflect the market price of risk (Franke et al. (2019). The estimation is based on the following Euler discretization: Let us present a brief description on how to estimate the SVCJ model with MCMC (see also Johannes and Polson (2009), Tsay (2005) and Asgharian and Bengtsson (2006) for more details).
Define the parameter vector as as the latent variance, jump sizes and jump. Recall that Y t is the log-returns.
The MCMC method treats all components of Θ and X def = {X t } t=1,..,T as random variables. The fundamental quantity is the joint pdf p(Θ, X|Y ) of parameters and latent variables conditioned on data using the Bayes formula: The Bayes formula can be decomposed into three factors: p(Y |Θ, X), the likelihood of the data, p(X|Θ) the prior of the latent variables conditioned on the parameters and p(Θ) the prior of the parameters. The prior distribution p(Θ) has to be specified beforehand and is part of the model specification. In comfortable settings, the posterior variation of the parameters, given the data, is robust with respect to the prior.
The posterior is typically not available in the closed form, and therefore simulation is used to obtain random draws from it. This is done by generating a sequence of draws, which form a Markov chain whose equilibrium distribution equals the posterior distribution.
The point estimates of parameters and latent variables are then taken from their sample means.
We use the same priors specified in Asgharian and Nossman (2011), who estimate a large group of international equity market returns with jump-diffusion models using the MCMC Be(2, 40) (Beta Distribution). The full posterior distributions of the parameters and the latentstate variables can be found in Asgharian and Nossman (2011) and Asgharian and Bengtsson (2006). We have varied the variance of the priors and found stable outcomes, i.e., the reported mean of the posterior that is taken as an estimate of Θ is quite robust relative to changes in variance of the prior distributions. The posterior for all parameters except σ V and ρ are all conjugate (meaning that the posterior distribution is of the same type of distribution as the prior but with different parameters). The posterior for J t is a Bernoulli distribution. The jump sizes Z y t and Z v t follow a posterior normal distribution and a truncated normal distribution, respectively. Hence, it is straightforward to obtain draws for the joint distribution of J t , Z y t and Z v t .
However, the posteriors for ρ, σ V 2 and V t are nonstandard distributions and must be sampled using the Metropolis-Hastings algorithm. We use the random-walk method for ρ and V t , and independence sampling for σ 2 V . For the estimation of posterior moments, we perform 5000 iteratations, and in order to reduce the impact of the starting values, we allow for a burn-in for the first 1000 simulations.
The SVCJ model is known for being able to disentangle returns related to sudden unexpected jumps from large diffusive returns caused by periods of high volatility. For the BTC situation that we consider here, we are particularly interested in linking the latent historical jump times to news and known interventions.
number of iterations and i refers to each draw) indicate the posterior probability that there is a jump at time t. Unlike the "true" vector of jump times, it will not be a vector of ones and zero.
Following Johannes et al. (1999), we assert that a jump has occured on a specific date t if the estimated jump probability is sufficiently large, that is, greater than an approporiately chosen threshold value:J t = 1{Ĵ t > ζ}, t = 1, 2, ..., T In our empirical study, we choose ζ so that the number of inferred jump times divided by the number of observations is approximately equal to the estimate of λ.
We first estimate the BTC returns by taking the log first differences of prices, then use returns to estimate the SVCJ model. The parameter estimates (mean and variance of the posterior) of the SVCJ, SVJ and SV models for BTC are presented in Table 1. The estimate of µ is positive.
The correlation between returns and volatility ρ is significant and positive. This is remarkable and worth noting since it is different from a negative leverage effect observed over a sequence of studies in stock markets (see, e.g., French and Stambaugh (1987) and Schwert (1989)). The effect is named the "inverse leverage effect" and has been discovered in commodity markets (see Schwartz and Trolled (2009)). In other words, the "inverse leverage effect" (associated with a positive ρ) implies that increasing prices are associated with increasing volatility. The reason for this positive relationship between risk and returns might be due to BTC prices being different from conventional stock prices. The digital currency price may be dominated by the "noise trader" behavior described by Kyle (1985) and DeLong et al. (1990). Such investors, with no access to inside information, irrationally act on noise as if it were information that would give them an edge. This positive leverage effect has been also reported by such as Hou (2013) on other highly speculative markets, e.g., the Chinese stock markets.
Moreover, the estimates for the SVCJ model are much less extreme than for the SVJ and SVCJ models. More precisely, the volatility of variance σ v is substantially reduced from 0.017 (SV) to 0.011 (SVJ) and 0.008 (SVCJ). The mean of the jump size of the volatility µ v is significant and positive. The jump intensity λ is also significant. The jump correlation ρ j is negative but insignificant, which parallels the results of Eraker et al. (2003) and Chernov et al. (2003) for stock price dynamics. This effect might be due to the fact that even with a long data history, jumps are rare events. (The evidence is stronger for the BR specifications considered in Section 4.) In summary, the SVCJ model fits the data well by an MSE that is smaller than those of the SVJ and SV models. Figure 2 shows the estimated jumps in returns (first row) and the estimated jumps in volatility (middle row) together with the estimated volatility (last row). One sees that estimated jumps occur frequently for those of the returns and volatility. The estimated jumps size in returns and variance are different. Figure 3 presents the in-sample fitted volatility processes for the SVCJ and SVJ models, respectively. It is not hard to see that both models lead to a similar overall pattern for the volatility process, though the SVCJ model produces sharper peaks for BTC.
A useful model diagnosis is to examine the standardized residuals obtained from the discrete Notes: This table reports posterior means and 95% finite sample credibility intervals (in square brackets) for parameters of the SVCJ, SVJ, and SV models. All parameters are estimated using BTC daily returns calculated as the log-first difference based on the prices from 01/08/2014 to 29/09/2017. model, which estimates, The normality would be violated if the jumps are not perfectly estimated. However, several previous researches such as Larsson and Nossman (2011), Asgharian and Bengtsson (2006) and Asgharian and Nossman (2011) have estimated the SVCJ model with the MCMC in the equity market and use the normal plot as a diagnostic tool to visualize the model performance. We follow these literature calculating these standardized residuals based on the estimated parameters, then show the QQ plots of the standardized residuals from the fitting of different models in Figure 4. From these diagnostics, it is evident that the GARCH and even the SV models are misspecified. For the SVJ and SVCJ models, the QQ plot diagnostics are substantially improved. However, it is apparent that the SVCJ model is the preferred choice which is consistent with the MSE reported in Table 1.   Notes: This figure graphs the QQ plots versus standard normal for fitted standardized residuals from the SVCJ, SVJ and SV models using BTC daily returns calculated as the log-first difference based on the prices from 01/08/2014 to 29/09/2017. We also include the QQ plot for the GARCH model using the same sample period.
4 SV model with jumps: high frequency data 4.1 BR model in return-volatility co-jumps Imposing a specific structure in the stochastic process as documented in Section 3 may produce a specification error. Defining S t and σ t = √ V t as the price and volatility process, respectively, following the notation of BR, we therefore consider the BR affine jump-diffusion model: where ξ(·) is an increasingly monotonic function (we will choose it as log(·) in the following discussions), W = {W 1 , W 2 } is a bivariate standard Brownian motion vector and J = {J r,t , J σ,t , J r,σ,t } is a vector of mutually independent Poisson processes with constant intensities, which are denoted as λ r , λ σ and λ r,σ , respectively. Thus we allow for common and independent jumps in the system. The Poisson processes are also assumed to be independent from the Brownian motion.
The BR model is estimated through a GMM-like procedure based on infinitesimal cross-moments dubbed by the authors NIMM or Nonparametric Infinitesimal Method of Moments. We assume the distribution of the jumps to be normal, For any p 1 ≥ p 2 ≥ 0, the generic infinitesimal cross-moment of order p 1 and p 2 is defined as: In particular θ p 1 ,0 helps to identify features of the price process, and θ 0,p 2 helps to identify those of the variance process, while the genuine cross-moments with p 1 ≥ p 2 ≥ 1 are required to identify the common parameter shared by the two processes ρ 0 , ρ 1 , λ r,σ and ρ J .
To conduct the NIMM estimation in BR, we first need to estimate the cross-moments that are in theory functions of parameter of interest. The cross-moments are estimated via a nonparametric kernel method. In particular, denote the day index as t = 1, ..., T and the equispaced time index as i = 1, ..., N within each day. Denote r t,i,k as the high-frequency log returns for day t, knot i and minute k. We define the closing logarithmic prices as log(p t,i ) and logarithmic spot variance estimates asσ where ζ 1 ≈ 0.7979, θ t,i,k is a suitable threshold, and n j is the number of returns whose absolute value is greater than θ t,i,k . Then the generic cross-moment estimatorθ p 1 ,p 2 (σ) is defined aŝ where K(·) is a kernel function and h is the bandwidth. Finally, with the estimated crossmoments, one can estimate the parameters of interest via the NIMM method, see the details as in Bandi and Renò (2016) for the parametric estimation.

