PPricing cryptocurrency options
Ai Jun Hou ∗ , Weining Wang † , Cathy Y.H. Chen ‡ , Wolfgang Karl Härdle § September 24, 2020
Abstract
Cryptocurrencies, especially Bitcoin (BTC), which comprise a new digital asset class,have drawn extraordinary worldwide attention. The characteristics of the cryptocurrency/BTCinclude a high level of speculation, extreme volatility and price discontinuity. We proposea pricing mechanism based on a stochastic volatility with a correlated jump (SVCJ) modeland compare it to a flexible co-jump model by Bandi and Renò (2016). The estimationresults of both models confirm the impact of jumps and co-jumps on options obtained viasimulation and an analysis of the implied volatility curve. We show that a sizeable pro-portion of price jumps are significantly and contemporaneously anti-correlated with jumpsin volatility. Our study comprises pioneering research on pricing BTC options. We showhow the proposed pricing mechanism underlines the importance of jumps in cryptocurrencymarkets.
Key Words: CRIX, Bitcoin, Cryptocurrency, SVCJ, Option pricing, JumpsJEL Codes: C32, C58, C52Acknowledgement: We thank Bandi and Renò (2016) for providing us with the codes usedto implement the BR co-jumps model in this study. We acknowledge the helpful comments ∗ Stockholm Business School, Stockholm University. † Department of Economics and Related Studies, University of York. ‡ Adam Smith Business School, University of Glasgow. § School of Business and Economics, Ladislaus von Bortkiewicz Chair of Statistics, Humboldt-Universität zuBerlin. Sim Kee Boon Institute, Singapore Management University; Faculty of Mathematics and Physics, CharlesUniversity in Prague. a r X i v : . [ q -f i n . S T ] S e p rom the editor and two anonymous referees. We thank Xiaohao Ji for help with data pro-cessing. This research is supported by the Deutsche Forschungsgemeinschaft through the In-ternational Research Training Group 1792 "High Dimensional Nonstationary Time Series".(http://irtg1792.hu-berlin.de). In addition, it has been funded by the Natural Science Foun-dation of China (fund number 71528008). Ai Jun Hou acknowledges the financial support fromthe Jan Wallender and Tom Hedelius Foundation of Handelsbanken (P2019-0264). Härdle ac-knowledges the financial support from the Czech Science Foundation and the Yushan ScholarProgram. This is a post-peer-review, pre-copyedit version of an article published in the Journal of Finan-cial Econometrics. The final authenticated version is available online at: http://dx.doi.org/10.1093/jjfinec/nbaa006 . Bitcoin (BTC), the network-based decentralized digital currency and payment system, has gar-nered worldwide attention and interest since it was first introduced in 2009. The rapidly growingresearch related to BTC shows a prominent role of this new digital asset class in contemporaryfinancial markets. Several studies have suggested econometric methods to model the dynamicsof BTC prices, including cross-sectional regression models involving the major traded cryp-tocurrencies and also multivariate time-series models. Scaillet et al. (2019) show that jumpsare much more frequent in the BTC market than, for example, in the US equity market (seee.g., Bajgrowicz et al. (2015), Eraker (2004), Bandi and Renò (2016) and among others).Theseearlier studies suggest that jumps should be considered when modeling BTC prices. 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). For example, Hayes (2017) performs a regression using a cross-section dataset consisting of 66 traded digi-tal 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 longand short-run relationships between BTC prices and other related variables using an autoregressive distributed lagmodel.
First , as in the existing literature, theresults from the SVCJ and BR models indicate that jumps are present in the returns and varianceprocesses 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 thecorrelation between the return and volatility is significantly positive in the SVCJ model. How-ever, we cannot find significant negative relations between risk and return in the BR model. Thisimplies that a rise of price is not associated with a decrease in volatility, which is consistent withthe "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. Theparameter 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 theprice jump size and the volatility jump size turns out to be significant with a negative coefficientwith high-frequency data, while tending to be insignificant for the SVCJ fitting using dailyprices. 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 negativecorrelation between jump size only when augmenting return data with options data, and thenegative correlation between co-jump size being identified in the implied volatility smirk. Usinghigh-frequency data, Jocod and Todorov (2009) and Todorov and Tauchen (2010) also reportthat 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 volatilityand therefore overwhelmingly affected by jumps in the volatility processes. The results from theplots of implied volatility (IV) indicate that adding jumps in the return increases the slope of the4V curves. The greater steepness of the IV curve can be strengthened by the presence of jumpsin volatility. The presence of co-jumps enlarges the IV smile further. As evidenced from theIVs curve, options with a short time to maturity are more sensitive to jumps and co-jumps. Tofulfill a hedge or speculation need from institutional investors, we replicate the entire analysisfor the CRyptocurrency IndeX (CRIX), a market portfolio comprising leading cryptocurrencies(see more detail in ). A recent volatility index, VCRIX, created by Kimet al. (2019) also shows the evidence of jumps in CRIX.To summarize our contributions, this study is the first paper to extensively investigate thestochastic and econometric properties of BTC and incorporate these properties in the BTCoptions pricing. Our results have practical relevance in terms of model selection for charac-terizing the BTC dynamics. We document the necessity of incorporating jumps in the returnsand 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 im-portant for policymakers to design appropriate regulations for trading BTC derivatives and forinstitutional investors to launch effective risk management and efficient portfolio strategies.The paper is organized as follows. Section 2 briefly introduces the BTC market. Section 3studies the BTC return and variance dynamics with the SV, SVJ and SVCJ models. Fitting ofthe 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 fewpreliminary econometrics analysis and estimation results for the CRIX are in the Appendix.The codes for this research can be found in . We start by briefly introducing BTC. BTC was the first open source distributed cryptocurrencyreleased in 2009, after it was introduced in a paper “Bitcoin: A Peer-to-Peer Electronic