2021

[1]Jovanka Matic

[1]BRC Blockchain Research Center, International Research Training Group 1792,
Humboldt-Universität zu Berlin, Dorotheenstr. 1, Berlin, 10117, Germany

2]Department of Business and Economics, Berlin School of Economics and Law,
Badensche Str. 52, Berlin, 10825, Germany

3]BRC Blockchain Research Center, International Research Training Group 1792,
Humboldt-Universität zu Berlin, Dorotheenstr. 1, Berlin, 10117, Germany; 777
Sim Kee Boon Institute, Singapore Management University, 50 Stamford Rd,
Singapore, 178899, Singapore;Asian Competitiveness Institute, National
University of Singapore469C Bukit Timah Road, Singapore, 259772, Singapore;
Dept Information Science and Finance, National Yang-Ming Chiao Tung
University1001 University Road, Hsinchu, 300093, Taiwan; Dept Mathematics and
Physics, Charles University, Ke Karlovu 2027/3, Prague 2, 121 16 , Czech
Republic

# Hedging Cryptocurrency options

*<EMAIL_ADDRESS>Natalie Packham<EMAIL_ADDRESS>Wolfgang Karl Härdle<EMAIL_ADDRESS>*  [  [

###### Abstract

The cryptocurrency market is volatile, non-stationary and non-continuous.
Together with liquid derivatives markets, this poses a unique opportunity to
study risk management, especially the hedging of options, in a turbulent
market. We study the hedge behaviour and effectiveness for the class of affine
jump diffusion models and infinite activity Lévy processes. First, market data
is calibrated to stochastic volatility inspired (SVI)-implied volatility
surfaces to price options. To cover a wide range of market dynamics, we
generate Monte Carlo price paths using an SVCJ model (stochastic volatility
with correlated jumps), a close-to-actual-market GARCH-filtered kernel density
estimation as well as a historical backtest. In all three settings, options
are dynamically hedged with Delta, Delta-Gamma, Delta-Vega and Minimum
Variance strategies. Including a wide range of market models allows to
understand the trade-off in the hedge performance between complete, but overly
parsimonious models, and more complex, but incomplete models. The calibration
results reveal a strong indication for stochastic volatility, low jump
frequency and evidence of infinite activity. Short-dated options are less
sensitive to volatility or Gamma hedges. For longer-dated options, tail risk
is consistently reduced by multiple-instrument hedges, in particular by
employing complete market models with stochastic volatility.

###### keywords:

cryptocurrency options, hedging, bitcoin, digital finance, volatile markets

##  1 Introduction

Consider the problem of hedging contingent claims written on cryptocurrencies
(CC). The dynamics of this new expanding market is characterized by high
volatility, as is evident from the Cryptocurrency volatility index  VCRIX (see
Kim et al. (2021)) and large price jumps (Scaillet et al., 2018). We approach
hedging options written on Bitcoin (BTC) with models from the class of affine
jump diffusion models and infinite activity Lévy processes. Similarly to
Branger et al. (2012), we assess the hedge performance of implausible, yet
complete as well as plausible, but incomplete asset pricing models. Since
April 2019, contingent claims written on BTC and Ethereum (ETH) have been
actively traded on Deribit (www.deribit.com). The Chicago Merchantile Exchange
(CME) introduced options on BTC futures in January 2020. In contrast to
traditional asset classes such as equity or fixed income, the market for CC
options has only recently emerged and is still gaining liquidity, see e.g.
(Trimborn and Härdle, 2018) for an early description of the market.
Cryptocurrency markets are known to exhibit high volatility and frequent
jumps, see e.g. market crashes on 12 March 2020, 19 May 2021, 17 June 2022,
posing challenges to valuation and risk management. This erratic price
behaviour may be attributed to the lack of a fundamental value, see e.g. Biais
et al. (2022), Athey et al. (2016) and Makarov (2020).

As the option market is still immature and illiquid, in the sense that quotes
for many specific strikes or maturities are not directly observable or may be
stale, we derive options prices by interpolating prices from stochastic
volatility inspired (SVI) parametrized implied volatility (IV) surfaces
(Gatheral, 2004).

Aside from conducting a historical backtest, and in order to capture a variety
of market dynamics, the BTC market is imitated with two different Monte Carlo
simulation approaches. In a parametric price path generation approach, we
assume that the data-generating process is described by a SVCJ model. The
second scenario generation method is based on GARCH-filtered Kernel-density
estimation (GARCH-KDE), which can be thought of as a smooth historical
simulation taking into account the historical volatility dynamics, and which
is therefore close to actual market dynamics.

Under each of the two different market simulation methods, options are hedged
by a hedger employing models of different complexity. This deliberately
includes models that are “misspecified” in the sense that relevant risk
factors may be omitted (Branger et al., 2012). On the other hand, those models
are possibly parsimonious enough to yield a complete market. It is known that,
when comparing the hedge performance to a more realistic, albeit incomplete
market model, the simpler model may outperform the complex model (Detering and
Packham, 2015). In our context, a model is “misspecified” if it contains fewer
or different parameters than the SVCJ model. Specifically, as models included
in the class of SVCJ models, we consider the Black and Scholes (1973) (BS)
model, the Merton (1976) jump-diffusion model (JD), the Heston (1993)
stochastic volatility model (SV), the stochastic volatility with jumps model
(SVJ) (Bates, 1996) and the SVCJ model itself. Infinite activity Lévy hedge
models under consideration are the Variance-Gamma $(VG)$ model (Madan et al.,
1998) and the CGMY model (Carr et al., 2002). Options are hedged dynamically
with the following hedge strategies: Delta ($\Delta$), Delta-Gamma
($\Delta$-$\Gamma$), Delta-Vega ($\Delta$-$\mathcal{V}$) and minimum variance
strategies (MV).

To gain further insights, we separate the full time period, ranging from April
2019 to June 2020, into 3 different market scenarios with a bullish market
behavior, calm circumstances with low volatility and a stressed scenario
during the SARS-COV-2 crisis. In addition to evaluating the hedge performance,
we aim to identify BTC risk-drivers such as jumps. This contributes to the
understanding of what actually drives fluctuations on this market. A
historical backtest of the hedge performance, in the spirit of Detering and
Packham (2016) and Ting and Ewald (2013), completes and confirms the findings
of the SVCJ- and GARCH-KDE approaches.

A number of papers investigate the still young market of CC options. Trimborn
and Härdle (2018) describe the CC market dynamics via the cryptocurrency index
CRIX. Madan et al. (2019) price BTC options and calibrate parameters for a
number of option pricing models, including the Black-Scholes, stochastic
volatility and infinite activity models. Hou et al. (2020) price CRIX options
under the assumption that the dynamics of the underlying are driven by the
(SVCJ) model introduced in Duffie et al. (2000) and Eraker et al. (2003). The
literature on the aspects of risk management in CC markets is scarce but
growing. Dyhrberg (2016), Bouri et al. (2017) and Selmi et al. (2018)
investigate the role of BTC as a hedge instrument on traditional markets.
Sebastião and Godinho (2020) and Alexander et al. (2021) investigate the hedge
effectiveness of BTC futures, while Nekhili and Sultan (2021) hedge BTC risk
with conventional assets. To the best of our knowledge, hedging of CC options
has not yet been investigated in this depth and detail. The aspect of risk
management and the understanding of the dynamics of CCs is therefore a central
contribution of this study.

The remainder of the paper is structured as follows: Section 2 describes the
methodology, decomposed into market scenario generation, option valuation and
hedge routine. The hedge routine presents the hedge models and explains the
model parameter calibration and hedge strategy choices. In Section 3, we
present and evaluate the results of the hedge routine and in Section 4, we
conclude. The code is available as quantlets, accessible through Quantlet
under the name  hedging_cc.

##  2 Methodology

In this section, we introduce the methodology, comprising market scenario
generation, option valuation and hedging. We take an option seller’s
perspective and sell 1- and 3-month (M) contingent claims. The choice is
justified by the total trading volume of BTC options. Nearly $80\%$ of the
trading volume consists of options expiring in at most 1 month. Almost all
remaining options expire in 3 months or less (Alexander and Imeraj, 2022).

###  2.1 Synthetic market data generation

We describe how to generate synthetic market data, which serves as the input
for the main analysis. The principal goal of synthetic scenario generation is
to imitate the BTC market behavior, especially retaining its statistical
properties, with the added flexibility of Monte Carlo simulation to create a
large amount of plausible scenarios. In addition, we consider two simulation
methods capturing different statistical properties. They represent a trade-off
between a parametric model with valuable and traceable risk-factor information
and a flexible non-parametric closer-to-actual-market approach. The parametric
model is simulated under the risk neutral measure $\mathbb{Q}$ with a forward
looking perspective. The non-parametric simulation relates to the past market
behavior performed under the physical measure $\mathbb{P}$. The time frame
under consideration is from $1^{st}$ April 2019 to $30^{th}$ June 2020. The
BTC market behavior in this time period is time-varying. This makes it
convenient to segregate the time frame into three disjoint market segments
from April to September 2019 (bullish), October 2019 to February 2020 (calm)
and March to June 2020 (covid), respectively. Bearing in mind that we are
going to hedge 1-month and 3-month options, the minimal segment length is
chosen to exceed three months. A graphical representation of the BTC closing
price trajectory is illustrated in Figure 1 with the corresponding summary
statistics in Table 1. The first interval is labeled as the bullish segment,
because, to a great extent, the market behaves upward-trending. The second
period is labeled as the calm period. With an overall standard deviation of
$\hat{\sigma}=756.55$, price movements are more stagnant compared to the
bullish segment.

Figure 1: BTC closing price from 1st April 2019 to 30th June 2020, where the
blue trajectory represents the bullish market behavior, the black path the
calm period and red path the stressed scenario during  the Corona crisis.
LoadBTC

behavior | $\hat{\mu}$ | $\hat{\sigma}$ | min | $q^{25}$ | $q^{50}$ | $q^{75}$ | max
---|---|---|---|---|---|---|---
bullish | 0.0038 | 0.0428 | -0.1518 | -0.0157 | 0.0050 | 0.0227 | 0.1600
calm | 0.0009 | 0.0290 | -0.0723 | -0.0162 | -0.0015 | 0.0098 | 0.1448
covid | 0.0012 | 0.0490 | -0.4647 | -0.0107 | 0.0009 | 0.0162 | 0.1671

Table 1:  Summary statistics of the bullish, calm and covid market log returns
$R_{t}$.  hedging_cc

The last segment is the Corona crisis or stressed scenario, where financial
markets, especially CC markets, experienced high volatility. A notable mention
is the behavior of BTC on $12^{th}$ March 2020, where its price dropped by
nearly $50\%$.

We now turn to a formal mathematical framework. Let the BTC market be a
continuous-time, frictionless financial market. Borrowing and short-selling
are permitted. The constant risk-free interest rate $r\geq 0$ and the time
horizon $T<\infty$ are fixed. On a filtered probability space
$\left(\Omega,\mathcal{F},\left(\mathcal{F}_{t}\right)_{t\in[0,T]},\mathbb{P}\right)$,
the asset price process and the risk-free asset are defined by adapted
semimartingales $(S_{t})_{t\geq 0}$ and $\left(B_{t}\right)_{t\geq 0}$, where
$B_{0}=1$ and $B_{t}=e^{rt},\ t\geq 0$, respectively. The filtration is
assumed to satisfy the usual conditions (e.g. (Protter, 2005)). To ensure the
absence of arbitrage, we assume the existence of a risk-neutral measure
$\mathbb{Q}$. We consider an option writer’s perspective and short a European
call option. The price of the option with strike $K$ and time-to-maturity
(TTM) $\tau=T-t$ at time $t<T$ is $C(t,\tau,K)$. For multiple-instrument
hedges, we further assume the existence of a liquidly traded call option
$C_{2}(t,\tau,K_{2})$, $K_{2}\not=K$, suitable for hedging. The dynamic, self-
finance hedging strategy
$\xi=(\xi^{0},\xi^{1})=(\xi^{0}_{t},\xi^{1}_{t}))_{0\leq t\leq T}$ is an
$\mathcal{F}$-predictable process, where $\xi^{0}_{t}$ and $\xi^{1}_{t}$
denote the amounts in the risk-free security and the asset, respectively. The
resulting portfolio process $\Pi=(\Pi_{t})_{t\geq 0}$ is admissible and self-
financing. The evolution of the value process $\Pi$ is reviewed in detail in
Appendices A.1, A.2 and A.3.

