# Pricing cryptocurrency options

Ai Jun Hou, Weining Wang, Cathy Y.H. Chen, Wolfgang Karl Härdle 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 zu Berlin. Sim Kee Boon Institute, Singapore
Management University; Faculty of Mathematics and Physics, Charles University
in Prague.

###### Abstract

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 Renò, (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.

Key Words: CRIX, Bitcoin, Cryptocurrency, SVCJ, Option pricing, Jumps
JEL Codes: C32, C58, C52

Acknowledgement: We thank Bandi and Renò, (2016) for providing us with the
codes used to implement the BR co-jumps model in this study. We acknowledge
the helpful comments from the editor and two anonymous referees. We thank
Xiaohao Ji for help with data processing. This research is supported by the
Deutsche Forschungsgemeinschaft through the International Research Training
Group 1792 "High Dimensional Nonstationary Time Series". (http://irtg1792.hu-
berlin.de). In addition, it has been funded by the Natural Science Foundation
of China (fund number 71528008). Ai Jun Hou acknowledges the financial support
from the Jan Wallender and Tom Hedelius Foundation of Handelsbanken
(P2019-0264). Härdle acknowledges the financial support from the Czech Science
Foundation and the Yushan Scholar Program.
This is a post-peer-review, pre-copyedit version of an article published in
the Journal of Financial Econometrics. The final authenticated version is
available online at: http://dx.doi.org/10.1093/jjfinec/nbaa006.

##  1 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.111 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). 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.222For 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. 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.

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 nonparametric 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 ($\rho_{j}$) 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.

##  2 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.

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 (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
$t$.

##  3 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.

###  3.1 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:

$\displaystyle d\log S_{t}=\mu dt+\sqrt{V_{t}}dW_{t}^{(S)}+Z_{t}^{y}dN_{t}$
(1) $\displaystyle dV_{t}=\kappa(\theta-
V_{t})dt+\sigma_{V}\sqrt{V_{t}}dW_{t}^{(V)}+Z_{t}^{v}dN_{t}$ (2)
$\displaystyle{\sf Cov}(dW_{t}^{(S)},dW_{t}^{(V)})=\rho dt$ (3)
$\displaystyle\operatorname{P}(dN_{t}=1)=\lambda dt.$ (4)

Like in the Cox-Ingersoll-Ross model, $\kappa$ and $\theta$ 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
$\rho$. $N_{t}$ is a pure jump process with a constant mean jump-arrival rate
${\lambda}$. The random jump sizes are $Z_{t}^{y}$ and $Z_{t}^{v}$. Since the
jump-driving Poisson process is the same in both (1), (2), the jump sizes can
be correlated. The random jump size $Z_{t}^{y}$ conditional on $Z_{t}^{v}$, is
assumed to have a Gaussian distribution with a mean of
$\mu_{y}+\rho_{j}Z_{t}^{v}$ and standard deviation set to $\sigma_{y}$. The
jump in volatility $Z_{t}^{v}$ is assumed to follow an exponential
distribution with mean $\mu_{v}$:

$\displaystyle
Z_{t}^{y}|Z_{t}^{v}\sim\;\operatorname{N}(\mu_{y}+\rho_{j}Z_{t}^{v},\sigma^{2}_{y});\;\;\;Z_{t}^{v}\sim\;\exp(\mu_{v}).$
(5)

The correlation $\rho$ 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 $\rho_{j}$ takes care of that. The SV
process $\sqrt{V}_{t}$ is modelled as a square root process. With no jumps in
the volatility, the parameter $\theta$ is the long-run mean of $V_{t}$, and
the process reverts to this level at a speed governed by the parameter
$\kappa$. The parameter $\sigma_{V}$ 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 $\mu$ measures the
expected log-return.

SVCJ is a rich model since it encompasses the SV and SVJ approaches. If we set
$Z_{t}^{v}=0$ in (5), then jumps are only present in prices, we obtain the SVJ
model of Bates, (1996). Taking $\lambda=0$ such that jumps are not present,
the model reduces to the pure SV model originally proposed by Heston, (1993).
If we set $\kappa=\theta=\sigma_{V}=0$ and define $Z_{t}^{v}=0$, the model
reduces to the pure jump diffusion introduced in Merton, (1976).

###  3.2 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:

$\displaystyle\mathrm{}Y_{t}$ $\displaystyle=$
$\displaystyle\mu+\sqrt{V_{t-1}}\varepsilon_{t}^{y}+Z_{t}^{y}J_{t}$ (6)
$\displaystyle V_{t}$ $\displaystyle=$ $\displaystyle\alpha+\beta
V_{t-1}+\sigma_{V}\sqrt{V_{t-1}}\varepsilon_{t}^{v}+Z_{t}^{v}J_{t},$ (7)

where $Y_{t+1}=\log(S_{t+1}/S_{t})$ is the log return, $\alpha=\kappa\theta$,
$\beta=1-\kappa$ and $\varepsilon_{t}^{y}$, $\varepsilon_{t}^{v}$ are the
$\operatorname{N}(0,1)$ variables with correlation $\rho$. $J_{t}$ is a
Bernoulli random variable with $\operatorname{p}(J_{t}=1)=\lambda$ and the
jump sizes $Z_{t}^{y}$ and $Z_{t}^{v}$ are distributed as specified in (5).
The daily data sample from 01/08/2014 to 29/09/2017 is used to estimate the
model. All returns are in decimal form.