CashSystem” by a developer under the pseudonym Satoshi Nakamoto. It is a digital, decentralized,5artially anonymous currency, not backed by any government or other legal entity. The systemhas 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 currenciesare managed by a central bank, while BTCs are not regulated by any authority; instead, theyare maintained by a decentralized community. The transactions of bitcoins are recorded inthe 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 carriedout more economically and efficiently.Our empirical analyses are carried out based on both daily closing (SVCJ model) prices andfive minutes intra-daily (BR model) prices. The data cover the period from 1 August 2014 to29 September 2017 and are collected from Bloomberg. The dynamics of BTC daily prices (leftpanel) and BTC returns (right panel) are depicted in Figure 1. It shows that the BTC returnis clearly more volatile than the stock return, along with more frequent jumps or the scatteredvolatility spikes. Bitcoin’s price spent most of the year 2015 relatively stable. The BTC price inthe first four months of 2016 was in the range of 400-460 USD. It moved upward dramaticallyafter 2016 and increased to almost 5000 USD by the end of our sample period in 2017. At thetime 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 ob-served after March 2017 was due to the widespread interest in cryptocurrencies (CCs). Thesubsequent drop in June 2017 was caused by a sequence of political interventions. Severalgovernmental announcements of bans on initial coin offerings (ICOs) have spurred intensivemovements on CC markets. For example, the Chinese SEC (Securities and Exchange Commis-sion) denied permission for a bitcoin ETF on March 10, 2017; and Bitcoin crashed down afterChina banned initial coin offerings on September 4, 2017. The large upward movements inBTC prices caused the returns of BTC displaying extremely high volatility and with scatteredspikes/jumps. Several large jumps triggered by a series of big events in the BTC market can bedetected from the returns series, see also Kim et al. (2019). We have implemented a number of6ime series models to the BTC returns and the results are shown in Appendix 8.1 and Appendix8.2 . We find that the standard set of stationary models, such as ARIMA and GARCH, cannotfit the BTC returns well due to the presence of jumps.Figure 1: BTC Prices and Returns − . − . . . . . Notes : This figure graphs the BTC daily price (left panel) from 01/08/2014 to 29/09/2017 and BTC returns (rightpanel). The returns ( R t ) are calculated as R t = log( P t ) − log( P t − ) , where P t is the BTC price at time t . In this section, we estimate the SVCJ model using BTC prices. We begin with a simple SVCJjump specification, and switch to the BR model in Section 4. We focus the analysis on BTCand then introduce CRIX in Section 6.
In order to estimate the BTC dynamics with the SV and SVCJ models regarding returns andvolatility, we employ the continuous time model of Duffie et al. (2000) that encompasses thestandard 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 volatility7rocess. The SVCJ dynamics are as follows: d log S t = µdt + √ V t dW ( S ) t + Z yt dN t (1) dV t = κ ( θ − V t ) dt + σ V √ V t dW ( V ) t + Z vt dN t (2) Cov ( dW ( S ) t , dW ( V ) t ) = ρdt (3) P( dN t = 1) = λdt. (4)Like in the Cox-Ingersoll-Ross model, κ and θ are the mean reversion rate and mean rever-sion level, respectively. W ( S ) and W ( V ) are two correlated standard Brownian motions withcorrelation denoted as ρ . N t is a pure jump process with a constant mean jump-arrival rate λ .The random jump sizes are Z yt and Z vt . Since the jump-driving Poisson process is the same inboth (1), (2), the jump sizes can be correlated. The random jump size Z yt conditional on Z vt , isassumed to have a Gaussian distribution with a mean of µ y + ρ j Z vt and standard deviation setto σ y . The jump in volatility Z vt is assumed to follow an exponential distribution with mean µ v : Z yt | Z vt ∼ N( µ y + ρ j Z vt , σ y ); Z vt ∼ exp( µ v ) . (5)The correlation ρ between the diffusion terms is introduced to capture the possible leverageeffects between returns and volatility. The jumps may be correlated as well. The correlationterm ρ j takes care of that. The SV process √ V t is modelled as a square root process. With nojumps in the volatility, the parameter θ is the long-run mean of V t , and the process reverts to thislevel at a speed governed by the parameter κ . The parameter σ V is referred to as the volatilityof volatility, and it measures the variance responsiveness to diffusive volatility shocks. In theabsence 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 vt = 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 proposedby Heston (1993). If we set κ = θ = σ V = 0 and define Z vt = 0 , the model reduces to the pure8ump diffusion introduced in Merton (1976). There are plenty of different methods to estimate the diffusion process to real data. The gen-erality of simulation-based methods offers obvious advantages over the method of simulatedmoments 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). Forexample, Jacquier et al. (1994) show that MCMC is particularly well suited to deal with SVmodels. Eraker et al. (2003) and Eraker (2004) identify several advantages of using the MCMCapproach over other estimation models because MCMC methods are computationally efficientand the estimating is more flexible when using simulations. The MCMC method also providesmore accurate estimates of latent volatility, jump sizes, jump times, etc. A general discussionand review of the MCMC estimation of continuous-time models can be found in Johannes andPolson (2009).For the reasons discussed above, we estimate the SVCJ model using the MCMC method. Doingthis allows for a wide class of numerical fitting procedures that can be steered by a variationof the priors. Given that there are no BTC options yet, the MCMC method is more flexible inestimating the stochastic variance jumps and thus able to reflect the market price of risk (Frankeet al. (2019). The estimation is based on the following Euler discretization: Y t = µ + (cid:112) V t − ε yt + Z yt J t (6) V t = α + βV t − + σ V (cid:112) V t − ε vt + Z vt J t , (7)where Y t +1 = log( S t +1 /S t ) is the log return, α = κθ , β = 1 − κ and ε yt , ε vt are the N(0 , variables with correlation ρ . J t is a Bernoulli random variable with p( J t = 1) = λ and the jumpsizes Z yt and Z vt are distributed as specified in (5). The daily data sample from 01/08/2014 to29/09/2017 is used to estimate the model. All returns are in decimal form.9et us present a brief description on how to estimate the SVCJ model with MCMC (see also Jo-hannes and Polson (2009), Tsay (2005) and Asgharian and Bengtsson (2006) for more details).Define the parameter vector as Θ = { µ, µ y , σ y , λ, α, β, σ v , ρ, ρ j , µ v } and X t = { V t , Z yt , Z vt , J t } 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. Thefundamental quantity is the joint pdf p (Θ , X | Y ) of parameters and latent variables conditionedon data using the Bayes formula: p(Θ , X | Y ) = p( Y | Θ , X ) p( X | Θ) p(Θ) . (8)The Bayes formula can be decomposed into three factors: p( Y | Θ , X ) , the likelihood of thedata, p( X | Θ) the prior of the latent variables conditioned on the parameters and p(Θ) the priorof the parameters. The prior distribution p(Θ) has to be specified beforehand and is part of themodel specification. In comfortable settings, the posterior variation of the parameters, given thedata, is robust with respect to the prior.The posterior is typically not available in the closed form, and therefore simulation is used toobtain random draws from it. This is done by generating a sequence of draws, { Θ ( i ) , X ( i ) t } Ni =1 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 largegroup of international equity market returns with jump-diffusion models using the MCMCmethod, i.e., µ ∼ N(0 , , ( α, β ) ∼ N(0 × , I × ) , σ V ∼ IG(2 . , . , µ y ∼ N(0 , , σ y ∼ IG(10 , , ρ ∼ U( − , , ρ j ∼ N(0 , . , µ V ∼ IG(10 , (Inverse Gaussian) and λ ∼ Be(2 , (Beta Distribution). The full posterior distributions of the parameters and the latent-state 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 reportedmean of the posterior that is taken as an estimate of Θ is quite robust relative to changes invariance of the prior distributions. The posterior for all parameters except σ V and ρ are all10onjugate (meaning that the posterior distribution is of the same type of distribution as the priorbut with different parameters). The posterior for J t is a Bernoulli distribution. The jump sizes Z yt and Z vt follow a posterior normal distribution and a truncated normal distribution, respec-tively. Hence, it is straightforward to obtain draws for the joint distribution of J t , Z yt and Z vt .However, the posteriors for ρ , σ V and V t are nonstandard distributions and must be sampledusing the Metropolis-Hastings algorithm. We use the random-walk method for ρ and V t , andindependence sampling for σ V . For the estimation of posterior moments, we perform 5000 iter-atations, and in order to reduce the impact of the starting values, we allow for a burn-in for thefirst 1000 simulations.The SVCJ model is known for being able to disentangle returns related to sudden unexpectedjumps from large diffusive returns caused by periods of high volatility. For the BTC situationthat we consider here, we are particularly interested in linking the latent historical jump timesto news and known interventions. The estimates ˆ J t def = (1 /N ) (cid:80) Ni =1 J it (where N is the totalnumber of iterations and i refers to each draw) indicate the posterior probability that there is ajump 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 theestimated jump probability is sufficiently large, that is, greater than an approporiately chosenthreshold value: ˜ J t = 1 { ˆ J t > ζ } , t = 1 , , ..., T (9)In our empirical study, we choose ζ so that the number of inferred jump times divided by thenumber 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 returnsto estimate the SVCJ model. The parameter estimates (mean and variance of the posterior) ofthe 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 remarkableand worth noting since it is different from a negative leverage effect observed over a sequence11f studies in stock markets (see, e.g., French and Stambaugh (1987) and Schwert (1989)). Theeffect 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" (associatedwith a positive ρ ) implies that increasing prices are associated with increasing volatility. Thereason for this positive relationship between risk and returns might be due to BTC prices beingdifferent 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 thatwould 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 SVCJmodels. 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 significantand positive. The jump intensity λ is also significant. The jump correlation ρ j is negative butinsignificant, which parallels the results of Eraker et al. (2003) and Chernov et al. (2003) forstock 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 Section4.) In summary, the SVCJ model fits the data well by an MSE that is smaller than those of theSVJ 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 jumpsoccur frequently for those of the returns and volatility. The estimated jumps size in returns andvariance are different. Figure 3 presents the in-sample fitted volatility processes for the SVCJand SVJ models, respectively. It is not hard to see that both models lead to a similar overallpattern 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 discrete12able 1: BTC parameters for SVCJ, SVJ and SV models SV CJ SV J SV µ µ y -0.084 -0.562 -[-0.837, 0.670] [-1.280, 0.155] - σ y λ α β -0.132 -0.116 -0.033[-0.151 -0.114] [-0.137 -0.094] [-0.052 -0.013] ρ σ v ρ j -0.573 - -[-1.832, 0.685] - - µ v MSE
Notes : This table reports posterior means and 95% finite sample credibilityintervals (in square brackets) for parameters of the SVCJ, SVJ, and SV mod-els. All parameters are estimated using BTC daily returns calculated as thelog-first difference based on the prices from 01/08/2014 to 29/09/2017. model, which estimates, ε yt = Y t − µ − Z yt J t (cid:112) V t − (10)The normality would be violated if the jumps are not perfectly estimated. However, several pre-vious researches such as Larsson and Nossman (2011), Asgharian and Bengtsson (2006) andAsgharian and Nossman (2011) have estimated the SVCJ model with the MCMC in the equitymarket and use the normal plot as a diagnostic tool to visualize the model performance. Wefollow these literature calculating these standardized residuals based on the estimated parame-ters, then show the QQ plots of the standardized residuals from the fitting of different modelsin Figure 4. From these diagnostics, it is evident that the GARCH and even the SV modelsare misspecified. For the SVJ and SVCJ models, the QQ plot diagnostics are substantially im-proved. However, it is apparent that the SVCJ model is the preferred choice which is consistentwith the MSE reported in Table 1. 13igure 2: Jumps estimated in returns and volatility from the SVCJ model Jul14 Feb15 Aug15 Mar16 Sep16 Apr17 Nov17−505
Jumps in returns
Jul14 Feb15 Aug15 Mar16 Sep16 Apr17 Nov17012
Jumps in volatility
Jul14 Feb15 Aug15 Mar16 Sep16 Apr17 Nov17024 volatility
Notes : This figure graphs the estimated jumps in returns and volatility from the SVCJ model. The model isestimated using BTC daily returns calculated as the log-first difference based on the prices from 01/08/2014 to29/09/2017. The first-, second-, and third-subfigures plot jumps in returns, jumps in volatility and the estimatedvolatility, respectively.