For the Monte Carlo simulation, the finite time horizon is partitioned into
daily steps of size $\delta t=\frac{1}{365}$. The number of trajectories of
the asset price process is set to $n=100,000$.

####  2.1.1 SVCJ model

The parametric scenario generation approach assumes that the dynamics of the
asset price process $(S_{t})_{t\geq 0}$ and the volatility process
$(V_{t})_{t\geq 0}$ are described by the SVCJ model introduced in Duffie et
al. (2000). This particular choice is motivated by the methodology in Hou et
al. (2020), where the model is applied to pricing options on the CRIX . A high
degree of free parameters enables to model various market dynamics. More
specifically, the risk-neutral model dynamics are (Broadie and Kaya, 2005)

$\displaystyle dS_{t}$
$\displaystyle=(r-\lambda\bar{\mu})S_{t}dt+\sqrt{V_{t}}S_{t}\left[\rho
dW_{t}^{(1)}+\sqrt{1-\rho^{2}}dW_{t}^{(2)}\right]+Z^{s}_{t}dN_{t}$ (1)
$\displaystyle dV_{t}$ $\displaystyle=\kappa\left(\theta-
V_{t}\right)dt+\sigma^{v}\sqrt{V_{t}}dW_{t}^{(1)}+Z^{v}_{t}dN_{t}$

where $W_{t}^{(1)},W_{t}^{(2)}$ are two independent standard Wiener processes.
The scale of $V_{t}$ is given by $\sigma^{v}$, the mean reversion speed is
denoted by $\kappa$ and $\theta$ is the mean reversion level. The model allows
for simultaneous arrivals of jumps in returns and jumps in volatility governed
by the Poisson process $(N_{t})_{t\geq 0}$ with constant intensity $\lambda$.
The jump sizes in volatility $Z^{v}_{t}$ are exponentially distributed
$Z^{v}_{t}\sim\varepsilon\left(\mu^{v}\right)$ and the jumps sizes of the
asset price are conditionally normally distributed

$\Xi\stackrel{{\scriptstyle def}}{{=}}Z^{s}_{t}|Z^{v}_{t}\sim
N\left(\overline{\mu}^{s}+\rho^{j}Z^{v}_{t},\left(\sigma^{s}\right)^{2}\right)$
(2)

where $\overline{\mu}^{s}$ is the conditional mean jump size in the asset
price given by

$\overline{\mu}^{s}=\frac{\exp\left\\{\mu^{s}+\frac{\left(\sigma^{s}\right)^{2}}{2}\right\\}}{1-\rho^{j}\mu^{v}}-1$

In detail, $\sigma^{s}$ denotes the jump size standard deviation. The
unconditional mean is denoted by $\mu^{s}$, which is related to the jump
compensator $\lambda{\mu}^{*}$ by
$\mu^{s}=\log\left[(1+{\mu}^{*})\left(1-\rho^{j}\mu^{v}\right)\right]-\frac{1}{2}\left(\sigma^{s}\right)^{2}$.
The correlation parameter $\rho^{j}$ governs the correlations between jump
sizes. From an empirical point of view, in most markets, jumps occur seldomly
and are difficult to detect, which, as a consequence, makes the calibration of
$\rho^{j}$ unreliable (Broadie et al., 2007). Chernov et al. (2003), Broadie
et al. (2007), Eraker et al. (2003), Eraker (2004) and Branger et al. (2009)
therefore recommend to set $\rho^{j}=0$. In fact, this finding extends to the
BTC market, see Hou et al. (2020), who find that ”the jump correlation
$\rho^{j}$ is negative but statistically insignificant…”. Our main results are
therefore calculated assuming $\rho^{j}=0$. Nonetheless, we add some insights
into calibrating $\rho^{j}$ and hedging with the calibrated parameter in
Section 3.5.1. Note that despite a correlation of zero, the SVCJ model does
not reduce to an SVJ model, as it still features jumps in the volatility.

The resulting paths are simulated according to the Euler-Maruyama
discretization of (1) suggested in Belaygorod (2005). The corresponding model
parameters are re-calibrated on a daily basis according to the methodology
described in Section 2.3.2.

####  2.1.2 GARCH-KDE approach

Compared to the empirical price process, the SVCJ may appear quite
restrictive: aside from being an incomplete market model, the price dynamics
are limited by the specification of the stochastic volatility component as
well as the jump intensity and size. The semi-parametric method loosens the
assumptions by generating scenarios using GARCH-filtered kernel density
estimation (GARCH-KDE) as in e.g. McNeil and Frey (2000). Let $(R_{t})_{t\geq
0}$ denote BTC log-returns and $(\hat{\sigma}_{t})_{t\geq 0}$ the estimated
GARCH(1,1) volatility, (Bollerslev, 1986). The kernel density estimation is
performed on ”de-garched” residuals

$\hat{z}_{t}=\displaystyle\frac{R_{t}}{\hat{\sigma}_{t}}.$ (3)

The rationale is to capture the time-variation of volatility by the GARCH
filter and perform kernel density estimation on standardised residuals. The
estimated density function is

$\widehat{f}^{h}(z)=\frac{1}{nh}\sum_{t=1}^{n}\mathcal{K}\left(\frac{\hat{z}_{t}-z}{h}\right),$
(4)

where $\mathcal{K}\left(\cdot\right)$ denotes the Gaussian Kernel. The
resulting generated paths are $(S(0,i),\ldots,S(T,i))$, $i=1,\ldots,n$,
with111The simulated, discretised prices are denoted by $S(t,i)$ are opposed
to $S_{t}$, which refers to the continuous-time process.

$S(t,i)=\displaystyle
S(0)\exp\left[\sum_{k=1}^{t}\hat{\sigma}_{k}\hat{z}_{k}\right],\quad
t=0,\ldots,T.$ (5)

Throughout the paper, the parametric and the semi-parametric method are
referred to as the SVCJ and GARCH-KDE framework, respectively.

###  2.2 Valuation

This section describes how option prices are derived from the implied
volatility surface. As the market for CC claims, during the time period of our
dataset, is still relatively immature with only a limited number of actively
traded options on Deribit and the Chicago Mercantile Exchange, arbitrage-free
option prices are derived through the stochastic volatility inspired (SVI)
parameterization of the volatility surface of Gatheral and Jacquier (2014).
Let $\sigma^{\mathrm{BS}}(k,\tau)$ denote the BS IV with log-moneyness
$k=\ln\left(K/S_{0}\right)$ and total implied variance
$w(k,\tau)=\\{\sigma^{\mathrm{BS}}(k,\tau)\\}^{2}\tau$. For a fixed $\tau$,
the raw SVI parameterization of a total implied variance smile, as initially
presented in Gatheral (2004), is

$w(k,\chi_{R})=a+b\left\\{\rho^{SVI}(k-m)+\sqrt{(k-m)^{2}+\left(\sigma^{SVI}\right)^{2}}\right\\}.$
(6)

In the parameter set $\chi_{R}=\\{a,b,\rho^{SVI},m,\sigma^{SVI}\\}$,
$a\in\mathbb{R}$ governs the general level of variance, $b\geq 0$ regulates
the slopes of the wings, $\rho^{SVI}\in[-1,1]$ controls the skew,
$m\in\mathbb{R}$ enables horizontal smile shifts and $\sigma^{SVI}>0$ is the
ATM curvature of the smile (Gatheral and Jacquier, 2014). For each maturity,
the smile is recalibrated daily. The implied volatility is obtained by a
simple root-finding procedure, whereas the parameters $\chi_{R}$ are
calibrated according to the optimization technique explained in Section 2.3.2.
In addition, the calibration is subject to non-linear constraints prescribed
in Gatheral and Jacquier (2014). These constraints ensure convexity of the
option price, which rules out butterfly arbitrage. Calendar spread arbitrage
is avoided by penalizing fitted smiles, which induce a decrease in the level
of the total implied variance for a given strike level. For interpolation, the
at-the-money (ATM) total implied variance $\theta_{T}^{SVI}=w(0,T)$ is
interpolated for $t_{1}<T<t_{2}$ as in Gatheral and Jacquier (2014), where
$t_{1},t_{2}$ refer to time points at which implied volatilities are observed.
The resulting option price is the convex combination

$C(T,K)=\alpha_{T}C(t_{1},K)+\left(1-\alpha_{T}\right)C(t_{2},K),$ (7)

where
$\alpha_{T}=\displaystyle\frac{\sqrt{\theta_{t_{2}}^{SVI}}-\sqrt{\theta_{T}^{SVI}}}{\sqrt{\theta_{t_{2}}^{SVI}}-\sqrt{\theta_{t_{1}}^{SVI}}}\in\left[0,1\right]$.

###  2.3 Hedge routine

This section describes the models selected to hedge BTC options as well as the
model parameter calibration procedure. Given these model classes, hedge
strategies are chosen for the hedge routine.

####  2.3.1 Hedge models

For hedging purposes, the choice of a hedge model faces the trade-off between
sufficient complexity to describe the actual market dynamics and market
completeness (Detering and Packham, 2015). In practice, a trader may for
instance initiate hedging with an evidently wrong but simple model, such as
the complete BS option pricing model. A lower number of parameters provides a
parsimonious setup with potentially manageable explanatory power. In our
setting, a European option is hedged employing models of increasing
complexity. In the following, the model granularity is gradually extended by
the addition of risk-factors such as local volatility, jumps, stochastic
volatility and others. This covers the empirical finding of the previous
literature on CC’s, e.g. (Kim et al., 2021; Scaillet et al., 2018).
Accordingly, the hedge models selected encompass affine jump diffusion models
and infinite activity Levy processes.

The class of affine jump diffusion models covers well-known models nested in
(1). Due to its popularity in the financial world, the simple but complete BS
option pricing is selected as one of the hedge models. The volatility is
constant with $V_{t}=\sigma$ and there are no discontinuities from jumps
$N_{t}=0$. A slightly more complex model is the JD model. It assumes constant
volatility with $V_{t}=\theta$, $\sigma^{v}=0$ and extends the BS model by
allowing for jumps in returns. The jump size is $\operatorname{log}\xi\sim
N\left(\mu^{s},\left(\delta^{s}\right)^{2}\right)$ distributed.

Evidence for stochastic volatility motivates the choice of the SV model. The
jump component is excluded with $\lambda=0$ and $N_{t}=0$. We also examine the
SVCJ model itself as a model used for hedging. It serves as the most general
model and its hedge performance provides a meaningful insight for the
comparison of the SVCJ and GARCH-KDE framework, while in the SVCJ framework,
it provides “anticipated” hedge results (cf. Branger et al. (2012)). Due to
the jump scarcity and latent nature of the variance process $V_{t}$, we also
consider the SVJ model for hedging. In difference to the SVCJ model, this
model has jumps in returns but no jumps in volatility.

In contrast to affine jump processes, there exists a well-established class of
processes that do not entail a continuous martingale component. Instead, the
dynamics are captured by a right-continuous pure jump process, such as the
Variance Gamma (VG) model (Madan et al., 1998). The underlying $S_{t}$ evolves
as

$\displaystyle dS_{t}$ $\displaystyle=rS_{t-}dt+S_{t-}dX^{\mathrm{VG}}_{t}$
(8) $\displaystyle X^{\mathrm{VG}}_{t}$
$\displaystyle=\theta^{VG}G_{t}+\sigma^{VG}W_{G_{t}},$

with the characteristic function of the VG-process $X^{\mathrm{VG}}_{t}$ given
by

$\varphi_{\mathrm{VG}}(u\mathchar
24635\relax\;\sigma^{VG},\nu,\theta^{VG})=\left(1-\mathrm{i}u\theta^{VG}\nu+\frac{1}{2}\left(\sigma^{VG}\right)^{2}\nu
u^{2}\right)^{-1/\nu},$ (9)

where $r$ is the risk-free rate, $W_{t}$ is a Wiener process and $G_{t}$ is a
Gamma process. The overall volatility level is represented by $\sigma^{VG}$;
$\theta^{VG}$ governs the symmetry of the distribution and therefore controls
the implied volatility skew; $\nu$ controls for tails, kurtosis and thus
regulates the shape of the volatility surface. An alternative representation
of the $VG$ process appealing for practical interpretation is given by the
characteristic function

$\varphi_{\mathrm{VG}}(u\mathchar
24635\relax\;C,G,M)=\left(\frac{GM}{GM+(M-G)\mathrm{i}u+u^{2}}\right)^{C},$
(10)

where $C,\ G,\ M>0$. The detailed link between (9) and (10) is described in
Appendix A.4. An increase in $G$ ($M$) increases the size of upward jumps
(downward jumps). Accordingly, $\theta^{VG}$, $M$ and $G$ account for the
skewness of the distribution. An increase in $C$ widens the Lévy-measure. An
extension of the VG model is the CGMY model by Carr et al. (2002). On a finite
time interval, the additional parameter $Y$ permits infinite variation as well
as finite or infinite activity. Formally, in (8) the source of randomness is
replaced by a CGMY process $X^{CGMY}_{t}$ with the characteristic function

$\varphi_{\mathrm{CGMY}}\left(u\mathchar
24635\relax\;C,G,M,Y\right)=\exp\left[Ct\Gamma(-Y)\left\\{(M-\mathrm{i}u)^{Y}-M^{Y}+(G+\mathrm{i}u)^{Y}-G^{Y}\right\\}\right].$
(11)

The $X^{VG}_{t}$-process in the representation in Equation (9) is a special
case of the CGMY process for $Y=1$. On a finite time interval, the behavior of
the path depends on $Y$. For $Y<0$, there is a finite number of jumps, else
infinite activity. In case of $Y\in(1,2]$, there is also infinite variation.

####  2.3.2 Calibration routine

The model parameters are calibrated following the FFT option pricing technique
of Carr and Madan (1999). The price of a European-style option $C(T,K)$ is
given by

$\displaystyle C(T,K)$
$\displaystyle=\frac{1}{\pi}\operatorname{e}^{-\alpha\ln
K}\int_{0}^{\infty}\operatorname{e}^{-\mathrm{i}v\ln
K}\psi_{T}(v)dv,\quad\text{ with}$ (12) $\displaystyle\psi_{T}(v)$
$\displaystyle=\frac{\exp^{-rT}\phi_{T}(v-(\alpha+1)\mathrm{i})}{\alpha^{2}+\alpha-v^{2}+\mathrm{i}(2\alpha+1)v},$

where $\phi_{T}$ is the characteristic function of the $\alpha$-damped option
price $c_{T}\left[\ln(K)\right]=e^{\alpha\ln K}C(T,K),\ \alpha>0$. The ill-
posed nature of calibration can lead to extreme values of the model
parameters. This is avoided by employing a Tikhonov $L_{2}$-regularization
(Tikhonov et al., 2011). At the cost of accepting some bias, this penalizes
unrealistic values of the model parameters by giving preference to parameters
with smaller norms. Calibration is performed by the optimizer

$\displaystyle\theta^{*}$
$\displaystyle=\underset{\theta\in\Theta}{\operatorname{argmin}}R(\theta)$
(13) $\displaystyle R(\theta)$
$\displaystyle=\sqrt{\frac{1}{n}\sum_{i}\\{IV^{Model}(T^{i},K^{i},\theta)-IV^{Market}((T^{i},K^{i}))\\}^{2}}+\theta^{\top}\Gamma\theta,$

where $IV^{Model}\left(\cdot\right),IV^{Market}\left(\cdot\right)$ describe
model and market implied volatilities for maturity and strike $T^{i},K^{i}$.
$\Gamma$ is a diagonal positive semi-definite matrix. It corresponds to the
Tikhonov $L_{2}$-regularization, which gives preference to parameters with
smaller norms. The entries in the matrix $\Gamma$ are chosen individually for
each parameter to ensure that they maintain the same reasonable order of
magnitude.

The parameter space $\Theta\subset\mathbb{R}^{d}$ of each model in scope is
subject to linear inequality constraints. Given that the objective is not
necessarily convex, it may have multiple local minima. In order to explore the
entire parameter space, simplex-based algorithms are more appropriate than
local gradient-based techniques. In our case, we employ the Sequential Least
Squares Programming optimization (Kraft, 1988) routine. We adjust for time
effects by calibrating parameters on the IV surface instead of option prices.

We impose liquidity and moneyness cut-offs. Claims must have a positive
trading volume and an absolute BS Delta in $[0.25,0.75]$. This filters options
that are close to ATM as is custom in FX trading, see Clark (2011).

####  2.3.3 Hedging strategies

Any hedging strategy’s target is to protect against market movements and to
minimize Profit-and-Loss (P&L) of the hedged position. Hedges either reduce
risk by eliminating market-risk-related sensitivities
$\left(\Delta,\Gamma,\mathcal{V}\right)=\displaystyle\left(\frac{\partial
C}{\partial S},\frac{\partial^{2}C}{\partial^{2}S},\frac{\partial
C}{\partial\sigma}\right)$ or by minimising a risk measures, such as a hedged
position’s variance. Broadly, hedging strategies are split into single- and
multiple instrument hedges. Single instrument hedges incorporate the
$\Delta$\- and MV-hedging. Föllmer and Sondermann (1986)’s MV hedge aims to
find the strategy that minimizes the mean-squared error under $\mathbb{Q}$

$\left(\Pi_{0},\xi^{MV}_{t}\right)=\underset{\Pi_{0},\xi^{1}_{t}}{\text{argmin}}\
\mbox{\sf
E}_{\mathbb{Q}}\left[\left(C_{T}-\Pi_{0}-\int_{0}^{T}\xi^{1}_{u}dS_{u}\right)^{2}\right].$
(14)

Under the assumption of symmetric losses and gains, the minimizing strategy is
denoted by $\xi^{MV}_{t}$. The $\Delta$-hedge targets to protect the position
against first-order changes in the underlying $(S_{t})_{t\leq T}$.

In addition to hedging $\Delta$, multiple instrument hedges eliminate higher-
order sensitivities or sensitivities of risk factors other than the
underlying, e.g. $\sigma$. To achieve $\Delta$-$\Gamma$\- or
$\Delta$-$\mathcal{V}$-neutrality, an additional liquid option
$C_{2}(S(t),T,K_{1})$ with strike $K_{1}\neq K$ is priced from the SVI
parameterized IV surface, as explained in Section 2.2. For performance
comparison of linear and non-linear effects, the dynamic $\Delta$\- and
$\Delta$-$\Gamma$-hedging strategies are applied to all hedge models. The
$\Delta$-$\mathcal{V}$-hedge is only considered for affine jump diffusion
models. The technical aspects of the dynamic hedging strategies are described
in Appendices A.2 and A.3. The calibrated model parameters are used to compute
hedging strategies $(\xi_{t})_{0\leq t\leq T}$ for each model. Table 2
summarizes the hedging strategies applied to the respective hedge models.

model | strategies applied
---|---
BS |  $\Delta_{BS}$, $\Delta$-$\Gamma_{BS}$, $\Delta$-$\mathcal{V}_{BS}$
SV |  $\Delta$-$\mathcal{V}_{SV}$, $\Delta_{SV}$, $\Delta$-$\Gamma_{SV}$, MV
JD |  $\Delta_{JD}$, $\Delta$-$\Gamma_{JD}$, $\Delta$-$\mathcal{V}_{JD}$, MV
SVJ |  $\Delta_{SVJ}$, $\Delta$-$\Gamma_{SVJ}$, $\Delta$-$\mathcal{V}_{SVJ}$, MV
SVCJ |  $\Delta_{SVCJ}$, $\Delta$-$\Gamma_{SVCJ}$ $\Delta$-$\mathcal{V}_{SVCJ}$, MV
VG |  $\Delta_{VG}$, $\Delta$-$\Gamma_{VG}$, MV
CGMY |  $\Delta_{CGMY}$, $\Delta$-$\Gamma_{VG}$, MV

Table 2: Hedge strategy summary applied to the hedge models described in
Section 2.3.1.

The methods for computing sensitivities depends on the model. Where possible,
analytic formulas are used (e.g. BS-model). In cases where not analytic
formulas are available, e.g. the VG-model, finite differences are applied to
FFT-generated option prices.

####  2.3.4 Backtesting hedges on historical data

In addition to evaluating the hedges in Monte Carlo simulations, the hedging
strategies are backtested on the historical BTC price path. The principal idea
is to write an at-the-money option with fixed expiry (2 months in our setting)
each day. Each option is hedged by a self-financing hedging strategy with
daily rebalancing. At expiry, the P&L is recorded. This gives a sample of
P&L’s on real data. Details of the self-financing strategy are given in
Appendix A.1. The choice of 2-month expiry allows to construct P&L samples of
size 60 for each market regime (bullish, calm, Covid).

This setup follows the empirical study in Detering and Packham (2016). A
similar type of backtest, recording daily P&L instead of terminal P&L is
conducted in Ting and Ewald (2013). Daily P&L, however, depends on the option
price and is therefore model-dependent.

####  2.3.5 Hedge performance measures

Each model’s hedge performance is evaluated by indicators derived from the
relative P&L

$\pi^{rel}=e^{-rT}\frac{\Pi_{T}}{C(T,K)}.$ (15)

In a perfect hedge in a complete market, we have $\Pi_{T}=0$, and therefore
$\pi^{rel}=0$. However, in practice, due to model incompleteness,
discretization and model uncertainty, $\pi^{rel}\neq 0$. We evaluate the hedge
performance with the relative hedge error $\varepsilon^{hedge}$ as applied in
e.g. Poulsen et al. (2009), defined as

$\varepsilon^{hedge}=100\sqrt{\operatorname{Var}\left(\pi^{rel}\right)}.$ (16)

The rationale behind $\varepsilon^{hedge}$ is that standard deviation
represents a measure of uncertainty. A sophisticated hedge strategy reduces or
ideally eliminates uncertainty (Branger et al., 2012). The tail behavior is
captured by the expected shortfall

$\text{ES}_{\alpha}=\mathbb{E}\left[\pi^{rel}\mid\pi^{rel}>F_{\pi^{rel}}^{(-1)}(\beta)\right],$
(17)

where $F_{\pi^{rel}}^{(-1)}(\beta)$ denotes an $\beta$-quantile. In the
empirical part, these measures are estimated via the empirical distributions
from Monte Carlo, resp. historical simulation.

##  3 Empirical results

###  3.1 Data

The models are calibrated on the market prices of European-style Deribit
options written on BTC futures. The number of liquidly traded instruments
varies significantly with maturity. Therefore, the data is filtered with
liquidity cut-offs.

###  3.2 Option pricing

Option prices are obtained on every day of the hedging period. This is
necessary for the calculation of the initial value of the hedging portfolio
and to perform multi-asset dynamic hedging. Each option is priced according to
the IV surface on the given day. If the option is not traded for the given
strike or maturity, the SVI parametrized IV surface is interpolated in an
arbitrage-free way. For illustration, we take a look at CC option prices at
the beginning of each market period. Figure 2 displays the SVI parametrized
interpolated IV surfaces for SVI parameters listed in Table 15. The resulting
option prices used in the hedging routine are displayed in Table 3.

| $F_{0}$ | 1 M | 3 M
---|---|---|---
BULLISH | 4088.16 | 206.38 | 417.87
CALM | 8367.51 | 838.01 | 1449.82
COVID | 9804.85 | 610.36 | 1201.46

Table 3:  Interpolated 1-month and 3-month ATM option prices. $F(0)$ denotes
the price of the underlying at $t=0.$ hedging_cc

Recall that for a given IV surface the SVI parameters related by the formula
(6) are calibrated for each TTM. The temporal dynamics of the SVI parameters
provide the following insights: parameter $a$ increases with TTM, which aligns
with the increase of the ATM total variance as TTM rises. Parameter
$\sigma^{SVI}$ decreases with TTM, indicating decrease of the ATM curvature.
Increasing values of parameter $b$ indicate higher slopes of the wings as TTM
increases. Skewness, expressed in terms of the parameter $\rho^{SVI}$, varies
across market segments. Usually negative values of $\rho^{SVI}$ indicate a
preference for OTM puts over OTM calls. In the bullish period, skewness is
close to zero across most maturities.