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
$\Theta=\\{\mu,\mu_{y},\sigma_{y},\lambda,\alpha,\beta,\sigma_{v},\rho,\rho_{j},\mu_{v}\\}$
and $X_{t}=\\{V_{t},Z_{t}^{y},Z_{t}^{v},J_{t}\\}$ as the latent variance, jump
sizes and jump. Recall that $Y_{t}$ is the log-returns.

The MCMC method treats all components of $\Theta$ and
$X\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{X_{t}\\}_{t=1,..,T}$ as random
variables. The fundamental quantity is the joint pdf $p(\Theta,X|Y)$ of
parameters and latent variables conditioned on data using the Bayes formula:

$\displaystyle\operatorname{p}(\Theta,X|Y)=\operatorname{p}(Y|\Theta,X)\operatorname{p}(X|\Theta)\operatorname{p}(\Theta).$
(8)

The Bayes formula can be decomposed into three factors:
$\operatorname{p}(Y|\Theta,X)$, the likelihood of the data,
$\operatorname{p}(X|\Theta)$ the prior of the latent variables conditioned on
the parameters and $\operatorname{p}(\Theta)$ the prior of the parameters. The
prior distribution $\operatorname{p}(\Theta)$ 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, $\\{\Theta^{(i)},X_{t}^{(i)}\\}_{i=1}^{N}$ 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 method, i.e.,
$\mu\sim\operatorname{N}(0,25),(\alpha,\beta)\sim\operatorname{N}(0_{2\times
1},\mathbf{I}_{2\times
2}),\sigma_{2}^{V}\sim\operatorname{IG}(2.5,0.1),\mu_{y}\sim\operatorname{N}(0,100),\sigma_{2}^{y}\sim\operatorname{IG}(10,40),\rho\sim\operatorname{U}(-1,1),\rho_{j}\sim\operatorname{N}(0,0.5),\mu_{V}\sim\operatorname{IG}(10,20)$
(Inverse Gaussian) and $\lambda\sim\operatorname{Be}(2,40)$ (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 reported mean of the posterior that is taken
as an estimate of $\Theta$ is quite robust relative to changes in variance of
the prior distributions. The posterior for all parameters except $\sigma_{V}$
and $\rho$ 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 $Z_{t}^{y}$
and $Z_{t}^{v}$ follow a posterior normal distribution and a truncated normal
distribution, respectively. Hence, it is straightforward to obtain draws for
the joint distribution of $J_{t}$, $Z_{t}^{y}$ and $Z_{t}^{v}$. However, the
posteriors for $\rho$, $\sigma_{2}^{V}$ and $V_{t}$ are nonstandard
distributions and must be sampled using the Metropolis-Hastings algorithm. We
use the random-walk method for $\rho$ and $V_{t}$ , and independence sampling
for $\sigma_{V}^{2}$. 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. The estimates
$\hat{J_{t}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(1/N)\sum_{i=1}^{N}J_{t}^{i}$
(where $N$ is the total 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:

$\displaystyle\tilde{J}_{t}=1\\{\hat{J_{t}}>\zeta\\},\;\;\;\;t=1,2,...,T$ (9)

In our empirical study, we choose $\zeta$ so that the number of inferred jump
times divided by the number of observations is approximately equal to the
estimate of $\lambda$.

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 $\mu$ is positive. The correlation
between returns and volatility $\rho$ 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 $\rho$) 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 $\sigma_{v}$
is substantially reduced from 0.017 (SV) to 0.011 (SVJ) and 0.008 (SVCJ). The
mean of the jump size of the volatility $\mu_{v}$ is significant and positive.
The jump intensity $\lambda$ is also significant. The jump correlation
$\rho_{j}$ 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.