Correspondence between SVCJ and BR model
In this section, we fit the BR model using high-frequency data and discuss the comparison with the estimation of the SVCJ model. We collect high-frequency BTC prices from Bloomberg. The price data range is from 31/07/2014 to 29/07/2017, and we collect raw data at a frequency of 60 seconds 24 hours a day. Following Section 4.1, we aggregate the logarithm returns of Bitcoin over a 60-minute time range, namely r t,i,k = log S t,i,k − log S t,i,k−1 , with k = 1, · · · , 60. In addition, we also obtain the spot variance estimates for each day t and each knot i by applying the jump robust threshold bipower variation estimator as in Equation (14).
To compare the data of the high-frequency aggregated volatility and the daily Bitcoin volatility, we plot the averaged daily spot volatility from the high-frequency data and the daily spot volatility estimates from the SVCJ model together as in Figure 5. We observe that the two sequences sometimes peak at different time points despite that the general pattern agrees.
In Table 2 Both parameters are insignificant at the 95% level of confidence. We do not see an obvious tendency for the jumps to be downward, as observed in Bandi and Renò (2016).
We find that the leverage ρ 0 is estimated to be negative, i.e., −0.1485, though insignificant. The leverage would increase with an increasing volatility level as ρ 1 is estimated to be significant and with a value of 0.9292. The standard deviation of the jumps in return σ J,r is estimated to be significant with the value of 0.6890. When fitting a nonlinear structure to the standard deviation of the common price jumps, the parameters σ JJ,r,1 and σ JJ,r,2 are both significant. The standard deviation of jumps in volatility σ J,σ is estimated to be 0.8619 with significance. The standard deviation of the common volatility jump σ JJ,σ is estimated to be insignificant. Notably, the correlation of jumps ρ J is estimated to be negative and significant with a value of −0.5257, which is in line with BR. This negative and significant co-jump size correlation is discovered by Duffie et al. (2000), who conclude that the price and the volatility jump sizes are "nearly perfectly anti-correlated". Eraker (2004) finds a statistically significant correlation between the jump sizes only when employing option data in addition to stock returns data. Bandi and Renò (2016) also report a "nearly perfect anti-correlation" of -1.