(a) bullish

(b) calm

(c) covid

Figure 2: Market IVs in red and interpolated IV surface in blue on (a)
$1^{st}$ April 2019 (b) $1^{st}$ October 2019 (c) $1^{st}$ February 2020.
Fitted smiles with very short maturities less than 1 week are excluded from
plots, because they are not relevant for the hedging routine. Calibrated SVI
parameters shorter maturities are given in Table 15.  hedging_cc

###  3.3 Scenario generation results

period | mean | std | skew | kurt | $q^{25}$ | $q^{50}$ | $q^{75}$
---|---|---|---|---|---|---|---
BULLISH | 0.13 | 0.99 | 0.17 | 0.87 | -0.44 | 0.15 | 0.66
CALM | -0.02 | 0.74 | 0.34 | 0.12 | -0.51 | -0.06 | 0.38
COVID | 0.05 | 0.70 | -0.04 | 0.23 | -0.34 | 0.04 | 0.47

Table 4: Summary statistics of estimated historical densities $\hat{z_{t}}$
defined in (3) for a respective scenario.  hedging_cc

For the GARCH-KDE approach, the estimated residual distributions
$\widehat{f}^{h}(z)$ from (4) are displayed in Figure 3. The empirical moments
and quantiles are listed in Table 4. Figure 12 illustrates the GARCH(1, 1)
volatility estimates of BTC returns and the 7-day historical volatility. As a
consequence of de-garching, all three distributions are roughly symmetric and
mean-zero. Deviations are direct results from market moves: the upward-moving
market behavior in the bullish period leads to a left-skewed residual
distribution. High drops in the stressed period result in a negatively skewed
distribution.

Figure 3: Estimated residual density $\widehat{f}^{h}(z)$ in (4) during
bullish market behavior, calm period and the stressed scenario during the
Corona crisis for $h=0.2$.  hedging_cc

To demonstrate that the GARCH-KDE method is an appropriate method of sampling
”close-to-actual-market” paths, the boxplots in Figure 4 illustrate the
distributions of one simulated GARCH-KDE path and the corresponding historical
distribution. The strength of the GARCH-KDE approach, of course, lies in the
fact that through Monte Carlo simulation, the analysis is not restricted to
one path.

(a) bullish

(b) calm

(c) covid

Figure 4: Distributions of one sampled GARCH-KDE path and historical returns.
hedging_cc

SVCJ paths are simulated with daily re-calibrated parameters, which are
summarized in Appendix Table 6. Selected statistical properties of both
scenario generation approaches are given in Table 14. We observe differences
in tails, extreme values and standard deviation. Discrepancies in
$\hat{\sigma}$ are natural consequences from different methodological
assumptions. The SVCJ approach assumes volatility to be stochastic, whereas
GARCH-KDE models $\sigma_{t}$ with GARCH(1,1). Discrepancies in path extremes
result from the SVCJ model assumptions on return jump size $\Xi$ in (2). In
the calibration routine, the $L_{2}$-regularization is applied to control
extreme parameter values. Yet, estimated return jump sizes can be very large.
Resulting Euler discretized paths contain trajectories with extreme moves of
the underlying. These are e.g. extremely low and high prices during the calm
and stressed scenario displayed in Table 14. The sometimes erratic BTC price
evolution suggests that such price moves are entirely implausible.

segment | $\hat{\mu}$ | $\hat{\sigma}$ | min | $q^{1}$ | $q^{50}$ | $q^{99}$ | max
---|---|---|---|---|---|---|---
bullish | -0.03 | 0.18 | -0.39 | -0.37 | -0.00 | 0.46 | 0.61
calm | -0.23 | 0.24 | -0.44 | -0.43 | -0.34 | 0.53 | 0.58
covid | -0.28 | 0.17 | -0.49 | -0.48 | -0.33 | 0.11 | 0.67

Table 5: Summary statistics of calibrated SVCJ jump size $\Xi$ per market
segment.  hedging_cc

period | $\kappa$ | $\rho$ | $V_{0}$ | $\theta$ | $\sigma$ | $\lambda$ | $\mu_{y}$ | $\sigma_{y}$ | $\mu_{v}$
---|---|---|---|---|---|---|---|---|---
$BS_{bullish}$ | - | - | - | - | 0.84 | - | - | - | -
$BS_{calm}$ | - | - | - | - | 0.68 | - | - | - | -
$BS_{covid}$ | - | - | - | - | 0.78 | - | - | - | -
$Merton_{bullish}$ | - | - | - | - | 0.17 | 0.11 | 0.0 | 0.82 | -
$Merton_{calm}$ | - | - | - | - | 0.42 | 0.72 | 0.0 | 0.55 | -
$Merton_{covid}$ - | - | - | - |  | 0.48 | 0.40 | 0.0 | 0.69 | -
$SV_{bullish}$ | 0.75 | 0.16 | 0.76 | 0.42 | 0.82 | - | - | - | -
$SV_{calm}$ | 1.60 | 0.17 | 0.35 | 1.10 | 0.68 | - | - | - | -
$SV_{covid}$ | 1.43 | 0.01 | 0.63 | 0.95 | 0.56 | - | - | - | -
$SVJ_{bullish}$ | 0.72 | 0.15 | 0.75 | 0.42 | 0.80 | 0.16 | 0.01 | 0.0 | -
$SVJ_{calm}$ | 1.28 | 0.18 | 0.33 | 1.05 | 0.68 | 0.37 | 0.01 | 0.0 | -
$SVJ_{covid}$ | 0.98 | 0.14 | 0.50 | 0.74 | 0.72 | 0.86 | -0.15 | 0.0 | -
$SVCJ_{bullish}$ | 0.51 | 0.14 | 0.74 | 0.09 | 0.88 | 0.31 | -0.04 | 0.0 | 0.45
$SVCJ_{calm}$ | 0.75 | 0.28 | 0.30 | 0.38 | 0.83 | 0.85 | -0.30 | 0.0 | 0.99
$SVCJ_{covid}$ | 0.61 | 0.22 | 0.52 | 0.18 | 0.89 | 1.04 | -0.35 | 0.0 | 0.54

Table 6: Average calibrated SVCJ parameters with market segments.  hedging_cc

###  3.4 Calibration results

In each period, calibration is performed daily using instruments satisfying
the liquidity and moneyness requirements specified in Section 2.3.2. For an
overview, average numbers of options per maturity range used for calibration
are summarized in Table 7. As a consequence of the moneyness requirement, more
longer-dated options are selected. The average parameter values per period are
summarized in Table 6. Section 3.4.1 and 3.4.2 provide a detailed perspective
on the dynamics of the calibrated parameters. Calibration is carried out on
the market’s mid IVs. Of course, ignoring bid-ask spreads and the possibility
of stale prices may produce arbitrage opportunities as well as spikes in
parameters and calibration errors. However, this is considered a minor issue
and ignored. RMSE’s for the models are illustrated in Appendix C.3. Naturally,
the model fit improves with increasing model complexity. Hence, the BS model
has the highest RMSE values on average while the SVCJ model has the lowest.

segment / maturity |  $\leq 1$ W | $(1\text{ W},2\text{ W}]$ | $(2\text{ W},3\text{ M}]$ | $(3\text{ M},6\text{ M}]$ | $(6\text{ M},9\text{ M}]$
---|---|---|---|---|---
bullish | 2.77 | 1.72 | 4.61 | 7.14 | 2.53
calm | 2.53 | 2.24 | 3.75 | 4.28 | 3.18
crisis | 3.00 | 3.03 | 4.44 | 5.58 | 5.33

Table 7: Overview of average maturity counts of all options in a daily IV
surface fullfiling the liquidity and moneyness requirements (Section 2.3.2).
hedging_cc

####  3.4.1 Affine jump diffusion models

The calibrated parameter $\sigma^{BS}$ provides meaningful insights into
market expectations. Levels vary in the range $\sigma^{BS}\in[50\ \%,175\
\%]$, with summary statistics for this parameter provided in Table 8. Due to
the volatile nature of the CC markets, levels of $\sigma^{BS}$ are generally
higher than in traditional markets (Madan et al., 2019). In comparison, the
VIX index in the time period 1990-2021 ranges between 9.5% and 60%, with the
95%-quantile at 33.5%. Figure 5 shows the dynamics of $\sigma^{BS}$ over the
entire time frame. In the bullish period, volatility levels rise up to
$120\%$. In the calm period, as expected, the levels are lower than in the
other two periods with $\sigma^{BS}\in[0.61,0.91]$.

behavior | mean | std. dev. | min | $q^{25}$ | $q^{50}$ | $q^{75}$ | max
---|---|---|---|---|---|---|---
bullish | 0.84 | 0.16 | 0.50 | 0.72 | 0.85 | 0.97 | 1.20
calm | 0.68 | 0.06 | 0.61 | 0.64 | 0.66 | 0.70 | 0.89
stressed | 0.78 | 0.21 | 0.57 | 0.63 | 0.73 | 0.87 | 1.75

Table 8: Summary statistics of daily $\sigma^{BS}$ calibration. hedging_cc
Figure 5: Daily calibration $\sigma^{BS}$ segregated by market segment in
chronological order. Volatility levels are very high compared to equities or
indices such as S & P 500.  hedging_cc

Figure 6 plots the calibrated parameters $\sigma^{JD}$ and $\lambda^{JD}$ of
the JD model over time. In general, levels of $\sigma^{JD}$ are lower than
$\sigma^{BS}$, clearly visible during the calm and stressed scenario. As the
JD model is an extension of the BS model, higher levels of $\sigma^{BS}$ are
partially compensated by the jump component. On many days, $\sigma^{JD}$ is
close to $\sigma^{BS}$. The reason for this are generally low values of the
annual jump intensity $\lambda^{JD}$ and jump size $\mu_{y}$. On average, the
JD model expects less than one jump in returns per year.

Figure 6: Interplay between $\sigma^{JD}$ and $\lambda^{JD}$ segregated by
market segment in chronological order. Mostly, for high levels of
$\sigma^{JD}$ we observe low levels of $\lambda^{JD}$ and vice versa.
hedging_cc

The evolution of $\lambda^{JD}$ is compared to the jump intensities of
extended models $\lambda^{SVJ}$ and $\lambda^{SVCJ}$ in Figure Appendix 13.
Throughout, annualised jump intensities are low with mostly
$\lambda^{SV(C)J}\leq 2.5$. Overall, the conclusion is that jumps are
infrequent. We observe contrasting levels of $\lambda^{SVCJ}$ and
$\lambda^{JD}$. They are not directly comparable, as the jump intensity
$\lambda^{SVCJ}$ contributes to simultaneous jumps in returns and stochastic
volatility, while $\lambda^{JD}$ and $\lambda^{SVJ}$ corresponds solely to
jumps in returns. For example, levels of $\lambda^{SVCJ}$ in the calm period
are high whereas $\lambda^{SVJ}$ is close to zero.

The plausibility of the stochastic volatility assumption is analyzed by the
evolution and levels of $\sigma^{v}$. In most periods, levels of $\sigma^{v}$
are higher compared to traditional markets. In the broad picture, the
evolution of $\sigma^{v}$ does not depend on model choice a shown in Figure
Appendix 14. Table 16 summarizes statistical properties of this parameter by
model and market segment. In the bullish and calm period, the indication for
stochastic volatility is strong with vol-of-vol levels at $q^{50}\geq 80\%$
and $q^{50}\geq 75\%$, respectively. In the stressed period, levels of
$\sigma^{v}$ in SV,SVJ,SVCJ remain high at $q^{50}\geq 73\%$.

Empirical evidence suggests that in traditional markets the correlation
parameter $\rho^{SV(CJ)}$ is usually negative. Specifically, when prices fall,
volatility increases. However, across all three market segments and models,
$\rho^{SV(CJ)}$ is mainly positive and close to zero as illustrated in Figure
Appendix 15. Hou et al. (2020) name this phenomenon the inverse leverage
effect in CC markets, that was previous reported on commodity markets by
Schwartz and Trolle (2009).