Table 1: BTC parameters for SVCJ, SVJ and SV models

|  $SVCJ$ |  |  $SVJ$ |  |  SV
---|---|---|---|---|---
$\mu$ |  0.041 |  |  0.029 |  |  0.030
|  [0.022, 0.060] |  |  [0.011, 0.046] |  |  [0.014, 0.046]
$\mu_{y}$ |  -0.084 |  |  -0.562 |  |  -
|  [-0.837, 0.670] |  |  [-1.280, 0.155] |  |  -
$\sigma_{y}$ |  2.155 |  |  2.685 |  |  -
|  [1.142, 3.168] |  |  [1.519,3.850] |  |  -
$\lambda$ |  0.041 |  |  0.029 |  |  -
|  [0.025, 0.056] |  |  [0.019, 0.047] |  |  -
$\alpha$ |  0.010 |  |  0.010 |  |  0.009
|  [0.008, 0.012] |  |  [0.006, 0.015] |  |  [0.006, 0.012]
$\beta$ |  -0.132 |  |  -0.116 |  |  -0.033
|  [-0.151 -0.114] |  |  [-0.137 -0.094] |  |  [-0.052 -0.013]
$\rho$ |  0.407 |  |  0.321 |  |  0.169
|  [0.232, 0.583] |  |  [0.225, 0.417] |  |  [0.066, 0.271]
$\sigma_{v}$ |  0.008 |  |  0.011 |  |  0.017
|  [0.007 0.010] |  |  [0.007 0.014] |  |  [0.014 0.021]
$\rho_{j}$ |  -0.573 |  |  - |  |  -
|  [-1.832, 0.685] |  |  - |  |  -
$\mu_{v}$ |  0.620 |  |  - |  |  -
|  [0.426, 0.813] |  |  - |  |  -
$MSE$ |  0.735 |  |  0.757 |  |  0.763

  * •

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.

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 model, which estimates,

$\displaystyle\varepsilon_{t}^{y}=\frac{Y_{t}-\mu-
Z_{t}^{y}J_{t}}{\sqrt{V_{t-1}}}$ (10)

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.

Figure 2: Jumps estimated in returns and volatility from the SVCJ model

Notes: This figure graphs the estimated jumps in returns and volatility from
the SVCJ model. The model is estimated using BTC daily returns calculated as
the log-first difference based on the prices from 01/08/2014 to 29/09/2017.
The first-, second-, and third-subfigures plot jumps in returns, jumps in
volatility and the estimated volatility, respectively.

Figure 3: Estimated volatility from the SVCJ and SVJ models

Notes: This figure plots the estimated volatility from the SVCJ (dotted blue)
and SVJ (solid black) models. All models are estimated using BTC daily returns
calculated as the log-first difference based on the prices from 01/08/2014 to
29/09/2017.

Figure 4:  QQ plots for the SVCJ, SVJ and SV models

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
$\sigma_{t}=\sqrt{V}_{t}$ as the price and volatility process, respectively,
following the notation of BR, we therefore consider the BR affine jump-
diffusion model:

$\begin{split}d\log(S_{t})&=\mu_{r}dt+\sigma_{t}\\{\rho_{t}dW_{t}^{1}+\sqrt{1-\rho_{t}^{2}}dW_{t}^{2}\\}\\\
&+c_{r,t}^{J}dJ_{r,t}+c_{r,t}^{J\\!J}dJ_{r,\sigma,t},\\\
d\xi(\sigma_{t}^{2})&=\\{m_{0}+m_{1}\log(\sigma_{t}^{2})\\}dt+\Lambda
dW_{t}^{1}+c_{\sigma,t}^{J}dJ_{\sigma,t}+c_{\sigma,t}^{J\\!J}dJ_{r,\sigma,t},\\\
\rho_{t}&=\mbox{max}\\{\min(\rho_{0}+\rho_{1}\sigma_{t},1),-1\\},\end{split}$
(11)

where $\xi(\cdot)$ is an increasingly monotonic function (we will choose it as
$\log(\cdot)$ in the following discussions), $W=\\{W^{1},W^{2}\\}$ is a
bivariate standard Brownian motion vector and
$J=\\{J_{r,t},J_{\sigma,t},J_{r,\sigma,t}\\}$ is a vector of mutually
independent Poisson processes with constant intensities, which are denoted as
$\lambda_{r}$, $\lambda_{\sigma}$ and $\lambda_{r,\sigma}$, 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, i.e.
$(c_{r,t}^{J},c_{\sigma,t}^{J})\sim\mathop{\mbox{\sf N}}(\mu^{J},\Sigma^{J})$
and $(c_{r,t}^{J\\!J},c_{\sigma,t}^{J\\!J})\sim\mathop{\mbox{\sf
N}}(\mu^{J\\!J},\Sigma^{J\\!J})$, with

$\begin{split}&\mu^{J}=\begin{bmatrix}\mu_{J,r}\\\
\mu_{J,\sigma}\end{bmatrix},\quad\mu^{J\\!J}=\begin{bmatrix}\mu_{J\\!J,r,0}+\mu_{J\\!J,r,1}\sigma_{t}\\\
\mu_{J\\!J,\sigma}\end{bmatrix},\\\
&\Sigma^{J}=\begin{bmatrix}\sigma^{2}_{J,r}&0\\\
0&\sigma^{2}_{J,\sigma}\end{bmatrix},\quad\Sigma^{J\\!J}=\begin{bmatrix}(\sigma_{J\\!J,r,0}+\sigma_{J\\!J,r,1}\sigma_{t}^{\sigma_{JJ,r,2}})^{2}&\rho_{J}(\sigma_{J\\!J,r,0}+\sigma_{J\\!J,r,1}\sigma_{t}^{\sigma_{JJ,r,2}})\sigma_{J\\!J,\sigma}\\\
\rho_{J}(\sigma_{J\\!J,r,0}+\sigma_{J\\!J,r,1}\sigma_{t}^{\sigma_{JJ,r,2}})\sigma_{J\\!J,\sigma}&\sigma^{2}_{J\\!J,\sigma}\end{bmatrix}.\end{split}$
(12)