Option pricing
In the previous sections, we have shown that the SVCJ and the the BR models can well describe the log returns dynamics of BTC. In this section, we discuss option pricing for BTC based on the SVCJ and BR models, respectively.

BTC options
After fitting the SVCJ and the BR model, we advance with a numerical technique called Crude Monte Carlo (CMC) to approximate the BTC option prices. Derivative securities such as futures and options are priced under a probability measure Q commonly referred to as the "risk neutral" or martingale measure. Since our purpose is to explore the impact of model choice on option prices, we follow Eraker et al. (2003) and set the risk premia to zero. This choice can be disputed, but for the lack of existence of the officially traded options a justifiable path to pricing BTC contingent claims. Suppose we have an option with a payoff at time of maturity T as C(T ), and typically for call option C(T ) = (S T − K) + . The price of this option at time t is denoted as: where F t is a set that represents information up to time t. We approximate the European option prices of BTC using the CMC technique. The CMC simulation is done for 20000 iterations to approximate the option price using the parameters reported in Table 1 for the SVCJ, SVJ, and SV models and in Table 2 for the BR (assuming a daily interval) model. Since no BTC option market exists yet, we do not have real market option prices for comparison. Thus, we chose July 2017 randomly as the experimental month in our option-pricing simulation analysis. -12 using the intra-daily BTC returns. For each parameter, we report the estimate and the corresponding 95% finite sample credibility intervals in parentheses. The full model is shown in the forth column, and the second and third columns report the same model with the restriction of no co-jumps and no independent jumps, respectively.
Throughout our entire analysis of option pricing, the moneyness for strike K and S at t is defined to be K/S t . The pricing formula is a function of moneyness and time to maturity τ = (T − t) where T is the maturity day.
In Figure 6, we plot the simulated volatility of various models based on the parameters reported in Table 1 (for the SVCJ, SVJ and SV models) and in  Table 3. We see   Table 1 and Table 2 for the month of July 2017. The x-axis notes the dates in July 2017. The blue/red/orange/purple line plots the volatility from the SVJ/SVCJ/SV/BR models. Figure 8 displays the estimated BTC call option price differences between the SVCJ and SVJ models with respect to changes in moneyness and across time to maturity for July 2017. It is not hard to see that the pattern is similar to the fitted volatility shown in Figure 6. The difference between the SVCJ and the SVJ model is similar besides on July 15 when there is a large spike in the estimated volatility. Therefore, the price differences between the SVCJ and SVJ models are mainly caused by the jumps in the volatility process and the volatility level, which reflects the necessity of adopting the SVCJ model in practice.