Figure 3: Estimated volatility from the SVCJ and SVJ models
Jul14 Feb15 Aug15 Mar16 Sep16 Apr17 Nov1700.511.522.5 SVCJSVJ
Notes : This figure plots the estimated volatility from the SVCJ (dotted blue) and SVJ (solid black) models. Allmodels are estimated using BTC daily returns calculated as the log-first difference based on the prices from01/08/2014 to 29/09/2017.
Notes : This figure graphs the QQ plots versus standard normal for fitted standardized residuals from the SVCJ, SVJand SV models using BTC daily returns calculated as the log-first difference based on the prices from 01/08/2014to 29/09/2017. We also include the QQ plot for the GARCH model using the same sample period.
Imposing a specific structure in the stochastic process as documented in Section 3 may producea 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: d log( S t ) = µ r dt + σ t { ρ t dW t + (cid:112) − ρ t dW t } + c Jr,t dJ r,t + c JJr,t dJ r,σ,t ,dξ ( σ t ) = { m + m log( σ t ) } dt + Λ dW t + c Jσ,t dJ σ,t + c JJσ,t dJ r,σ,t ,ρ t = max { min( ρ + ρ σ t , , − } , (11)where ξ ( · ) is an increasingly monotonic function (we will choose it as log( · ) in the follow-ing discussions), W = { W , W } 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 inten-sities, which are denoted as λ r , λ σ and λ r,σ , respectively. Thus we allow for common andindependent jumps in the system. The Poisson processes are also assumed to be independentfrom the Brownian motion.The BR model is estimated through a GMM-like procedure based on infinitesimal cross-momentsdubbed by the authors NIMM or Nonparametric Infinitesimal Method of Moments. We as-sume the distribution of the jumps to be normal, i.e. ( c Jr,t , c
Jσ,t ) ∼ N ( µ J , Σ J ) and ( c JJr,t , c
JJσ,t ) ∼ N ( µ JJ , Σ JJ ) , with µ J = µ J,r µ J,σ , µ JJ = µ JJ,r, + µ JJ,r, σ t µ JJ,σ , Σ J = σ J,r σ J,σ , Σ JJ = ( σ JJ,r, + σ JJ,r, σ σ JJ,r, t ) ρ J ( σ JJ,r, + σ JJ,r, σ σ JJ,r, t ) σ JJ,σ ρ J ( σ JJ,r, + σ JJ,r, σ σ JJ,r, t ) σ JJ,σ σ JJ,σ . (12)For any p ≥ p ≥ , the generic infinitesimal cross-moment of order p and p is defined as: θ p ,p ( σ ) = lim ∆ → E { [log( S t +∆ ) − log( S t )] p [log( σ t +∆ ) − log( σ t )] p | σ t = σ } . (13)In particular θ p , helps to identify features of the price process, and θ ,p helps to identify thoseof the variance process, while the genuine cross-moments with p ≥ p ≥ are required toidentify the common parameter shared by the two processes ρ , ρ , λ r,σ and ρ J .To conduct the NIMM estimation in BR, we first need to estimate the cross-moments that are intheory functions of parameter of interest. The cross-moments are estimated via a nonparametrickernel method. In particular, denote the day index as t = 1 , ..., T and the equispaced time indexas 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 variance16stimates as ˆ σ t,i = TT − − n j ζ − T (cid:88) k =2 | r t,i,k || r t,i,k − | {| r t,i,k |≤ θ t,i,k } {| r t,i,k − |≤ θ t,i,k − } , (14)where ζ ≈ . , θ t,i,k is a suitable threshold, and n j is the number of returns whose absolutevalue is greater than θ t,i,k . Then the generic cross-moment estimator ˆ θ p ,p ( σ ) is defined as ˆ θ p ,p ( σ ) = (cid:80) T − t =1 (cid:80) Ni =1 K ( ˆ σ t,i − σh ) { log( S t +1 ,i ) − log( S t,i ) } p { log(ˆ σ t +1 ,i ) − log(ˆ σ t,i ) } p ∆ (cid:80) Tt =1 (cid:80) Ni =1 K ( ˆ σ t,i − σh ) (15)where K ( · ) is a kernel function and h is the bandwidth. Finally, with the estimated cross-moments, one can estimate the parameters of interest via the NIMM method, see the details asin Bandi and Renò (2016) for the parametric estimation. In this section, we fit the BR model using high-frequency data and discuss the comparison withthe estimation of the SVCJ model. We collect high-frequency BTC prices from Bloomberg. Theprice data range is from / / to / / , and we collect raw data at a frequency of seconds 24 hours a day. Following Section 4.1, we aggregate the logarithm returns of Bitcoinover a -minute time range, namely r t,i,k = log S t,i,k − log S t,i,k − , with k = 1 , · · · , . Inaddition, we also obtain the spot variance estimates for each day t and each knot i by applyingthe jump robust threshold bipower variation estimator as in Equation (14).To compare the data of the high-frequency aggregated volatility and the daily Bitcoin volatil-ity, we plot the averaged daily spot volatility from the high-frequency data and the daily spotvolatility estimates from the SVCJ model together as in Figure 5. We observe that the twosequences sometimes peak at different time points despite that the general pattern agrees.In Table 2, we show the full model estimation results. The drift parameter µ r is estimated tobe small and insignificant. The linear mean reversions, which can be seen as m and m , are17igure 5: The averaged daily spot volatility and the daily spot volatility estimates. Notes: This figure plots the averaged daily spot volatility from the high-frequency data (dotted line in blue) andthe daily spot volatility estimates (solid line in black) both negative. However they are both insignificant. The volatility of volatility Λ is estimatedto be very significant with a value of . . The averaged number of independent jumps involatility is estimated at an annual rate of . ∗ , which is around . The estimatednumber of co-jumps is around . ∗ ≈ . The mean of the independent variance jumpsis significant at a level of − . . µ JJ,r, is small (-0.0187) and negative, and µ JJ,r, is . .Both parameters are insignificant at the level of confidence. We do not see an obvioustendency for the jumps to be downward, as observed in Bandi and Renò (2016).We find that the leverage ρ is estimated to be negative, i.e., − . , though insignificant. Theleverage would increase with an increasing volatility level as ρ is estimated to be significantand with a value of . . The standard deviation of the jumps in return σ J,r is estimated to besignificant with the value of . . When fitting a nonlinear structure to the standard deviationof the common price jumps, the parameters σ JJ,r, and σ JJ,r, are both significant. The standarddeviation of jumps in volatility σ J,σ is estimated to be . with significance. The standarddeviation of the common volatility jump σ JJ,σ is estimated to be insignificant. Notably, thecorrelation of jumps ρ J is estimated to be negative and significant with a value of − . ,which is in line with BR. This negative and significant co-jump size correlation is discoveredby Duffie et al. (2000), who conclude that the price and the volatility jump sizes are "nearly18erfectly anti-correlated". Eraker (2004) finds a statistically significant correlation between thejump 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. In the previous sections, we have shown that the SVCJ and the the BR models can well describethe log returns dynamics of BTC. In this section, we discuss option pricing for BTC based onthe SVCJ and BR models, respectively.