For any $p_{1}\geq p_{2}\geq 0$, the generic infinitesimal cross-moment of
order $p_{1}$ and $p_{2}$ is defined as:

$\theta_{p_{1},p_{2}}(\sigma)=\lim_{\Delta\to
0}\frac{1}{\Delta}\mathop{\mbox{\sf
E}}\\{[\log(S_{t+\\!\Delta})-\log(S_{t})]^{p_{1}}[\log(\sigma^{2}_{t+\\!\Delta})-\log(\sigma^{2}_{t})]^{p_{2}}|\sigma_{t}=\sigma\\}.$
(13)

In particular $\theta_{p_{1},0}$ helps to identify features of the price
process, and $\theta_{0,p_{2}}$ helps to identify those of the variance
process, while the genuine cross-moments with $p_{1}\geq p_{2}\geq 1$ are
required to identify the common parameter shared by the two processes
$\rho_{0},\rho_{1}$, $\lambda_{r,\sigma}$ and $\rho_{J}$.

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 as

$\hat{\sigma}_{t,i}^{2}=\frac{T}{T-1-n_{j}}\zeta_{1}^{-2}\sum_{k=2}^{T}|r_{t,i,k}||r_{t,i,k-1}|\text{1}_{\\{|r_{t,i,k}|\leq\theta_{t,i,k}\\}}\text{1}_{\\{|r_{t,i,k-1}|\leq\theta_{t,i,k-1}\\}},$
(14)

where $\zeta_{1}\approx 0.7979$, $\theta_{t,i,k}$ is a suitable threshold, and
$n_{j}$ is the number of returns whose absolute value is greater than
$\theta_{t,i,k}$. Then the generic cross-moment estimator
$\hat{\theta}_{p_{1},p_{2}}(\sigma)$ is defined as

$\hat{\theta}_{p_{1},p_{2}}(\sigma)=\frac{\sum_{t=1}^{T-1}\sum_{i=1}^{N}K(\frac{\hat{\sigma}_{t,i}-\sigma}{h})\\{\log(S_{t+1,i})-\log(S_{t,i})\\}^{p_{1}}\\{\log(\hat{\sigma}^{2}_{t+1,i})-\log(\hat{\sigma}^{2}_{t,i})\\}^{p_{2}}}{\Delta\sum_{t=1}^{T}\sum_{i=1}^{N}K(\frac{\hat{\sigma}_{t,i}-\sigma}{h})}$
(15)

where $K(\cdot)$ 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 as in Bandi and Renò, (2016) for the
parametric estimation.

###  4.2 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 $k=1,\cdots,60$. 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.

Figure 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) and the daily spot volatility estimates
(solid line in black)

In Table 2, we show the full model estimation results. The drift parameter
$\mu_{r}$ is estimated to be small and insignificant. The linear mean
reversions, which can be seen as $m_{0}$ and $m_{1}$, are both negative.
However they are both insignificant. The volatility of volatility $\Lambda$ is
estimated to be very significant with a value of $0.6766$. The averaged number
of independent jumps in volatility is estimated at an annual rate of
$0.0519*252$, which is around $13$. The estimated number of co-jumps is around
$0.0584*252\approx 17$. The mean of the independent variance jumps is
significant at a level of $-0.2783$. $\mu_{JJ,r,0}$ is small (-0.0187) and
negative, and $\mu_{JJ,r,1}$ is $0.1265$. 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 $\rho_{0}$ is estimated to be negative, i.e.,
$-0.1485$, though insignificant. The leverage would increase with an
increasing volatility level as $\rho_{1}$ is estimated to be significant and
with a value of $0.9292$. The standard deviation of the jumps in return
$\sigma_{J,r}$ 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 $\sigma_{JJ,r,1}$ and $\sigma_{JJ,r,2}$ are both
significant. The standard deviation of jumps in volatility $\sigma_{J,\sigma}$
is estimated to be $0.8619$ with significance. The standard deviation of the
common volatility jump $\sigma_{JJ,\sigma}$ is estimated to be insignificant.
Notably, the correlation of jumps $\rho_{J}$ 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.