This relationship in the CC markets is also supported by the correlation
between the  CRIX and the  VCRIX under the physical measure $\mathbb{P}$.
Pearson’s correlation coefficient is $\rho^{pearson}=0.51$ in the bullish and
$\rho^{pearson}=0.64$ in the calm period, respectively. In the stressed
segment, correlation is negative with $\rho^{pearson}=-0.73$.

####  3.4.2 VG and CGMY

The prospect of infinite variation is evaluated by the calibration of the CGMY
model with average calibrated parameters in Table 9. Precisely, we are
interested in the evolution of the infinite activity parameter $Y^{CGMY}$
portrayed in Figure 7. As in each market segment we mostly have $Y^{CGMY}>0$,
there is evidence for infinite activity. In the bullish period, there is also
evidence of infinite variation, as we mostly have for $Y^{CGMY}\in(1,2]$ (Carr
et al., 2002).

market segment | C | G | M | Y
---|---|---|---|---
$CGMY_{bullish}$ | 4.24 | 22.21 | 24.79 | 1.20
$CGMY_{calm}$ | 10.37 | 7.67 | 9.30 | 0.14
$CGMY_{covid}$ | 7.94 | 11.38 | 17.24 | 0.68

Table 9: Average calibrated parameters of the CGMY model segregated by market
segment.  hedging_cc Figure 7: Daily $Y^{CGMY}$ calibration segregated by
market segment. Often, we observe $Y^{CGMY}>0$. This provides indication for
infinite activity. As $Y^{CGMY}\in(1,2]$ in the bullish segment, there is
evidence for infinite variation.  hedging_cc

The bullish period catches high magnitudes of jump size direction increase
parameters $G_{CGMY}$ and $M_{CGMY}$, reflecting the nature of this market
segment. Similarly, the increase in decreased jump size parameter $M_{CGMY}$
is mainly higher in the stressed scenario. A graphical illustration is given
in Figure Appendix 17. The VG is calibrated under representation (9). Overall,
volatility levels of $\sigma^{VG}$ are comparable to $\sigma^{BS}$, as
illustrated in Figure Appendix 16.

###  3.5 Hedge results

At the beginning of each market period, we short 1- and 3-month ATM options
with option premiums listed in Table 3. As outlined earlier, the price process
is simulated in both SVCJ and the GARCH-KDE setting. The exposure in each
option is dynamically hedged using the strategies summarized in Table 2. The
hedge performance is evaluated in terms of the hedge error $\varepsilon^{rel}$
and tail measures $\text{ES}^{5\%}$ and $\text{ES}^{95\%}$. The hedge results
are shown in Tables 10 and 11. For a concise graphical representation, the
best performing hedge strategies across models are compared in boxplots
displayed in Figures 8 and Figures 9. For each model, the best performing
strategy is selected according to $\operatorname{ES}^{5\%}$, as this provides
a trade-off between an extreme, yet plausible tail summary.

These are the main findings: First, with some exceptions, using multiple
instruments for hedging, i.e., Delta-Gamma and Delta-Vega hedges, when
compared to a simple Delta-hedge lead to a substantial reduction in tail risk.
Hence, whenever liquidly traded options are available for hedging, they should
be used.

Exceptions are the calm and COVID periods in the GARCH-KDE approach for the
short-maturity option as well as the calm period and GARCH-KDE approach for
the long-date option – here, no significant improvement is achieved by
including a second hedge instrument. In any case, no deterioration takes place
when using a second security for hedging. Contrary to the SVCJ approach, which
models several risk factors (jumps, stochastic volatility) explicitly, the
GARCH-KDE approach, with a smooth KDE density, exhibits less sensitivity to
concrete risk factors (e.g. Vega) in the calm period, even despite the GARCH
filter, see Figure 12.

Second, for short-dated options, no substantial differences occur in the
optimal hedging strategies across models. The sole exception is worse
performance of the VG- and CGMY-models in the calm period when price paths are
generated in the SVCJ model.

Third, turning to the long-dated option, although not always best performing,
it can be said that stochastic volatility models perform consistently well.
Amongst the stochastic volatility models, the SV model as the simplest model,
does not underperform and sometimes even is the best-performing model. For the
choice of a SV hedge model, the $\Delta^{SV}$-$\mathcal{V}^{SV}$ hedge is a
replicating strategy (Kurpiel and Roncalli, 1999) and performs often better
than other models under the same or different strategies. As calibrated jump
intensities $\lambda^{SVJ}$ and $\lambda^{SVCJ}$ are low, the SVJ or SVCJ are
often similar to the SV leading to comparable hedge results.

The simulated hedge results are confirmed in the historical hedge backtest. As
before, with expections (calm period), hedges involving multiple hedge
instruments consistently achieve desirable variance and tail risk reduction.
For example, in the bullish period, the $\Delta$-$\mathcal{V}_{SV}$ strategy
strikingly outperforms other best performing strategies.

(a) GARCH-KDE bullish

(b) SVCJ bullish

(c) GARCH-KDE calm

(d) SVCJ calm

(e) GARCH-KDE Covid

(f) SVCJ Covid

Figure 8: 1-month option hedge performance boxplots of $\pi^{rel}$ under (a)
GARCH-KDE and (b) SVCJ market simulation. For illustrative purposes
$\pi^{rel}$ is truncated at $q^{5}$ and $q^{95}$. The vertical axis portrays
$\Delta^{BS}$ hedge results compared each model’s best performing strategy.
This best performing strategy is selected according to the minimal
$\text{ES}^{5\%}$.

|  |  |  | bullish |  |  |  |
---|---|---|---|---|---|---|---|---
GARCH-KDE | $\Delta_{BS}$ |  $\Delta$-$\Gamma_{BS}$ |  $\Delta$-$\mathcal{V}_{SV}$ |  $\Delta$-$\Gamma_{JD}$ |  $\Delta$-$\mathcal{V}_{SVJ}$ |  $\Delta$-$\mathcal{V}_{SVCJ}$ |  $\Delta$-$\Gamma_{VG}$ |  $\Delta$-$\Gamma_{CGMY}$
Min | -3.35 | -2.58 | -2.62 | -2.48 | -2.63 | -2.59 | -2.51 | -2.53
$\text{ES}^{5\%}$ | -1.75 | -1.34 | -1.32 | -1.21 | -1.32 | -1.27 | -1.24 | -1.27
$\text{ES}^{95\%}$ | 1.17 | 1.49 | 1.51 | 1.65 | 1.5 | 1.57 | 1.64 | 1.61
Max | 3.31 | 5.32 | 5.29 | 5.33 | 5.28 | 5.35 | 4.77 | 5.05
$\varepsilon^{rel}$ | 63.14 | 59.55 | 59.39 | 60.43 | 59.40 | 59.75 | 60.97 | 60.87
SVCJ | $\Delta_{BS}$ |  $\Delta$-$\mathcal{V}_{BS}$ |  $\Delta$-$\mathcal{V}_{SV}$ |  $\Delta$-$\Gamma_{JD}$ |  $\Delta$-$\mathcal{V}_{SVJ}$ |  $\Delta$-$\mathcal{V}_{SVCJ}$ |  $\Delta$-$\Gamma_{VG}$ |  $\Delta$-$\Gamma_{CGMY}$
Min | -11.35 | -9.46 | -9.65 | -9.69 | -9.65 | -9.58 | -8.13 | -8.07
$\text{ES}^{5\%}$ | -1.48 | -1.16 | -1.16 | -1.06 | -1.16 | -1.12 | -1.08 | -1.10
$\text{ES}^{95\%}$ | 1.02 | 0.98 | 0.98 | 1.11 | 0.98 | 1.04 | 1.12 | 1.10
Max | 18.69 | 20.15 | 20.46 | 20.51 | 20.46 | 20.58 | 22.56 | 24.47
$\varepsilon^{rel}$ | 56.12 | 50.7 | 50.2 | 51.36 | 49.86 | 50.37 | 52.32 | 52.56
|  |  |  | calm |  |  |  |
GARCH-KDE | $\Delta_{BS}$ | $\Delta_{BS}$ | $\text{MV}_{SV}$ | $\Delta_{JD}$ | $\text{MV}_{SVJ}$ | $\text{MV}_{SVCJ}$ | $\Delta_{VG}$ | $\Delta_{CGMY}$
Min | -0.94 | -0.94 | -1.01 | -1.07 | -1.03 | -1.1 | -1.16 | -1.18
$\text{ES}^{5\%}$ | -0.16 | -0.16 | -0.17 | -0.19 | -0.15 | -0.15 | -0.2 | -0.2
$\text{ES}^{95\%}$ | 1.04 | 1.04 | 1.05 | 1.03 | 1.07 | 1.09 | 1.08 | 1.08
Max | 1.77 | 1.77 | 1.81 | 1.8 | 1.91 | 1.86 | 1.8 | 1.81
$\varepsilon^{rel}$ | 25.44 | 25.44 | 25.52 | 25.97 | 25.78 | 26.01 | 26.8 | 26.87
SVCJ | $\Delta_{BS}$ |  $\Delta$-$\Gamma_{BS}$ |  $\Delta$-$\mathcal{V}_{SV}$ |  $\Delta$-$\Gamma_{JD}$ |  $\Delta$-$\mathcal{V}_{SVJ}$ |  $\Delta$-$\mathcal{V}_{SVCJ}$ |  $\Delta$-$\Gamma_{VG}$ |  $\Delta$-$\Gamma_{CGMY}$
Min | -8.07 | -4.45 | -4.45 | -5.07 | -4.45 | -4.46 | -5.04 | -6.24
$\text{ES}^{5\%}$ | -2.20 | -1.01 | -1.00 | -1.01 | -0.96 | -1.01 | -1.19 | -1.14
$\text{ES}^{95\%}$ | 1.13 | 1.12 | 1.12 | 1.13 | 1.09 | 1.13 | 1.15 | 1.17
Max | 8.81 | 8.86 | 8.88 | 12.07 | 8.88 | 9.69 | 8.73 | 9.95
$\varepsilon^{rel}$ | 67.72 | 43.78 | 43.66 | 44.58 | 42.29 | 44.34 | 48.69 | 48.24
|  |  |  | covid |  |  |  |
GARCH-KDE | $\Delta_{BS}$ | $\Delta_{BS}$ | $\text{MV}_{SV}$ | $\Delta_{JD}$ | $\text{MV}_{SVJ}$ | $\text{MV}_{SVCJ}$ | $\Delta_{VG}$ | $\Delta_{CGMY}$
Min | -1.39 | -1.39 | -1.28 | -1.38 | -1.29 | -1.23 | -1.39 | -1.39
$\text{ES}^{5\%}$ | -0.49 | -0.49 | -0.46 | -0.55 | -0.51 | -0.39 | -0.48 | -0.48
$\text{ES}^{95\%}$ | 0.88 | 0.88 | 0.89 | 0.83 | 0.87 | 0.96 | 0.88 | 0.88
Max | 1.37 | 1.37 | 1.39 | 1.33 | 1.38 | 1.54 | 1.44 | 1.43
$\varepsilon^{rel}$ | 30.21 | 30.21 | 29.52 | 30.3 | 30.08 | 30.78 | 29.62 | 29.56
SVCJ | $\Delta_{BS}$ |  $\Delta$-$\Gamma_{BS}$ |  $\Delta$-$\mathcal{V}_{SV}$ |  $\Delta$-$\Gamma_{JD}$ |  $\Delta$-$\Gamma_{SVJ}$ |  $\Delta$-$\Gamma_{SVCJ}$ |  $\Delta$-$\Gamma_{VG}$ |  $\Delta$-$\Gamma_{CGMY}$
Min | -16.51 | -10.93 | -10.88 | -14.36 | -14.92 | -29.05 | -24.66 | -17.07
$\text{ES}^{5\%}$ | -3.13 | -1.64 | -1.72 | -1.76 | -1.76 | -1.84 | -1.85 | -1.75
$\text{ES}^{95\%}$ | 1.08 | 0.98 | 1.01 | 1.09 | 1.08 | 1.11 | 1.00 | 1.06
Max | 7.74 | 8.92 | 7.00 | 21.48 | 14.13 | 20.24 | 11.11 | 11.54
$\varepsilon^{rel}$ | 88.09 | 56.03 | 57.62 | 60.19 | 60.53 | 63.85 | 61.3 | 58.33

Table 10: 1-month ATM hedge performances with the best and worst performing
strategy. The table corresponds to results in Figure 8.