After fitting the SVCJ and the BR model, we advance with a numerical technique called CrudeMonte Carlo (CMC) to approximate the BTC option prices. Derivative securities such as futuresand 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 optionprices, we follow Eraker et al. (2003) and set the risk premia to zero. This choice can bedisputed, but for the lack of existence of the officially traded options a justifiable path to pricingBTC 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 isdenoted as: E Q [exp {− r ( T − t ) } C ( T ) |F t ] , (16)where F t is a set that represents information up to time t . We approximate the European optionprices of BTC using the CMC technique. The CMC simulation is done for 20000 iterationsto 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 BTCoption market exists yet, we do not have real market option prices for comparison. Thus, wechose July 2017 randomly as the experimental month in our option-pricing simulation analysis.19able 2: BR parametric estimates and their confidence inter-vals. The parametric model is specified as in Equation 11-12. Thefirst column specifies that J r,σ = 0 (no co-jumps) and the secondcolumn specifies that J r = J σ = 0 (no independent jumps). no cojumps no ind. jumps full model µ r ρ ρ -0.3744 -0.2237 0.9292[-0.8513, 0.1025] [-0.7088, 0.2614] [0.5884, 1.2699] m -0.0500 -0.0500 -0.0495[-0.1275, 0.0275] [-0.1275, 0.0275] [-0.1475, 0.0485] m -0.0168 -0.0125 -0.0600[-0.2128, 0.1792] [-0.2085, 0.1835] [-0.2560, 0.1360] Λ µ J,r µ JJ,r, µ JJ,r, σ J,r σ JJ,r, σ JJ,r, σ JJ,r, µ J,σ -0.5000 0 -0.2783[-0.5364,-0.4636] - [-0.4992, -0.0574] µ JJ,σ σ J,σ σ JJ,σ ρ J λ r λ σ λ r,σ Notes : This table reports the parameter estimates of the model specified in Equation 11-12 usingthe intra-daily BTC returns. For each parameter, we report the estimate and the corresponding finite sample credibility intervals in parentheses. The full model is shown in the forth col-umn, and the second and third columns report the same model with the restriction of no co-jumpsand no independent jumps, respectively.
Throughout our entire analysis of option pricing, the moneyness for strike K and S at t isdefined to be K/S t . The pricing formula is a function of moneyness and time to maturity20 = ( T − t ) where T is the maturity day.In Figure 6, we plot the simulated volatility of various models based on the parameters reportedin Table 1 (for the SVCJ, SVJ and SV models) and in Table 2 (for the BR model) for themonth of July 2017. It can be seen from this figure that the approximated volatility on 15July, 2017, had a large jump (there was a large increase observed on 15 July, 2017, in the BTChistorical prices). The sudden jump is perfectly captured by the BR, SVCJ and SVJ models,while the SV model cannot characterize the volatility as well as the other three models. TheBR model estimates the jump more than the SVCJ and the SV model, this could be attributed tothe uncorrelated jumps, which is not considered by the SVCJ and the SVJ models. Assuminga BTC spot price S t = 2250, the estimated BTC call option prices across moneyness and timeto maturity on July 17, 2017, obtained using the SVCJ model are presented in Table 3. We seethat, for example, a call option on BTC with the strike K = 1250 and time to maturity of 90days would be traded at 1157.95 on 17 July, 2017.To further understand how the option price changes with respect to changes in time to maturityand moneyness for different models, we show in Figure 7 the one-dimensional contour plot ofthe option prices surface across time to maturity and moneyness estimated from the SVCJ, SVJ,SV and the BR models for the month of July 2017. When examining moneyness, the time tomaturity is fixed at 30 days, and when looking at the time to maturity, moneyness is fixed atat-the-money (ATM). We can see from the contour plot that the relationship between the optionprice and the time to maturity or moneyness varies over time for all four models. The BR modeland the SVCJ models have more volatile patterns than those of the SCJ and SV models. Thisfigure conveys a homogeneous message as we can see from Figure 6 in the volatility plots. Forexample, for the BTC price, we see a drastic change in the contour structure on, e.g., 15 July,2017 as the price suddenly drops from 2232.65 USD on / / to 1993.26 USD. Thesudden drop in price should be attributed to the big jump in volatility shown in Figure 6, andwe can also observe this jump on 15 July in Figure 7. We have also calculated option prices for the SVJ, SV models. These results are available upon requests. Thecodes for this research can be found in . Notes : This figure plots the estimated volatility of the SVCJ, SVJ, SV and the BR models. The volatility isapproximated based on the parameters reported in Table 1 and Table 2 for the month of July 2017. The x-axis notesthe 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 SVJmodels with respect to changes in moneyness and across time to maturity for July 2017. It isnot hard to see that the pattern is similar to the fitted volatility shown in Figure 6. The differencebetween the SVCJ and the SVJ model is similar besides on July 15 when there is a large spikein the estimated volatility. Therefore, the price differences between the SVCJ and SVJ modelsare mainly caused by the jumps in the volatility process and the volatility level, which reflectsthe necessity of adopting the SVCJ model in practice.