Table 2: BR parametric estimates and their $95\%$ confidence intervals. The
parametric model is specified as in Equation 11-12. The first column specifies
that $J_{r,\sigma}=0$ (no co-jumps) and the second column specifies that
$J_{r}=J_{\sigma}=0$ (no independent jumps).

|  no cojumps |  |  no ind. jumps |  |  full model
---|---|---|---|---|---
$\mu_{r}$ |  0.0021 |  |  0.0027 |  |  0.0082
|  [-0.1939, 0.1981] |  |  [-0.1933, 0.1987] |  |  [-0.0444, 0.0608]
$\rho_{0}$ |  0.0044 |  |  -0.0148 |  |  -0.1485
|  [-0.1150, 0.1237] |  |  [-0.1401, 0.1105] |  |  [-0.4851, 0.1882]
$\rho_{1}$ |  -0.3744 |  |  -0.2237 |  |  0.9292
|  [-0.8513, 0.1025] |  |  [-0.7088, 0.2614] |  |  [0.5884, 1.2699]
$m_{0}$ |  -0.0500 |  |  -0.0500 |  |  -0.0495
|  [-0.1275, 0.0275] |  |  [-0.1275, 0.0275] |  |  [-0.1475, 0.0485]
$m_{1}$ |  -0.0168 |  |  -0.0125 |  |  -0.0600
|  [-0.2128, 0.1792] |  |  [-0.2085, 0.1835] |  |  [-0.2560, 0.1360]
$\Lambda$ |  0.7634 |  |  0.7853 |  |  0.6766
|  [0.5674, 0.9594] |  |  [0.5893, 0.9813] |  |  [0.6570, 0.6963]
$\mu_{J,r}$ |  0.1577 |  |  0 |  |  2.5486
|  [0.0372, 0.2782] |  |  - |  |  [2.3526, 2.7446]
$\mu_{JJ,r,0}$ |  0 |  |  -0.0804 |  |  -0.0187
|  - |  |  [-0.5383, 0.3774] |  |  [-0.1085, 0.0711]
$\mu_{JJ,r,1}$ |  0 |  |  0.0192 |  |  0.1265
|  - |  |  [-0.6850, 0.7234] |  |  [-0.4183, 0.6713]
$\sigma_{J,r}$ |  0.6801 |  |  0 |  |  0.6890
|  [0.5453, 0.8148] |  |  - |  |  [0.4930, 0.8850]
$\sigma_{JJ,r,0}$ |  0 |  |  0.0864 |  |  0.0043
|  - |  |  [-0.3242, 0.4971] |  |  [-0.5459, 0.5544]
$\sigma_{JJ,r,1}$ |  0 |  |  1.8713 |  |  1.2159
|  - |  |  [1.8436, 1.8991] |  |  [1.0199, 1.4119]
$\sigma_{JJ,r,2}$ |  0 |  |  2.6521 |  |  3.9590
|  - |  |  [2.5377, 2.7664] |  |  [3.7630, 4.1550]
$\mu_{J,\sigma}$ |  -0.5000 |  |  0 |  |  -0.2783
|  [-0.5364,-0.4636] |  |  - |  |  [-0.4992, -0.0574]
$\mu_{JJ,\sigma}$ |  0 |  |  -1.9181 |  |  -0.4927
|  - |  |  [-2.0805,-1.7557] |  |  [-0.6429, -0.3425]
$\sigma_{J,\sigma}$ |  0.7945 |  |  0 |  |  0.8619
|  [0.7379, 0.8511] |  |  - |  |  [0.7237, 1.0001]
$\sigma_{JJ,\sigma}$ |  0 |  |  1.0705 |  |  0.0717
|  - |  |  [0.8716, 1.2693] |  |  [-0.0767, 0.2202]
$\rho_{J}$ |  0 |  |  -1.0000 |  |  -0.5257
|  - |  |  [-1.4648, -0.5351] |  |  [-0.7217, -0.3297]
$\lambda_{r}$ |  0.0002 |  |  0 |  |  0.0000
|  [-0.1958, 0.1962] |  |  - |  |  [-0.1960, 0.1960]
$\lambda_{\sigma}$ |  0.0700 |  |  0 |  |  0.0519
|  [0.0504, 0.0896] |  |  - |  |  [0.0323, 0.0715]
$\lambda_{r,\sigma}$ |  0 |  |  0.0060 |  |  0.0584
|  - |  |  [-0.0136, 0.0256] |  |  [0.0564, 0.0603]

  * •

Notes: This table reports the parameter estimates of the model specified in
Equation 11-12 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.

##  5 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.

###  5.1 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:

$\mathop{\mbox{\sf E}}{{}_{Q}}[\exp\\{-r(T-t)\\}C(T)|\mathcal{F}_{t}],$ (16)

where $\mathcal{F}_{t}$ 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. 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 $\tau=(T-t)$
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
2 (for the BR model) for the month of July 2017. It can be seen from this
figure that the approximated volatility on 15 July, 2017, had a large jump
(there was a large increase observed on 15 July, 2017, in the BTC historical
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. The BR model estimates the jump more than the SVCJ and the SV
model, this could be attributed to the uncorrelated jumps, which is not
considered by the SVCJ and the SVJ models. Assuming a BTC spot price $S_{t}$ =
2250, the estimated BTC call option prices across moneyness and time to
maturity on July 17, 2017, obtained using the SVCJ model333We have also
calculated option prices for the SVJ, SV models. These results are available
upon requests. The codes for this research can be found in www.quantlet.de.
are presented in Table 3. We see that, for example, a call option on BTC with
the strike $K=1250$ and time to maturity of 90 days 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 maturity and moneyness for different models, we show in Figure 7 the
one-dimensional contour plot of the 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 to maturity is
fixed at 30 days, and when looking at the time to maturity, moneyness is fixed
at at-the-money (ATM). We can see from the contour plot that the relationship
between the option price and the time to maturity or moneyness varies over
time for all four models. The BR model and the SVCJ models have more volatile
patterns than those of the SCJ and SV models. This figure conveys a
homogeneous message as we can see from Figure 6 in the volatility plots. For
example, 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
$15/07/2017$ to 1993.26 USD. The sudden drop in price should be attributed to
the big jump in volatility shown in Figure 6, and we can also observe this
jump on 15 July in Figure 7.