BTC implied volatility smiles
It is well known that stochastic volatility determines excess kurtosis in the conditional distribution of returns. The excess kurtosis causes symmetrically higher implied Black Scholes volatility when strikes are away from the current prices, e.g., the level of moneyness is away from the ATM level. This phenomenon is called the "volatility smile". It is well documented in the existing literature that the effect is stronger for short and medium maturity options than Notes: This figure graphs the call option prices surface counter plot across different moneyness and different times to maturities for the month of July 2017, as shown in the right-hand side labels. When looking at moneyness, the time to maturity is fixed at 30 days, and when looking at the time to maturity, moneyness is ATM. The colour in the graph represents the price level; the brighter the colour, the higher the price.

Figure 8: Call option price differences between the SVCJ and SVJ models: BTC
Notes: This figure plots the option price differences between the SVCJ and SVJ models for July 2017. When looking at moneyness, the time to maturity is fixed at 30 days, and when looking at the time to maturity, moneyness is ATM. The colour in the graph represents the price difference level; the brighter the colour, the larger the difference between the price from the SVCJ and SVJ models. for long maturity options for which the conditional returns are closer to normal (Das and Sundaram (1999)). The presence of co-jumps, and the negative correlation between the presence of co-jumps sizes yield additional sources of skewness in the conditional distribution of stock returns (Bandi and Renò (2016)).
To further examine the option-pricing property of BTC, we approximate the implied Black Scholes volatility from various models for different degrees of moneyness (strike/spot) and different times to maturity. First, the European call option prices are simulated using the model parameters reported in Table 1 for the SVCJ, SVJ and SV models and Table 2 for the BR model.
Then the volatility from various models is implied from the Black Scholes model based on the options approximated from different models. We consider four times to maturity: one week, one month, three months and one year. We report the implied volatility surface as a function of moneyness and time to maturity. The results indicate that jumps in returns and volatility include important differences in the shape of the implied volatility (IV) curves, especially for the short maturities options. Figure 9 shows the IV curves for the SVCJ, SVJ and SV models for four different maturities and across moneyness. It can be seen from Figure 9, that adding jumps in returns steepens the slope of the IV curves. Jumps in volatility further steepen the IV curves. For short maturity options, the difference between the SVCJ, SVJ and SV models for far ITM options is quite large, with the SVCJ model giving the sharpest skewness among the three models. The difference between the SVCJ and SV volatility is approximately 2-3% for up to one month. All three models have a one-side volatility skewness. This could be due to the skewness in the conditional distribution of BTC returns (Das and Sundaram (1999)) and/or that the negative co-jump size yields an additional source of skewness (Bandi and Renò (2016)). As time to maturity increases, the volatility curve flattens for all models. According to Das and Sundaram (1999), jumps in returns result in a discrete mixture of normal distributions for returns, which easily generates unconditional and conditional non-normalities over short frequencies such as daily or weekly.
Over longer intervals, e.g., more than a month, a central-limit effect results in decreases in the amount of excess and kurtosis. Indeed, diffusive stochastic volatility models may generate very flat curves, such as a flat BTC IV for the three-month and the one-year times to maturity.
However, for the SVCJ model, the curve flattens at a slightly higher level. The implied volatility of the SVJ model is closer to the SVCJ model than the SV model. The difference between the SVCJ, SVJ and SV models becomes larger with short time to maturity options, i.e., the oneweek and one-month times to maturity. Similar results have been documented in other studies in which these models have been applied to equity index data. Eraker et al. (2003), Eraker (2004 and Duffie et al. (2000) find that jumps in returns and variance are important in capturing systematic variations in Black-Scholes volatility. In general, although the BTC market has the unique feature of having more jumps, which makes it different from other mature markets (e.g., equity), the option prices and the IV from the affine models generally follow the conventional characteristics reported from other option markets.
We have also estimated the BR IVs with the same time to maturity and moneyness used for the SVCJ IVs. We simulate the option prices using the model parameters reported in column jump sizes equaling -0.5257 or a correlation between jump sizes equaling to zero. The IVs as a function of moneyness from the BR model are plotted in Figure 10. We can see that the IVs of the BR model agree with the SVCJ model. We see a one-side volatility skewness, i.e., the ITM call option prices are higher than the OTM call options. However, due to the significantly negative jump-size correlation ρ J , the slope of the IVs from the BR full model is steeper than the BR model with a case of uncorrelated jump sizes. The impact of the negative jump size correlation is stronger for short time to maturity options, i.e., the one-week and one-month times to maturity. This is mentioned in the results of Duffie et al. (2000) as well, who find a superior fit of the IV smirk when calibrating a more negative correlation between jump sizes.
Similarly, Eraker (2004) finds a statistically significant correlation between jump size only when employing option data in addition to returns data. Bandi and Renò (2016) also shows that anticorrelated jump sizes are a fundamental property of prices and volatility. However, the use of high-frequency data is sufficient to reveal this property with no further need for option data.  Table 2. The full model uses parameters from column 4 of Table 2. A co-jumps correlation of 0 means that ρ J is set to zero while the other parameters remain the same as in the full model.