It is well known that stochastic volatility determines excess kurtosis in the conditional dis-tribution of returns. The excess kurtosis causes symmetrically higher implied Black Scholesvolatility when strikes are away from the current prices, e.g., the level of moneyness is awayfrom the ATM level. This phenomenon is called the "volatility smile". It is well documentedin the existing literature that the effect is stronger for short and medium maturity options than22igure 7: Call option prices across moneyness and time to maturity: BTC
Notes : This figure graphs the call option prices surface counter plot across different moneyness and different timesto maturities for the month of July 2017, as shown in the right-hand side labels. When looking at moneyness, thetime to maturity is fixed at 30 days, and when looking at the time to maturity, moneyness is ATM. The colour inthe 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. Whenlooking 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 largerthe difference between the price from the SVCJ and SVJ models. / / from the SVCJ model K \ τ Notes : This table reports the approximated call option prices at different time to maturity τ andstrike prices K the SVCJ model on 17/07/2017 based on the parameters reported in Table 1. Thenumbers in the first row are the time to maturity. The numbers in the first column are the strikeprices. The spot BTC price is assumed to be 2250. for long maturity options for which the conditional returns are closer to normal (Das and Sun-daram (1999)). The presence of co-jumps, and the negative correlation between the presenceof co-jumps sizes yield additional sources of skewness in the conditional distribution of stockreturns (Bandi and Renò (2016)).To further examine the option-pricing property of BTC, we approximate the implied BlackScholes volatility from various models for different degrees of moneyness (strike/spot) anddifferent times to maturity. First, the European call option prices are simulated using the modelparameters 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 theoptions 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 ofmoneyness and time to maturity. The results indicate that jumps in returns and volatility includeimportant differences in the shape of the implied volatility (IV) curves, especially for the shortmaturities options.Figure 9 shows the IV curves for the SVCJ, SVJ and SV models for four different maturities24nd across moneyness. It can be seen from Figure 9, that adding jumps in returns steepensthe slope of the IV curves. Jumps in volatility further steepen the IV curves. For short maturityoptions, 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 differencebetween the SVCJ and SV volatility is approximately 2-3% for up to one month. All threemodels have a one-side volatility skewness. This could be due to the skewness in the conditionaldistribution of BTC returns (Das and Sundaram (1999)) and/or that the negative co-jump sizeyields 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 inreturns result in a discrete mixture of normal distributions for returns, which easily generatesunconditional 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 theamount of excess and kurtosis. Indeed, diffusive stochastic volatility models may generate veryflat 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 volatilityof the SVJ model is closer to the SVCJ model than the SV model. The difference between theSVCJ, SVJ and SV models becomes larger with short time to maturity options, i.e., the one-week and one-month times to maturity. Similar results have been documented in other studiesin 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 capturingsystematic variations in Black- Scholes volatility. In general, although the BTC market has theunique 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 conventionalcharacteristics reported from other option markets.We have also estimated the BR IVs with the same time to maturity and moneyness used forthe SVCJ IVs. We simulate the option prices using the model parameters reported in column4 of Table 2. We distinguish the case of ρ J , which is set to be a model-fitted parameter fromthe SVCJ fit or to be zero, i.e., the IV surface corresponds to a case with a correlation between25igure 9: The IV for the BTC market: the SVCJ, SVJ and SV models Notes : This figure plots the Black Scholes IV for the BTC market based on the SVCJ, SVJ and SV models. Thex-axis shows moneyness and the y-axis shows the IV. Four times to maturity have been considered: one week,one month, three months and one year. The lines with ◦ , ∗ , (cid:5) plots the IVs of the SVCJ, SVJ and SV models,respectively. jump sizes equaling -0.5257 or a correlation between jump sizes equaling to zero. The IVs asa function of moneyness from the BR model are plotted in Figure 10. We can see that the IVsof the BR model agree with the SVCJ model. We see a one-side volatility skewness, i.e., theITM call option prices are higher than the OTM call options. However, due to the significantlynegative jump-size correlation ρ J , the slope of the IVs from the BR full model is steeper thanthe BR model with a case of uncorrelated jump sizes. The impact of the negative jump sizecorrelation is stronger for short time to maturity options, i.e., the one-week and one-monthtimes to maturity. This is mentioned in the results of Duffie et al. (2000) as well, who find asuperior 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 whenemploying option data in addition to returns data. Bandi and Renò (2016) also shows that anti-correlated jump sizes are a fundamental property of prices and volatility. However, the use ofhigh-frequency data is sufficient to reveal this property with no further need for option data.26igure 10: The IV for the BTC market: BR model Notes : This figure plots the implied Black Scholes volatility for the BTC option prices based on the BR model. Thex-axis shows moneyness, and the y-axis shows the IV. Four times to maturity have been considered: one week, onemonth, three months and one year. The IVs are based on the simulated option prices using the model parametersreported in Table 2. The full model uses parameters from column 4 of Table 2. A co-jumps correlation of 0 meansthat 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 endoge-nously determined number of constituents using some statistical criteria, is described in Härdleand Trimborn (2015) and further sharpened in Trimborn and Härdle (2018). It is constructed totrack the entire cryptocurrency market performance as closely as possible. The representativityand the tracking performance can be assured as CRIX considers a frequently changing marketstructure. The reallocation of the CRIX happens on a monthly and quarterly basis (see Trimbornand Härdle (2018) and thecrix.de for details). CRIX has been widely investigated in thepioneering research on cryptocurrencies, including by Chen et al. (2017), Hafner (2018), Chenand Hafner (2019) and da Gama Silva et al. (2019).There are two advantages of holding a portfolio comprising a wide variety of Cryptocurrencieslike CRIX. The first advantage is the diversification benefit. The evidence from Härdle et al.