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.

Figure 6: Estimated volatility of BTC for July 2017: BTC

Notes: This figure plots the estimated volatility of the SVCJ, SVJ, SV and the
BR models. The volatility is approximated based on the parameters reported in
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 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 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.

Table 3: Call option price of BTC on $17/07/2017$ from the SVCJ model

$K\backslash\tau$ | 1 | 7 | 30 | 60 | 90 | 180 | 360 | 720
---|---|---|---|---|---|---|---|---
1250.00 | 1069.18 | 1017.81 | 1099.87 | 1125.90 | 1157.95 | 1248.98 | 1361.04 | 1365.96
1350.00 | 959.02 | 959.02 | 1006.02 | 1066.67 | 1094.08 | 1224.48 | 1302.60 | 1316.03
1450.00 | 885.20 | 860.15 | 929.32 | 995.45 | 1046.89 | 1099.35 | 1258.83 | 1438.90
1550.00 | 802.38 | 791.34 | 901.27 | 950.34 | 1015.76 | 1114.94 | 1192.24 | 1332.08
1650.00 | 707.97 | 739.10 | 825.07 | 882.17 | 902.32 | 1062.17 | 1175.59 | 1282.36
1750.00 | 625.86 | 678.22 | 786.88 | 856.72 | 896.56 | 962.79 | 1192.61 | 1338.49
1850.00 | 552.26 | 618.94 | 697.11 | 785.62 | 862.83 | 897.74 | 1110.36 | 1289.51
1950.00 | 502.28 | 545.58 | 663.47 | 740.32 | 819.72 | 903.60 | 1052.09 | 1229.45
2050.00 | 425.46 | 511.28 | 629.14 | 741.65 | 772.51 | 905.30 | 1027.76 | 1193.43
2150.00 | 358.30 | 460.57 | 597.44 | 683.55 | 740.64 | 870.66 | 1036.76 | 1164.23
2250.00 | 302.88 | 408.62 | 543.02 | 633.31 | 720.57 | 872.42 | 938.68 | 1051.71
2350.00 | 265.91 | 378.10 | 492.86 | 594.01 | 651.03 | 783.37 | 887.62 | 1064.33
2450.00 | 211.26 | 347.79 | 470.85 | 580.30 | 657.43 | 761.39 | 940.90 | 1085.75
2550.00 | 193.69 | 304.13 | 437.06 | 547.15 | 608.36 | 766.19 | 914.62 | 1101.72
2650.00 | 156.38 | 266.64 | 421.86 | 518.27 | 571.42 | 719.92 | 827.17 | 992.20
2750.00 | 136.24 | 247.38 | 397.92 | 484.70 | 556.31 | 651.86 | 863.10 | 1066.75
2850.00 | 135.28 | 228.47 | 345.42 | 465.75 | 541.61 | 672.76 | 788.25 | 955.97
2950.00 | 100.02 | 202.57 | 341.11 | 413.75 | 488.15 | 627.52 | 780.53 | 917.27
3050.00 | 103.45 | 179.93 | 313.83 | 424.23 | 496.05 | 619.88 | 758.99 | 911.33
3150.00 | 82.59 | 162.72 | 290.90 | 371.20 | 450.85 | 593.10 | 752.88 | 888.89
3250.00 | 72.93 | 140.40 | 273.97 | 358.26 | 442.91 | 571.96 | 726.49 | 933.57

  * •

Notes: This table reports the approximated call option prices at different
time to maturity $\tau$ and strike prices $K$ the SVCJ model on 17/07/2017
based on the parameters reported in Table 1. The numbers in the first row are
the time to maturity. The numbers in the first column are the strike prices.
The spot BTC price is assumed to be 2250.

###  5.2 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 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 one-week 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 4 of Table 2. We distinguish the case of
$\rho_{J}$, which is set to be a model-fitted parameter from the SVCJ fit or
to be zero, i.e., the IV surface corresponds to a case with a correlation
between 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 $\rho_{J}$, 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 anti-
correlated 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.

Figure 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. The x-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 $\circ$, $\ast$, $\diamond$ plots the IVs
of the SVCJ, SVJ and SV models, respectively.

Figure 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. The x-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 IVs are based on the simulated
option prices using the model parameters reported in Table 2. The full model
uses parameters from column 4 of Table 2. A co-jumps correlation of 0 means
that $\rho_{J}$ is set to zero while the other parameters remain the same as
in the full model.