(a) GARCH-KDE bullish

(b) SVCJ bullish

(c) GARCH-KDE calm

(d) SVCJ calm

(e) GARCH-KDE covid

(f) SVCJ covid

Figure 9: 3-month option hedge performance boxplots of $\pi^{rel}$ under (a)
GARCH-KDE and (b) SVCJ market simulation. For illustrative purposes
$\pi^{rel}$ is truncated at $q^{5}$ and $q^{95}$. The vertical axis portrays
$\Delta^{BS}$ hedge results compared each model’s best performing strategy.
This best performing strategy is selected according to the minimal
$\text{ES}^{5\%}$.  hedging_cc

|  |  |  | bullish |  |  |  |
---|---|---|---|---|---|---|---|---
GARCH KDE | $\Delta_{BS}$ |  $\Delta$-$\Gamma_{BS}$ |  $\Delta$-$\mathcal{V}_{SV}$ |  $\Delta$-$\Gamma_{JD}$ |  $\Delta$-$\mathcal{V}_{SVJ}$ |  $\Delta$-$\mathcal{V}_{SVCJ}$ |  $\Delta$-$\Gamma_{VG}$ |  $\Delta$-$\Gamma_{CGMY}$
Min | -6.55 | -6.36 | -6.35 | -6.32 | -6.35 | -6.34 | -6.36 | -6.37
$\text{ES}^{5\%}$ | -2.38 | -1.99 | -1.95 | -1.96 | -1.97 | -1.95 | -1.98 | -1.99
$\text{ES}^{95\%}$ | 2.43 | 2.83 | 2.8 | 2.85 | 2.81 | 2.81 | 2.83 | 2.83
Max | 11.46 | 11.73 | 11.74 | 11.76 | 11.00 | 11.73 | 11.72 | 11.71
$\varepsilon^{rel}$ | 101.91 | 101.76 | 100.30 | 101.77 | 101.02 | 100.72 | 101.75 | 101.75
SVCJ | $\Delta_{BS}$ |  $\Delta$-$\Gamma_{BS}$ |  $\Delta$-$\mathcal{V}_{SV}$ |  $\Delta$-$\Gamma_{JD}$ |  $\Delta$-$\mathcal{V}_{SVJ}$ |  $\Delta$-$\mathcal{V}_{SVCJ}$ |  $\Delta$-$\Gamma_{VG}$ |  $\Delta$-$\Gamma_{CGMY}$
Min | -14.67 | -11.58 | -11.57 | -11.51 | -11.55 | -9.30 | -11.6 | -11.6
$\text{ES}^{5\%}$ | -1.10 | -0.64 | -0.63 | -0.62 | -0.63 | -0.62 | -0.63 | -0.63
$\text{ES}^{95\%}$ | 0.84 | 0.64 | 0.62 | 0.66 | 0.62 | 0.64 | 0.65 | 0.65
Max | 10.14 | 11.42 | 11.29 | 11.34 | 11.26 | 9.02 | 11.27 | 11.27
$\varepsilon^{rel}$ | 44.14 | 26.5 | 25.86 | 26.45 | 25.89 | 25.26 | 26.55 | 26.39
|  |  |  | calm |  |  |  |
GARCH-KDE | $\Delta_{BS}$ | $\Delta_{BS}$ | $\text{MV}_{SV}$ | $\Delta_{JD}$ | $\text{MV}_{SVJ}$ | $\Delta_{SVCJ}$ | $\Delta_{VG}$ | $\Delta_{CGMY}$
Min | -0.29 | -0.29 | -0.27 | -0.28 | -0.25 | -0.25 | -0.28 | -0.28
$\text{ES}^{5\%}$ | 0.18 | 0.18 | 0.20 | 0.15 | 0.19 | 0.20 | 0.19 | 0.19
$\text{ES}^{95\%}$ | 0.76 | 0.76 | 0.76 | 0.73 | 0.77 | 0.75 | 0.75 | 0.75
Max | 1.04 | 1.04 | 1.06 | 1.05 | 1.12 | 1.07 | 1.12 | 1.12
$\varepsilon^{rel}$ | 13.59 | 13.59 | 13.11 | 13.53 | 13.82 | 12.82 | 13.18 | 13.18
SVCJ | $\Delta_{BS}$ |  $\Delta$-$\Gamma_{BS}$ |  $\Delta$-$\mathcal{V}_{SV}$ |  $\Delta$-$\mathcal{V}_{JD}$ |  $\Delta$-$\mathcal{V}_{SVJ}$ |  $\Delta$-$\mathcal{V}_{SVCJ}$ |  $\Delta$-$\Gamma_{VG}$ |  $\Delta$-$\Gamma_{CGMY}$
Min | -12.63 | -8.68 | -12.75 | -6.32 | -7.79 | -12.75 | -12.73 | -12.74
$\text{ES}^{5\%}$ | -1.56 | -0.85 | -0.71 | -0.79 | -0.78 | -0.89 | -0.96 | -0.97
$\text{ES}^{95\%}$ | 0.88 | 0.82 | 0.69 | 0.77 | 0.79 | 0.88 | 0.89 | 0.90
Max | 7.74 | 5.19 | 7.79 | 4.15 | 7.78 | 8.99 | 8.97 | 9.25
$\varepsilon^{rel}$ | 53.39 | 33.36 | 28.28 | 31.01 | 31.26 | 36.05 | 38.82 | 39.09
|  |  |  | covid |  |  |  |
GARCH-KDE | $\Delta_{BS}$ |  $\Delta$-$\Gamma_{BS}$ |  $\Delta$-$\mathcal{V}_{SV}$ |  $\Delta$-$\Gamma_{JD}$ |  $\Delta$-$\Gamma_{SVJ}$ |  $\Delta$-$\Gamma_{SVCJ}$ |  $\Delta$-$\Gamma_{VG}$ |  $\Delta$-$\Gamma_{CGMY}$
Min | -4.36 | -2.69 | -2.64 | -2.64 | -2.44 | -2.58 | -2.7 | -2.71
$\text{ES}^{5\%}$ | -1.56 | -0.8 | -0.76 | -0.77 | -0.70 | -0.78 | -0.83 | -0.84
$\text{ES}^{95\%}$ | 0.6 | 0.93 | 0.9 | 0.97 | 1.11 | 1.00 | 0.91 | 0.9
Max | 3.88 | 3.33 | 3.32 | 4.52 | 4.57 | 4.45 | 4.49 | 4.55
$\varepsilon^{rel}$ | 50.06 | 34.48 | 33.09 | 34.57 | 40.02 | 37.4 | 34.63 | 34.67
SVCJ | $\Delta_{BS}$ |  $\Delta$-$\Gamma_{BS}$ |  $\Delta$-$\mathcal{V}_{SV}$ |  $\Delta$-$\Gamma_{JD}$ |  $\Delta$-$\Gamma_{SVJ}$ |  $\Delta$-$\Gamma_{SVCJ}$ |  $\Delta$-$\Gamma_{VG}$ |  $\Delta$-$\Gamma_{CGMY}$
Min | -13.53 | -7.89 | -7.9 | -14.3 | -11.76 | -11.75 | -20.99 | -11.72
$\text{ES}^{5\%}$ | -2.77 | -1.18 | -1.26 | -1.34 | -1.36 | -1.39 | -1.26 | -1.25
$\text{ES}^{95\%}$ | 0.87 | 0.71 | 0.68 | 0.78 | 0.94 | 0.93 | 0.73 | 0.73
Max | 13.48 | 10.78 | 10.77 | 13.60 | 13.66 | 13.6 | 13.67 | 13.65
$\varepsilon^{rel}$ | 88.42 | 38.24 | 39.34 | 43.95 | 48. | 49.06 | 42.99 | 41.27

Table 11: 3-month ATM hedge performance with the best and worst performing
strategy. The table corresponds to results in Figure 9.

(a) Bullish

(b) Calm

(c) Covid

Figure 10: Historical backtest hedge performance; $\pi^{rel}$ for 2-months ATM
options. For illustrative purposes $\pi^{rel}$ is truncated at $q^{5}$ and
$q^{95}$. The vertical axis portrays $\Delta^{BS}$ hedge results compared each
model’s best performing strategy. This best performing strategy is selected
according to the minimal $\text{ES}^{5\%}$.  hedging_cc

|  |  |  | bullish |  |  |  |
---|---|---|---|---|---|---|---|---
Backtest | $\Delta_{BS}$ |  $\Delta$-$\Gamma_{BS}$ |  $\Delta$-$\mathcal{V}_{SV}$ |  $\Delta$-$\mathcal{V}_{JD}$ |  $\Delta$-$\Gamma_{SVJ}$ |  $\Delta$-$\mathcal{V}_{SVCJ}$ |  $\Delta$-$\Gamma_{VG}$ |  $\Delta$-$\Gamma_{CGMY}$
Min | -4.34 | -3.85 | -1.35 | -3.85 | -3.85 | -2.27 | -3.86 | -3.86
$\text{ES}^{5\%}$ | -4.34 | -2.5 | -1.13 | -2.5 | -2.5 | -1.66 | -2.51 | -2.51
$\text{ES}^{95\%}$ | 1.75 | 2.20 | 2.09 | 2.20 | 2.09 | 2.08 | 2.19 | 2.19
Max | 2.11 | 2.56 | 2.56 | 2.56 | 2.56 | 2.55 | 2.55 | 2.55
$\varepsilon^{rel}$ | 98.75 | 101.15 | 73.56 | 101.25 | 96.69 | 81.22 | 101.13 | 101.09
|  |  |  | calm |  |  |  |
Backtest | $\Delta_{BS}$ |  $\Delta$-$\mathcal{V}_{BS}$ |  $\Delta$-$\mathcal{V}_{SV}$ | $\Delta_{JD}$ | $\text{MV}_{SVJ}$ | $\Delta_{SVCJ}$ | $\Delta_{VG}$ | $\Delta_{CGMY}$
Min | -0.20 | -0.48 | -0.16 | -0.15 | -0.15 | -0.13 | -0.15 | -0.15
$\text{ES}^{5\%}$ | -0.05 | -0.36 | -0.05 | -0.02 | -0.08 | -0.06 | -0.12 | -0.1
$\text{ES}^{95\%}$ | 0.62 | 0.32 | 0.62 | 0.6 | 0.58 | 0.62 | 0.68 | 0.67
Max | 0.82 | 0.41 | 0.84 | 0.75 | 0.78 | 0.82 | 0.88 | 0.90
$\varepsilon^{rel}$ | 14.31 | 13.22 | 14.37 | 13.48 | 14.43 | 14.52 | 16.53 | 16.30
|  |  |  | covid |  |  |  |
Backtest | $\Delta_{BS}$ |  $\Delta$-$\Gamma_{BS}$ |  $\Delta$-$\Gamma_{SV}$ |  $\Delta$-$\Gamma_{JD}$ |  $\Delta$-$\mathcal{V}_{SVJ}$ |  $\Delta$-$\Gamma_{SVCJ}$ |  $\Delta$-$\Gamma_{VG}$ |  $\Delta$-$\Gamma_{CGMY}$
Min | -1.96 | -1.33 | -1.23 | -1.19 | -1.32 | -1.36 | -1.27 | -1.26
$\text{ES}^{5\%}$ | -1.37 | -0.77 | -0.70 | -0.66 | -0.81 | -0.87 | -0.75 | -0.75
$\text{ES}^{95\%}$ | 0.70 | 0.51 | 0.56 | 0.58 | 0.56 | 0.50 | 0.52 | 0.51
Max | 0.78 | 0.69 | 0.82 | 0.91 | 0.72 | 0.70 | 0.76 | 0.76
$\varepsilon^{rel}$ | 49.47 | 27.68 | 27.34 | 27.62 | 29.97 | 30.45 | 27.10 | 27.11

Table 12: Backtest of hedging 2-month ATM options with the best and worst
performing strategies. The table corresponds to results in Figure 10.