The CRyptocurrency IndeX (CRIX)
The CRyptocurrency IndeX, a value-weighted cryptocurrency market index with an endogenously determined number of constituents using some statistical criteria, is described in Härdle and Trimborn (2015) and further sharpened in Trimborn and Härdle (2018). It is constructed to track the entire cryptocurrency market performance as closely as possible. The representativity and the tracking performance can be assured as CRIX considers a frequently changing market structure. The reallocation of the CRIX happens on a monthly and quarterly basis (see Trimborn and Härdle (2018)  Therefore we perform an analysis for CRIX. All econometric models have been estimated with the CRIX data. We summarize our major findings here and place the supplementary parts in the appendix. In brief, all the model parameters estimated with CRIX convey a similar configuration as estimated with BTC, e.g., the mean jump size of the CRIX volatility process reported in Table 7 is 0.709, which is 0.620 for BTC shown in Table 1. The estimated volatility from the SVCJ and SVJ models (see Figure 14) shows that the jumps are better captured by the SVCJ than the SVJ model. In addition, Figure 15 displays the call option prices surface contour plot from the SVJ, SV and SVCJ models with respect to changes in moneyness and time to maturity. It shows that the SVCJ model has more volatile patterns than those of the SVJ and SV models with the BTC options. In general, we confirm the consistency between BTC and the CRIX.

Conclusion
"The Internet is among the few things that humans have built that they do not truly understand" according to Schmidt and Cohen (2017). Cryptocurrency, a kind of innovative internet-based asset, brings new challenges but also new ways of thinking for economists, cliometricians and financial specialists. Unlike classic financial markets, the BTC market has a unique market microstructure created by a set of opaque, unregulated, decentralized and highly speculation driven markets.
This study provides a way of pricing cryptocurrency derivatives using advanced option-pricing models such as the SVCJ and BR models. We find that in general, the SVCJ model performs as well as the non-affine BR model. We especially find that the correlation between the jump sizes in returns and the volatility process is anti-correlated. The jump-size correlation is statistically (marginally) negative in the BR (SVCJ) model. Deviating from the equity market, we cannot obtain a significant negative "leverage effect" parameter ρ, which implies a nonnegative relation between returns and volatility. The reason for this relationship might be that BTC is different from the conventional stock market, not only because the BTC market is highly unregulated but also due to the fact that the BTC price is not informative (as there are no fundamentals allowing the BTC market to set a "fair" price) and is driven by emotion and sentiment. This speculative behaviour can be explained by the "noise trader" theory from Kyle (1985). The positive relation might result from the fact that BTC investors irrationally act on noise as if it were information that would give them an edge.
We find that option prices are very much driven by jumps in the returns and volatility processes and co-jumps between the returns and volatility. This can be seen from the shape of the IV curves. This study provides a grounding base, or an anchor, for future studies which aim to price cryptocurrency derivatives. This study provides useful information for establishing an options market for BTC in the near future.

Appendix
We provide preliminary fit results of econometric models on the Bitcoin time series. We also collect results on analysis of the CRIX.