(2019) shows that the correlations among the most leading coins are around 0.5, indicating apromising potential of diversification. The correlations among coins vary over time, as shownin Härdle et al. (2019). It shows that the diversification effect through forming a portfolio isbeneficial, although this effect may vary over time.The second advantage underscores that the efficient portfolio, like CRIX, entails a higher Sharperatio than that of BTC. From the view of institutional investors, a smart strategy is to hold a mar-ket portfolio comprising of the coins with sufficient liquidity and market capitalization to lever-age between profitability and risk-sharing. A simple calculation of the annual Sharpe ratio forboth BTC and CRIX-based portfolios sheds some light. The Sharpe ratios of CRIX in 2016 and2017 are respectively 0.094 and 0.194, however, the ratios of BTC are relatively lower (0.085 in2016 and 0.149 in 2017). It suggests that investors should rather look at all possible portfoliosin an investment opportunity set that potentially optimize their mean-variance preference.Given the merits of portfolio deployment over a single altcoin investment rule, institutionalinvestors may demand the corresponding derivatives for hedging position risk. The optionswith a cryptocurrency index (CRIX) as underlying may fulfill such needs in practice. Apart28rom hedging purposes, and for speculators without any position, such index options are quiteprecious and enable them to bet on future movement.Therefore we perform an analysis for CRIX. All econometric models have been estimated withthe CRIX data. We summarize our major findings here and place the supplementary partsin the appendix. In brief, all the model parameters estimated with CRIX convey a similarconfiguration as estimated with BTC, e.g., the mean jump size of the CRIX volatility processreported in Table 7 is 0.709, which is 0.620 for BTC shown in Table 1. The estimated volatilityfrom the SVCJ and SVJ models (see Figure 14) shows that the jumps are better captured by theSVCJ than the SVJ model. In addition, Figure 15 displays the call option prices surface contourplot from the SVJ, SV and SVCJ models with respect to changes in moneyness and time tomaturity. It shows that the SVCJ model has more volatile patterns than those of the SVJ andSV models with the BTC options. In general, we confirm the consistency between BTC and theCRIX. 29
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-basedasset, brings new challenges but also new ways of thinking for economists, cliometricians andfinancial specialists. Unlike classic financial markets, the BTC market has a unique marketmicrostructure created by a set of opaque, unregulated, decentralized and highly speculationdriven markets.This study provides a way of pricing cryptocurrency derivatives using advanced option-pricingmodels such as the SVCJ and BR models. We find that in general, the SVCJ model performs aswell as the non-affine BR model. We especially find that the correlation between the jump sizesin 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 cannotobtain a significant negative "leverage effect" parameter ρ , which implies a nonnegative relationbetween returns and volatility. The reason for this relationship might be that BTC is differentfrom the conventional stock market, not only because the BTC market is highly unregulated butalso due to the fact that the BTC price is not informative (as there are no fundamentals allowingthe BTC market to set a "fair" price) and is driven by emotion and sentiment. This speculativebehaviour can be explained by the "noise trader" theory from Kyle (1985). The positive relationmight result from the fact that BTC investors irrationally act on noise as if it were informationthat would give them an edge.We find that option prices are very much driven by jumps in the returns and volatility processesand co-jumps between the returns and volatility. This can be seen from the shape of the IVcurves. This study provides a grounding base, or an anchor, for future studies which aim toprice cryptocurrency derivatives. This study provides useful information for establishing anoptions market for BTC in the near future. 30 Appendix
We provide preliminary fit results of econometric models on the Bitcoin time series. We alsocollect results on analysis of the CRIX.
We first fit an ARIMA model. After an inspection through the ACF and PACF plot in Figure11, we start with an ARIMA( p, d, q ) model, a ( L )∆ y t = b L ε t (17)where y t is the variable of interest, ∆ y t = y t − y t − , L is the lag operator and ε t a stationaryerror term. Model selection criteria such as AIC or BIC indicates that the ARIMA( , , ) isthe model of choice. The parameters estimated from the ARIMA(2,0,2) are reported in Table 4.The significant negative signs in a and a indicate an overreaction, that is, a promising positivereturn today leads to a return reversal in the following two days or vice versa. Hence, the CCmarkets tend to overreact to good or bad news, and this overreaction can be corrected in thefollowing two days. An ARIMA model for the CC assets, therefore, suggests predictabilitydue to an “overreaction”. The Ljung-Box test confirms that there is no serial dependence inthe residuals based on the ARIMA( , , ) specification. Note that the squared residuals carryincremental information that is addressed in the following GARCH analysis. The GARCH model, introduced first by Bollerslev (1986), reflects the changes in the condi-tional volatility of the underlying asset in a parsimonious way. The volatility properties ofdigital currency assets have been studied in a vast amount of literature that applies GARCH-type methods (Hotz-Behofsits et al., 2018; Chu et al., 2017; Chan et al., 2017; Conrad et al.,31able 4: Estimation result of ARIMA(2,0,2)BitcoinCoefficients Estimate Standard error (Ro-bust)intercept c a -0.867 0.304 a -0.596 0.177 b b Notes : This table reports the parameter estimated from ARIMA (2,0,2) with BTCdaily returns. The residual distributions are assumed to be Gaussian. The maximizedlikelihood value is 2231.7. The AIC and BIC are -4451.4 and -4415.74, respectively.