##  6 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) and thecrix.de for
details). CRIX has been widely investigated in the pioneering research on
cryptocurrencies, including by Chen et al., (2017), Hafner, (2018), Chen and
Hafner, (2019) and da Gama Silva et al., (2019).

There are two advantages of holding a portfolio comprising a wide variety of
Cryptocurrencies like 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 a promising potential
of diversification. The correlations among coins vary over time, as shown in
Härdle et al., (2019). It shows that the diversification effect through
forming a portfolio is beneficial, although this effect may vary over time.

The second advantage underscores that the efficient portfolio, like CRIX,
entails a higher Sharpe ratio than that of BTC. From the view of institutional
investors, a smart strategy is to hold a market portfolio comprising of the
coins with sufficient liquidity and market capitalization to leverage between
profitability and risk-sharing. A simple calculation of the annual Sharpe
ratio for both BTC and CRIX-based portfolios sheds some light. The Sharpe
ratios of CRIX in 2016 and 2017 are respectively 0.094 and 0.194, however, the
ratios of BTC are relatively lower (0.085 in 2016 and 0.149 in 2017). It
suggests that investors should rather look at all possible portfolios in an
investment opportunity set that potentially optimize their mean-variance
preference.

Given the merits of portfolio deployment over a single altcoin investment
rule, institutional investors may demand the corresponding derivatives for
hedging position risk. The options with a cryptocurrency index (CRIX) as
underlying may fulfill such needs in practice. Apart from hedging purposes,
and for speculators without any position, such index options are quite
precious and enable them to bet on future movement.

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.

##  7 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
$\rho$, 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.

##  8 Appendix

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

###  8.1 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,

$a(L)\Delta y_{t}=b_{L}\varepsilon_{t}$ (17)

where $y_{t}$ is the variable of interest, $\Delta y_{t}=y_{t}-y_{t-1}$, $L$
is the lag operator and $\varepsilon_{t}$ 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.

Table 4: Estimation result of ARIMA(2,0,2)

|  |  Bitcoin |  |
---|---|---|---|---
Coefficients |  |  Estimate |  |  Standard error (Robust)
intercept $c$ |  |  0.002 |  |  0.001
$a_{1}$ |  |  -0.867 |  |  0.304
$a_{2}$ |  |  -0.596 |  |  0.177
$b_{1}$ |  |  0.868 |  |  0.321
$b_{2}$ |  |  0.539 |  |  0.190

Notes: This table reports the parameter estimated from ARIMA (2,0,2) with BTC
daily returns. The residual distributions are assumed to be Gaussian. The
maximized likelihood 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 calculated based on the price from 01/08/2014 to
29/09/2017. The x-axis plots the lags, and the y-axis plots the ACF and PACF
values.

###  8.2 GARCH Model

The GARCH model, introduced first by Bollerslev, (1986), reflects the changes
in the conditional volatility of the underlying asset in a parsimonious way.
The volatility properties of digital 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.,, 2018).

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:

$\displaystyle a(L)\Delta y_{t}$ $\displaystyle=$ $\displaystyle
b_{L}\varepsilon_{t}$ (18) $\displaystyle\varepsilon_{t}$ $\displaystyle=$
$\displaystyle Z_{t}\sigma_{t},\quad Z_{t}\sim t(\nu)$
$\displaystyle\sigma_{t}^{2}$ $\displaystyle=$
$\displaystyle\omega+\beta_{1}\sigma_{t-1}^{2}+\alpha_{1}\varepsilon_{t-1}^{2}$
(19)

where $\sigma_{t}^{2}$ represents the conditional variance of the process at
time $t$ and $t(\nu)$ refers to the zero-mean $t$ distribution with $\nu$
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 $\alpha_{1}+\beta_{1}<1$ is imposed. As
shown in Table 5, the $\beta_{1}$ 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 $\alpha_{1}$
coefficient and relatively lower weight on the $\beta_{1}$ 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 leverage effect is captured
by the exponential GARCH (EGARCH) model by Nelson, (1991),

$\displaystyle\varepsilon_{t}$ $\displaystyle=$ $\displaystyle
Z_{t}\sigma_{t}$ $\displaystyle Z_{t}$ $\displaystyle\sim$ $\displaystyle
t(\nu)$ $\displaystyle\log(\sigma_{t}^{2})$ $\displaystyle=$
$\displaystyle\omega+\sum_{i=1}^{p}\beta_{i}\log(\sigma_{t-i}^{2})+\sum_{j=1}^{q}g_{j}\left(Z_{t-j}\right)$
(20)

where $g_{j}\left(Z_{t}\right)=\alpha_{j}Z_{t}+\phi_{j}(\lvert
Z_{t-j}\rvert-{\mathop{\mbox{\sf E}}}\lvert Z_{t-j}\rvert)$ with
$j=1,2,\ldots,q$. When $\phi_{j}=0$, 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
$g_{j}\left(Z_{t}\right)$ must be a function of the magnitude and the sign of
$Z_{t}$. Over the range of $0<Z_{t}<\infty$, $g_{j}\left(Z_{t}\right)$ is
linear in $Z_{t}$ with slope $\alpha_{j}+\phi_{j}$, and over the range
$-\infty<Z_{t}\leq 0$, $g_{j}\left(Z_{t}\right)$ is linear in $Z_{t}$ with
slope $\alpha_{j}-\phi_{j}$.