ARIMA
We first fit an ARIMA model. After an inspection through the ACF and PACF plot in Figure   11, we start with an ARIMA(p, d, q) model, where y t is the variable of interest, ∆y t = y t − y t−1 , L is the lag operator and ε t a stationary error term. Model selection criteria such as AIC or BIC indicates that the ARIMA(2, 0, 2) is the model of choice. The parameters estimated from the ARIMA(2,0,2) are reported in Table 4.
The significant negative signs in a 1 and a 2 indicate an overreaction, that is, a promising positive return today leads to a return reversal in the following two days or vice versa. Hence, the CC markets tend to overreact to good or bad news, and this overreaction can be corrected in the following two days. An ARIMA model for the CC assets, therefore, suggests predictability due to an "overreaction". The Ljung-Box test confirms that there is no serial dependence in the residuals based on the ARIMA(2, 0, 2) specification. Note that the squared residuals carry incremental information that is addressed in the following GARCH analysis.

GARCH Model
The GARCH model, introduced first by Bollerslev (   Let us start with a GARCH-type model for characterizing the conditional variance process of BTC. The ARIMA-t-GARCH model with t-distributed innovations used to capture fat tails is as follows: where σ 2 t represents the conditional variance of the process at time t and t(ν) refers to the zeromean t distribution with ν degrees of freedom. The choice of the t-distribution rather than the Gaussian distribution is supported by Hotz-Behofsits et al. (2018) and Chan et al. (2017).
The covariance stationarity constraint α 1 + β 1 < 1 is imposed. As shown in Table 5, the β 1 estimate from BTC indicates a persistence in the variance process, but its value is relatively smaller than those estimated from the stock index returns (see Franke et al. (2019)). Typically, the persistence-of-volatility estimates are very near to one, showing that conditional models for stock index returns are very close to being integrated. By comparison, BTC places a relatively higher weight on the α 1 coefficient and relatively lower weight on the β 1 to imply a less-smooth volatility process and striking disturbances from the innovation term. This may further imply that the innovation is not pure white noise and can occasionally be contaminated by the presence of jumps.
In addition to the property of leptokurtosis, the leverage effect is commonly observed in practice. According to a large body of literature, starting with Engle and Ng (1993), the leverage effect refers to an asymmetric volatility response given a negative or positive shock. The lever- age effect is captured by the exponential GARCH (EGARCH) model by Nelson (1991), where g j (Z t ) = α j Z t + φ j (|Z t−j | − E|Z t−j |) with j = 1, 2, . . . , q. When φ j = 0, we have the logarithmic GARCH (LGARCH) model from Geweke (1986) and Pantula (1986). To accommodate the asymmetric relation between stock returns and volatility changes, the value of g j (Z t ) must be a function of the magnitude and the sign of Z t . Over the range of 0 < Z t < ∞, g j (Z t ) is linear in Z t with slope α j + φ j , and over the range −∞ < Z t ≤ 0, g j (Z t ) is linear in Z t with slope α j − φ j .
The estimation results based on the ARIMA(2,0,2)-t-EGARCH(1,1) model are reported in Table 6. The estimated α 1 is no longer significant, showing a vanished sign effect. However, a significant positive value of φ 1 indicates that the magnitude effect represented by φ 1 (|Z t−1 | − E|Z t−1 |) plays a bigger role in the innovation in log(σ 2 t ).
We compare the model performances between two types of GARCH models through information criteria, and a t-EGARCH(1,1) model is suggested. Note that, as shown in Figure 12, the QQ plots demonstrate a deviation from the student-t. In Chen et al. (2017), GARCH and variants such as t-GARCH, EGARCH have been reported, and, while they are seen to fit the dynamics of BTC nicely, they still could not handle the extreme tails in the residual distribution. Equipped with these findings and taking into account the occasional interventions, we opt for the models with jumps for better characterization of CC dynamics. The presence of jumps is indeed more likely in this decentralized, unregulated and illiquid market. Numerous political interventions also suggest the introduction of the jump component into a pricing model. Figure 12: The QQ plot for BTC based on the residuals of t-GARCH(1,1) model

CRIX
This appendix presents the empirical results of CRIX covering (1) jumps in returns and volatility from the SVCJ model shown in Figure 13 and (2) the estimated volatility from the SVCJ and SVJ models shown in Figure 14. (3) The estimated call options across moneyness and time to maturity in Figure 15. In general, a general consistency can be found between CRIX and BTC.
Other results are available upon request.  Notes: The table reports posterior means and 95% credibility intervals (in square brackets) for the parameters of the SVCJ, SVJ and SV models. All parameters are estimated using CRIX daily returns calculated as the log difference based on the prices from 01/08/2014 to 29/09/2017. Figure 15: Call option prices across moneyness and time to maturity: CRIX