####  3.5.1 Hedges with jump size correlation

The hedges above (Section 3.5) are performed under the assumption that the
jump size correlation parameter $\rho^{j}$ is zero. This assumption is
particularly well-founded on the BTC market, because jump size correlation
$\rho^{j}$ is reportedly insignificant (Hou et al., 2020). Nevertheless, we
investigate whether $\rho^{j}\neq 0$ impacts hedging and look for differences
to the main hedge results from Section 3.5. Therefore, the hedge routines are
repeated for a daily calibrated $\rho^{j}$ and for fixed parameter values
$\rho^{j}\in\\{-0.5,0.5\\}$. For comparison, we deliberately look at a few
examples. Note that changes to $\rho^{j}\neq 0$ impact the SVCJ Monte Carlo
simulation and the SVCJ hedge strategies (Table 2).

The calibration results from Figure 11 and Table 13 show that most calibrated
values lie close to $\rho^{j}=0$. As such, the SVCJ’s hedge performance
results are similar. An example are the SVCJ’s hedge performance comparisons
in the historical backtest in Table 17 to Table 19.

Figure 11: Distribution of daily calibrated $\rho^{j}$, where most values lie
close to $\rho^{j}=0$.  hedging_cc

segment | mean | std. dev. | min | $q^{25}$ | $q^{50}$ | $q^{75}$ | max
---|---|---|---|---|---|---|---
bullish | 0.01 | 0.07 | -0.27 | -0.01 | 0.00 | 0.02 | 0.50
calm | 0.03 | 0.07 | -0.09 | -0.01 | 0.01 | 0.06 | 0.34
covid | 0.04 | 0.13 | -0.12 | -0.01 | 0.00 | 0.07 | 0.84

Table 13: Summary statistics of daily $\rho^{j}$ calibration. hedging_cc

Appendix B, table 20 shows hedge results, when bumping the correlation to
$\rho^{j}=0.5$, resp. $\rho^{j}=-0.5$. Return jump sizes (2) depend on
$\rho^{j}$. Unsurprisingly, large correlation changes have a significant
impact on the hedge performance.

##  4 Conclusion

From a risk management perspective, CC markets are a highly interesting new
asset class: on the one hand CC prices are subject to extreme moves, jumps and
high volatility, while on the other hand, derivatives are actively traded –
and have been for several years – on several exchanges. This paper presents an
in-depth comparison of different hedging methods, providing concise answers to
the trade-off between hedging in a complete, albeit oversimplified model and
hedging in a more appropriate, albeit incomplete market model.

As a central part of the methodology, we simulate price paths given the
Bitcoin price history in two different ways: First, a semi-parametric approach
(under the physical measures $\mathbb{P}$) combines GARCH volatilities with
KDE estimates of the GARCH residuals. These paths are statistically close to
the actual market behaviour. Second, paths are generated (under the risk-
neutral measure $\mathbb{Q}$) in the parametric SVCJ model, where the SVCJ
model parameters include valuable information on the contributing risk factors
such as jumps. The time period under consideration features diverse market
behaviour, and as such, lends itself to being partitioned into “bullish”,
“calm” and “Covid-19” periods.

We hedge options with maturities of one and three months. If not directly
quoted on the BTC market, option prices are interpolated from an arbitrage-
free SVI-parametrization of the volatility surface. The options are then
hedged assuming risk managers use market models from the classes of affine
jump diffusion and infinite activity Lévy models, which feature risk factors
such as jumps and stochastic volatility. The calibration of these models
strongly support the following risk factors: stochastic volatility, infrequent
jumps, some indication for infinite activity and inverse leverage effects on
the market. Under GARCH-KDE and SVCJ, options are hedged with dynamic Delta,
Delta-Gamma, Delta-Vega and minimum variance hedging strategies.

For longer-dated options, multiple-instrument hedges lead to considerable tail
risk reduction. For the short-dated option, using multiple hedging instruments
did not significantly outperform a single-instrument hedge. This is in-line
with traditional markets, where even in highly volatile market periods, short-
dated options are less sensitive to volatility or Gamma effects. For longer-
dated options, multiple-instrument hedges consistently improve the hedge
quality. Hence, if several liquidly traded options are available for hedging,
they should be used. Among all models, persistently good hedge results are
achieved by hedging with stochastic volatility models. This demonstrates that
complete market models with stochastic volatility perform well, while models
allowing for jump risk, although more realistic, do not produce better hedges
due to the associated market incompleteness. These findings are confirmed for
a historical backtest, where a 2-month option is written every day, generating
a series of daily P&L’s from hedging at expiry.

## Statements and declarations

‘Statements and Declarations’: ’The code is available as quantlets, accessible
through Quantlet under the name  hedging_cc. Quotes and BTC prices are
provided by Tardis.dev and the Blockchain Research Center BRC. The data is
available upon request. Acknowledgments Financial support from the Deutsche
Forschungsgemeinschaft via the IRTG 1792 “High Dimensional Non Stationary Time
Series”, Humboldt-Universität zu Berlin, is gratefully acknowledged. The
European Union’s Horizon 2020 research and innovation program “FIN-TECH: A
Financial supervision and Technology compliance training programme” under the
grant agreement No 825215 (Topic: ICT-35-2018, Type of action: CSA), the
European Cooperation in Science & Technology COST Action grant CA19130 -
Fintech and Artificial Intelligence in Finance - Towards a transparent
financial industry, the Yushan Scholar Program of Taiwan and the Czech Science
Foundation’s grant no. 19-28231X / CAS: XDA 23020303 are greatly acknowledged.

##  Appendix A Hedging details

###  A.1 Hedge routine

We illustrate the dynamic hedging routine on a single instrument self-financed
hedging strategy $\xi$ and apply it analogously for all other hedging
strategies considered in this study. The simulated, discretised prices are
denoted by $S(t,i)$ are opposed to $S_{t}$, which refers to the continuous-
time process.

At time $t=0$ and for $B(0)=B_{0,i}=1$ the value of the portfolio for the
self-financed strategy $\xi$ is

$\displaystyle\Pi(0)$
$\displaystyle=C\left(0,S(0)\right)=\xi(0)S(0)+\left\\{C\left(0,S(0)\right)-\xi(0)S(0)\right\\}B(0)$
(18) $\displaystyle M(0)$ $\displaystyle=C\left(0,S(0)\right)-\xi(0)S(0)$

where $B(t)$ is a risk-free asset and $M(t)$ the money market account vector.
The value of the portfolio at time $t>0$ is

$\displaystyle M(t)$
$\displaystyle=M(t-dt)+\left\\{\xi(t-dt)-\xi(t)\right\\}\frac{S(t)}{B(t)}$
(19) $\displaystyle\Pi(t)$
$\displaystyle=\xi(t-dt)S(t)+M(t-dt)B(t-dt)e^{rdt}=\xi(t)S(t)+\underbrace{\frac{\Pi(t)-\xi(t)S(t)}{B(t)}}_{=M(t)}B(t)$

At maturity $T$, the final $PnL$ distribution vector is

$\Pi(T)=\xi(T-dt)S(t)+M(T-dt)B(t)$ (20)

###  A.2 Dynamic Delta-hedging

The option writer shorts the call $C(t)$, longs the underlying $S(t)$ and
sends the remainder to a money market account $B(t)$ for which

$dB(t)=rB(t)dt$

At time $t$, the value of portfolio $\Pi(t)$ is

$\Pi(t)=-C(t)+\Delta(t)S(t)+\frac{\left\\{C(t)-\Delta(t)S(t)\right\\}}{B(t)}B(t)$
(21)

The changes evolve through

$d\Pi(t)=-dC(t)+\Delta(t)dS(t)+\left\\{C(t)-\Delta(t)S(t)\right\\}rdt$ (22)

###  A.3 Dynamic multiple-instrument-hedging

We will explain the $\Delta-\mathcal{V}$ hedge in detail. The
$\Delta-\Gamma$\- hedge is performed accordingly. This strategy eliminates the
sensitivity to changes in the underlying and changes in volatility. The option
writer shorts the call option $C$, takes the position $\Delta$ in the asset
and $\Lambda$ in the second contingent claim. At time $t$, the value of the
portfolio is

$\Pi(t)=-C(t)+\Lambda C_{1}(t)+\Delta S(t)$ (23)

with the change in the portfolio $\Pi(t)$

$d\Pi(t)=\Delta(t)dS+\left\\{C(t)-\Delta S(t)-\Lambda C_{2}(t)\right\\}rdt-
dC(t)+\Lambda dC_{2}(t)$ (24)

That is

$\displaystyle d\Pi(t)$ $\displaystyle=\left(C(S,V,t)-\Delta S(t)-\Lambda
C_{2}(S,V,t)\right)rdt$ (25) $\displaystyle-\left(\frac{\partial C}{\partial
t}+\frac{1}{2}\frac{\partial^{2}C}{\partial
S^{2}}VS^{2}+\frac{1}{2}\frac{\partial^{2}C}{\partial
V^{2}}^{2}V+\frac{\partial^{2}C}{\partial V\partial S}\rho VS\right)dt$
$\displaystyle+\Lambda\left(\frac{\partial C_{2}}{\partial
t}+\frac{1}{2}\frac{\partial^{2}C_{2}}{\partial
S^{2}}VS^{2}+\frac{1}{2}\frac{\partial^{2}C_{2}}{\partial
V^{2}}^{2}V+\frac{\partial^{2}C_{2}}{\partial V\partial S}\rho VS\right)dt$
$\displaystyle+\left(\Lambda\frac{\partial C_{2}}{\partial S}-\frac{\partial
C}{\partial S}+\Delta\right)dS+\left(\Lambda\frac{\partial C_{2}}{\partial
V}-\frac{\partial C}{\partial V}\right)dV$

For the choice of

$\displaystyle\Delta$ $\displaystyle=\frac{\partial C}{\partial
S}-\Lambda\frac{\partial C_{2}}{\partial S}$ $\displaystyle\Lambda$
$\displaystyle=\frac{\partial C/\partial v}{\partial C_{2}/\partial v}$

the portfolio is $\Delta-\mathcal{V}$ hedged. Analogously, for the choice of

$\displaystyle\Delta$ $\displaystyle=\frac{\partial C}{\partial
S}-\Lambda\frac{\partial C_{2}}{\partial S}$ $\displaystyle\Lambda$
$\displaystyle=\frac{\partial^{2}C}{\partial^{2}S}$

this is a $\Delta-\Gamma$ hedge. For comparison, these hedges are applied to
all models in the class of affine jump diffusion models.

###  A.4 Alternative representation of the VG process

The alternative representation of the VG process has the characteristic
function

$\varphi_{\mathrm{VG}}(u\mathchar
24635\relax\;C,G,M)=\left(\frac{GM}{GM+(M-G)\mathrm{i}u+u^{2}}\right)^{C}$
(26)

where $C,G,M>0$ with

$\displaystyle C$ $\displaystyle=1/\nu$ (27) $\displaystyle G$
$\displaystyle=\left(\sqrt{\frac{1}{4}\left(\theta^{VG}\right)^{2}\nu^{2}+\frac{1}{2}\left(\sigma^{VG}\right)^{2}\nu}-\frac{1}{2}\theta^{VG}\nu\right)^{-1}$
$\displaystyle M$
$\displaystyle=\left(\sqrt{\frac{1}{4}\left(\theta^{VG}\right)^{2}v^{2}+\frac{1}{2}\left(\sigma^{VG}\right)^{2}\nu}+\frac{1}{2}\theta^{VG}\nu\right)^{-1}$

An increase in $G$ increases the size of upward jumps, while an increase in
$M$ increases the size of downward jumps. Accordingly, $\theta^{VG}$, $M$ and
$G$ account for the skewness of the distribution. $C$ governs the Levy-measure
by widening it with its increase and narrowing it with its decrease.