Figure 11: ACF and PACF of BTC
Notes : This figure plots the ACF and PACF for BTC returns. The returns are the log-first difference calculatedbased on the price from 01/08/2014 to 29/09/2017. The x-axis plots the lags, and the y-axis plots the ACF andPACF values. t -GARCH model with t -distributed innovations used to capture fat tails isas follows: a ( L )∆ y t = b L ε t (18) ε t = Z t σ t , Z t ∼ t ( ν ) σ t = ω + β σ t − + α ε t − (19)where σ t represents the conditional variance of the process at time t and t ( ν ) refers to the zero-mean t distribution with ν degrees of freedom. The choice of the t -distribution rather than theGaussian distribution is supported by Hotz-Behofsits et al. (2018) and Chan et al. (2017).The covariance stationarity constraint α + β < is imposed. As shown in Table 5, the β estimate from BTC indicates a persistence in the variance process, but its value is relativelysmaller 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 forstock index returns are very close to being integrated. By comparison, BTC places a relativelyhigher weight on the α coefficient and relatively lower weight on the β to imply a less-smoothvolatility process and striking disturbances from the innovation term. This may further implythat the innovation is not pure white noise and can occasionally be contaminated by the presenceof jumps.In addition to the property of leptokurtosis, the leverage effect is commonly observed in prac-tice. According to a large body of literature, starting with Engle and Ng (1993), the leverageeffect refers to an asymmetric volatility response given a negative or positive shock. The lever-33able 5: Estimated coefficients of t -GARCH(1,1)Coefficients Estimates Robust std t valueBTC ω . e −
05 1 . e −
05 2 . α . e −
01 4 . e −
02 5 . β . e −
01 5 . e −
02 14 . ν . e + 00 4 . e −
01 8 . Notes : This table reports the estimated parameters from the t-GARCH(1,1) model. The robustversion of standard errors (robust std) are based on the method of White (1982). age effect is captured by the exponential GARCH (EGARCH) model by Nelson (1991), ε t = Z t σ t Z t ∼ t ( ν )log( σ t ) = ω + p (cid:88) i =1 β i log( σ t − i ) + q (cid:88) j =1 g j (cid:0) Z t − j (cid:1) (20)where g j ( Z t ) = α j Z t + φ j ( | Z t − j | − E | Z t − j | ) with j = 1 , , . . . , q . When φ j = 0 , we havethe logarithmic GARCH (LGARCH) model from Geweke (1986) and Pantula (1986). To ac-commodate 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 < Z t < ∞ , g j ( Z t ) is linear in Z t with slope α j + φ j , and over the range −∞ < Z t ≤ , 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 Ta-ble 6. The estimated α is no longer significant, showing a vanished sign effect. However, asignificant positive value of φ indicates that the magnitude effect represented by φ ( | Z t − | − E | Z t − | ) plays a bigger role in the innovation in log( σ t ) .We compare the model performances between two types of GARCH models through informa-tion 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 andvariants such as t -GARCH, EGARCH have been reported, and, while they are seen to fit thedynamics of BTC nicely, they still could not handle the extreme tails in the residual distribution.34able 6: Estimated coefficients of t -EGARCH(1,1) modelCoefficients Estimates Robust std t valueBTC ω . e −
05 1 . e −
05 2 . α . e −
03 5 . e −
02 0 . β . e −
01 1 . e −
02 61 . φ . e −
01 6 . e −
02 6 . ν . e + 00 4 . e −
01 7 . Notes : This table reports the estimated parameters from the t-EGARCH(1,1) model. Therobust version of standard errors (robust std) are based on the method of White (1982).
Equipped with these findings and taking into account the occasional interventions, we opt forthe models with jumps for better characterization of CC dynamics. The presence of jumps isindeed more likely in this decentralized, unregulated and illiquid market. Numerous politicalinterventions 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 This appendix presents the empirical results of CRIX covering (1) jumps in returns and volatilityfrom the SVCJ model shown in Figure 13 and (2) the estimated volatility from the SVCJ andSVJ models shown in Figure 14. (3) The estimated call options across moneyness and time tomaturity in Figure 15. In general, a general consistency can be found between CRIX and BTC.Other results are available upon request. 35 ul14 Feb15 Aug15 Mar16 Sep16 Apr17 Nov17−505
Jumps in returns
Jul14 Feb15 Aug15 Mar16 Sep16 Apr17 Nov17024
Jumps in volatility
Jul14 Feb15 Aug15 Mar16 Sep16 Apr17 Nov17024 volatility
Figure 13: Jumps estimated in returns and volatility from the SVCJ model: CRIX
Jul14 Feb15 Aug15 Mar16 Sep16 Apr17 Nov1700.511.522.533.54 SVCJSVJ
Figure 14: Estimated volatility from the SVCJ and SVJ models: CRIX36able 7: Parameters for the SVCJ, SVJ and SV models: CRIX
SV CJ SV J SV µ µ y -0.0492 -0.515 -[-0.777, 0.678] [-1.110, 0.079] - σ y λ α β -0.188 -0.240 -0.038[-0.205, -0.170] [-0.383, -0.096] [-0.056 -0.020] ρ σ v ρ j -0.210 - -[-0.924, 0.503] - - µ v MSE
Notes : The table reports posterior means and 95% credibility intervals (insquare brackets) for the parameters of the SVCJ, SVJ and SV models. Allparameters are estimated using CRIX daily returns calculated as the log dif-ference based on the prices from 01/08/2014 to 29/09/2017.
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