The estimation results based on the ARIMA(2,0,2)-$t$-EGARCH(1,1) model are
reported in Table 6. The estimated $\alpha_{1}$ is no longer significant,
showing a vanished sign effect. However, a significant positive value of
$\phi_{1}$ indicates that the magnitude effect represented by $\phi_{1}(\lvert
Z_{t-1}\rvert-{\mathop{\mbox{\sf E}}}\lvert Z_{t-1}\rvert)$ plays a bigger
role in the innovation in $\log(\sigma_{t}^{2})$.

Table 5: Estimated coefficients of $t$-GARCH(1,1)

Coefficients |  |  Estimates |  |  Robust std |  |  $t$ value
---|---|---|---|---|---|---
BTC |  |  |  |  |  |
$\omega$ |  |  $3.92e-05$ |  |  $1.49e-05$ |  |  $\enspace 2.63$
$\alpha_{1}$ |  |  $2.28e-01$ |  |  $4.46e-02$ |  |  $\enspace 5.12$
$\beta_{1}$ |  |  $7.70e-01$ |  |  $5.13e-02$ |  |  $\enspace 14.98$
$\nu$ |  |  $3.64e+00$ |  |  $4.08e-01$ |  |  $\enspace 8.91$

Notes: This table reports the estimated parameters from the t-GARCH(1,1)
model. The robust version of standard errors (robust std) are based on the
method of White, (1982).

Table 6: Estimated coefficients of $t$-EGARCH(1,1) model

Coefficients |  |  Estimates |  |  Robust std |  |  $t$ value
---|---|---|---|---|---|---
BTC |  |  |  |  |  |
$\omega$ |  |  $3.84e-05$ |  |  $1.47e-05$ |  |  $\enspace 2.61$
$\alpha_{1}$ |  |  $1.05e-03$ |  |  $5.10e-02$ |  |  $\enspace 0.98$
$\beta_{1}$ |  |  $9.52e-01$ |  |  $1.54e-02$ |  |  $\enspace 61.73$
$\phi_{1}$ |  |  $4.16e-01$ |  |  $6.64e-02$ |  |  $\enspace 6.25$
$\nu$ |  |  $3.26e+00$ |  |  $4.16e-01$ |  |  $\enspace 7.82$

Notes: This table reports the estimated parameters from the t-EGARCH(1,1)
model. The robust version of standard errors (robust std) are based on the
method of White, (1982).

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
---

###  8.3 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.

Figure 13: Jumps estimated in returns and volatility from the SVCJ model: CRIX
Figure 14: Estimated volatility from the SVCJ and SVJ models: CRIX Table 7:
Parameters for the SVCJ, SVJ and SV models: CRIX

|  $SVCJ$ |  |  $SVJ$ |  |  SV
---|---|---|---|---|---
$\mu$ |  0.042 |  |  0.0437 |  |  0.017
|  [0.030, 0.054] |  |  [0.027, 0.061] |  |  [0.000 0.034]
$\mu_{y}$ |  -0.0492 |  |  -0.515 |  |  -
|  [-0.777, 0.678] |  |  [-1.110, 0.079] |  |  -
$\sigma_{y}$ |  2.061 |  |  2.851 |  |  -
|  [1.214, 2.907] |  |  [1.349, 4.354] |  |  -
$\lambda$ |  0.0515 |  |  0.035 |  |  -
|  [0.038, 0.065] |  |  [0.017, 0.052] |  |  -
$\alpha$ |  0.0102 |  |  0.026 |  |  0.010
|  [0.009, 0.012] |  |  [-0.012 0.063] |  |  [0.007 0.012]
$\beta$ |  -0.188 |  |  -0.240 |  |  -0.038
|  [-0.205, -0.170] |  |  [-0.383, -0.096] |  |  [-0.056 -0.020]
$\rho$ |  0.275 |  |  0.214 |  |  0.003
|  [0.140, 0.409] |  |  [0.014, 0.415] |  |  [-0.130 0.136]
$\sigma_{v}$ |  0.007 |  |  0.016 |  |  0.018
|  [0.005, 0.009] |  |  [-0.001, 0.033] |  |  [0.014 0.022]
$\rho_{j}$ |  -0.210 |  |  - |  |  -
|  [-0.924, 0.503] |  |  - |  |  -
$\mu_{v}$ |  0.709 |  |  - |  |  -
|  [0.535, 0.883] |  |  - |  |  -
$MSE$ |  0.673 |  |  0.707 |  |  0.736

  * •

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

Notes: This figure graphs the call option prices surface counterplot across
moneyness and time to maturities for the month of July 2017 for CRIX. 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.

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