##  Appendix B Tables

framework | $\hat{\mu}$ | $\hat{\sigma}$ | min | $q^{1}$ | $q^{50}$ | $q^{99}$ | max
---|---|---|---|---|---|---|---
$SVCJ_{{\text{BULLISH}}_{30}}$ | 4087.32 | 343.05 | 1352.90 | 3411.83 | 4065.04 | 5177.02 | 15819.48
$SVCJ_{\text{CALM}_{30}}$ | 8369.33 | 1650.21 | 646.68 | 3475.29 | 8367.51 | 13092.95 | 26271.20
$SVCJ_{\text{COVID}_{30}}$ | 9800.32 | 1269.66 | 1435.49 | 5406.93 | 9804.85 | 13341.72 | 41464.61
$KDE_{{BULLISH}_{30}}$ | 4393.95 | 606.01 | 2089.55 | 3237.62 | 4277.65 | 6248.48 | 10209.30
$KDE_{\text{CALM}_{30}}$ | 8359.21 | 746.38 | 4545.46 | 6608.25 | 8349.06 | 10524.45 | 15611.32
$KDE_{\text{COVID}_{30}}$ | 9933.81 | 836.48 | 5579.96 | 8007.32 | 9848.62 | 12365.51 | 16863.17
$SVCJ_{\text{BULLISH}_{90}}$ | 4087.50 | 657.29 | 419.77 | 2961.11 | 4001.56 | 6336.31 | 56189.20
$SVCJ_{\text{CALM}_{90}}$ | 8367.54 | 2982.34 | 37.40 | 2488.41 | 8124.74 | 18415.20 | 118249.15
$SVCJ_{\text{COVID}_{90}}$ | 9796.71 | 2456.05 | 119.85 | 3620.50 | 9682.93 | 17545.53 | 115020.35
$KDE_{\text{BULLISH}_{90}}$ | 5116.43 | 1419.86 | 1325.30 | 3038.41 | 4762.11 | 9988.82 | 28593.53
$KDE_{\text{CALM}_{90}}$ | 8345.58 | 1407.72 | 3034.41 | 5341.07 | 8274.30 | 12590.88 | 22406.78
$KDE_{\text{COVID}_{90}}$ | 10718.15 | 3457.73 | 1560.16 | 4729.19 | 10007.73 | 23519.87 | 81081.55

Table 14: Summary statistics of scenario generations framework per market
segment and maturity  hedging_cc

TTM | a | b | $\rho^{SVI}$ | m | $\sigma^{SVI}$ | penalty
---|---|---|---|---|---|---
0.01 | 0.17 | 0.10 | 0.00 | 0.00 | 1.00 | 24.53
0.03 | 0.003 | 0.01 | 0.15 | 0.01 | 0.17 | 0.00001
0.07 | 0.01 | 0.04 | 0.00 | -0.01 | 0.08 | 0.000004
0.24 | 0.02 | 0.10 | -0.11 | -0.01 | 0.45 | 0.001
0.49 | 0.01 | 0.17 | -0.02 | 0.04 | 0.77 | 0.002
0.74 | 0.14 | 0.09 | 0.00 | 0.01 | 0.93 | 0.03
0.01 | 0.001 | 0.05 | -0.13 | 0.02 | 0.08 | 0.09
0.03 | 0.01 | 0.05 | -0.39 | 0.01 | 0.16 | 0.01
0.07 | 0.01 | 0.10 | -0.02 | 0.12 | 0.32 | 0.02
0.16 | 0.06 | 0.15 | -0.50 | -0.17 | 0.54 | 0.01
0.24 | 0.04 | 0.19 | -0.27 | -0.10 | 0.76 | 0.03
0.49 | 0.18 | 0.21 | 0.23 | 0.38 | 1.00 | 0.01
0.02 | 0.004 | 0.02 | 0.50 | 0.02 | 0.01 | 0.03
0.04 | 0.003 | 0.05 | -0.07 | -0.03 | 0.11 | 0.01
0.07 | 0.01 | 0.08 | -0.09 | -0.05 | 0.15 | 0.02
0.15 | 0.02 | 0.13 | 0.19 | 0.07 | 0.29 | 0.04
0.40 | 0.06 | 0.20 | -0.15 | -0.21 | 0.56 | 0.01
0.65 | 0.14 | 0.18 | 0.16 | -0.12 | 0.88 | 0.02

Table 15: Calibrated SVI parameters at the beginning of the bullish, calm and
stressed segment.  hedging_cc

| SV | SVJ | SVCJ
---|---|---|---
$\hat{\mu}$ | 0.82 | 0.78 | 0.87
$\hat{\sigma}$ | 0.32 | 0.33 | 0.35
$\operatorname{min}$ | 0.00 | 0.00 | 0.00
$q^{25}$ | 0.62 | 0.62 | 0.69
$q^{50}$ | 0.84 | 0.81 | 0.92
$q^{75}$ | 1.04 | 0.99 | 1.06
$\operatorname{max}$ | 1.49 | 1.57 | 2.43
$\hat{\mu}$ | 0.68 | 0.72 | 0.90
$\hat{\sigma}$ | 0.30 | 0.36 | 0.37
$\operatorname{min}$ | 0.00 | 0.00 | 0.00
$q^{25}$ | 0.50 | 0.56 | 0.70
$q^{50}$ | 0.75 | 0.79 | 1.02
$q^{75}$ | 0.90 | 0.95 | 1.19
$\operatorname{max}$ | 1.43 | 1.40 | 1.44
$\hat{\mu}$ | 0.56 | 0.72 | 0.84
$\hat{\sigma}$ | 0.49 | 0.66 | 0.45
$\operatorname{min}$ | 0.00 | 0.00 | 0.00
$q^{25}$ | 0.27 | 0.29 | 0.61
$q^{50}$ | 0.50 | 0.73 | 0.88
$q^{75}$ | 0.78 | 1.01 | 1.04
$\operatorname{max}$ | 3.83 | 6.33 | 3.83

Table 16: Summary statistics of $\sigma^{v}$ for all 3 market segments and
models.  hedging_cc

$\rho^{j}$ | Measure | $\Delta$ |  $\Delta$-$\Gamma$ |  $\Delta$-$\mathcal{V}$ | MV
---|---|---|---|---|---
0 | $\text{ES}^{5\%}$ | -2.97 | -2.50 | -1.66 | -2.99
market | $\text{ES}^{5\%}$ | -2.99 | -2.49 | -2.49 | -2.97
0 | $\varepsilon^{rel}$ | 95.68 | 100.77 | 64.79 | 95.96
market | $\varepsilon^{rel}$ | 97.07 | 101.51 | 102.65 | 96.93

Table 17: Historical hedge backtest performance comparison for the SVCJ hedge
strategies (Table 2) when the jump size correlation parameter $\rho^{j}$ is
calibrated to the market $\rho^{j}=\text{'market'}$ vs. $\rho^{j}=0$ during
the bullish period.

$\rho^{j}$ | Measure | $\Delta$ |  $\Delta$-$\Gamma$ |  $\Delta$-$\mathcal{V}$ | MV
---|---|---|---|---|---
0 | $\text{ES}^{5\%}$ | -0.06 | -0.33 | -0.35 | -0.07
market | $\text{ES}^{5\%}$ | -0.07 | -0.33 | -0.32 | -0.06
0 | $\varepsilon^{rel}$ | 2.07 | 1.63 | 1.91 | 2.36
market | $\varepsilon^{rel}$ | 2.03 | 1.79 | 1.75 | 2.02

Table 18: Historical hedge backtest performance comparison for the SVCJ hedge
strategies (Table 2) when the jump size correlation parameter $\rho^{j}$ is
calibrated to the market $\rho^{j}=\text{'market'}$ vs. $\rho^{j}=0$ during
the calm period.

$\rho^{j}$ | Measure | $\Delta$ |  $\Delta$-$\Gamma$ |  $\Delta$-$\mathcal{V}$ | MV
---|---|---|---|---|---
0 | $\text{ES}^{5\%}$ | -1.38 | -0.87 | -0.89 | -1.52
market | $\text{ES}^{5\%}$ | -1.41 | -0.69 | -0.73 | -1.41
0 | $\varepsilon^{rel}$ | 25.66 | 9.12 | 9.43 | 37.59
market | $\varepsilon^{rel}$ | 25.60 | 7.83 | 8.17 | 25.51

Table 19: Historical hedge backtest performance comparison for the SVCJ hedge
strategies (Table 2) when the jump size correlation parameter $\rho^{j}$ is
calibrated to the market $\rho^{j}=\text{'market'}$ vs. $\rho^{j}=0$ during
the covid period.

|  |  | calm |  |  |  |
---|---|---|---|---|---|---|---
Approach | Maturity | $\rho^{j}$ | Measure | $\Delta$ |  $\Delta$-$\Gamma$ |  $\Delta$-$\mathcal{V}$ | MV
SVCJ | 1M | 0 | $\text{ES}^{5\%}$ | -2.31 | -1.01 | -1.01 | -2.37
SVCJ | 1M | 0.5 | $\text{ES}^{5\%}$ | -2.50 | -1.09 | -1.09 | -2.50
SVCJ | 1M | -0.5 | $\text{ES}^{5\%}$ | -6.33 | -1.17 | -1.17 | -6.46
SVCJ | 1 M | 0 | $\varepsilon^{rel}$ | 49.52 | 19.72 | 19.66 | 51.48
SVCJ | 1 M | 0.5 | $\varepsilon^{rel}$ | 129.27 | 85.27 | 83.75 | 128.77
SVCJ | 1 M | -0.5 | $\varepsilon^{rel}$ | 399.69 | 23.99 | 23.50 | 408.91
SVCJ | 3 M | 0 | $\text{ES}^{5\%}$ | -1.59 | -0.89 | -0.89 | -1.61
SVCJ | 3 M | 0.5 | $\text{ES}^{5\%}$ | -3.77 | -1.35 | -1.21 | -3.69
SVCJ | 3 M | -0.5 | $\text{ES}^{5\%}$ | -6.71 | -0.97 | -0.96 | -6.85
SVCJ | 3 M | 0 | $\varepsilon^{rel}$ | 29.86 | 13.32 | 12.99 | 31.20
SVCJ | 3 M | 0.5 | $\varepsilon^{rel}$ | 307.47 | 284.2 | 119.3 | 303.13
SVCJ | 3 M | -0.5 | $\varepsilon^{rel}$ | 445.28 | 16.03 | 16.04 | 457.92
GARCH-KDE | 1 M | 0 | $\text{ES}^{5\%}$ | -0.16 | -0.49 | -0.48 | -0.15
GARCH-KDE | 1 M | 0.5 | $\text{ES}^{5\%}$ | -0.25 | -0.49 | -0.49 | -0.24
GARCH-KDE | 1 M | -0.5 |  |  $\text{ES}^{5\%}$ -0.15 | -0.48 | -0.48 | -0.14
GARCH-KDE | 1 M | 0 | $\varepsilon^{rel}$ | 6.6 | 5.32 | 5.3 | 6.77
GARCH-KDE | 1 M | 0.5 | $\varepsilon^{rel}$ | 7.84 | 5.36 | 5.34 | 7.82
GARCH-KDE | 1 M | -0.5 | $\varepsilon^{rel}$ | 6.90 | 5.32 | 5.31 | 7.18
GARCH-KDE | 3 M | 0 | $\text{ES}^{5\%}$ | 0.20 | -0.20 | -0.20 | 0.20
GARCH-KDE | 3 M | 0.50 | $\text{ES}^{5\%}$ | -0.28 | -0.31 | -0.31 | -0.27
GARCH-KDE | 3 M | -0.5 | $\text{ES}^{5\%}$ | 0.13 | -0.18 | -0.18 | 0.09
GARCH-KDE | 3 M | 0 | $\varepsilon^{rel}$ | 1.64 | 0.79 | 0.79 | 2.0
GARCH-KDE | 3 M | 0.5 | $\varepsilon^{rel}$ | 12.20 | 1.88 | 1.86 | 12.17
GARCH-KDE | 3 M | -0.5 | $\varepsilon^{rel}$ | 2.81 | 0.81 | 0.81 | 3.46
Backtest | 2 M | 0 | $\text{ES}^{5\%}$ | -0.06 | -0.33 | -0.35 | -0.07
Backtest | 2 M | 0.5 | $\text{ES}^{5\%}$ | -0.83 | -1.47 | -1.32 | -0.87
Backtest | 2 M | 0.5 | $\text{ES}^{5\%}$ | -1.53 | -0.95 | -0.97 | -1.68
Backtest | 2 M | 0 | $\varepsilon^{rel}$ | 2.07 | 1.63 | 1.91 | 2.36
Backtest | 2 M | 0.5 | $\varepsilon^{rel}$ | 61.87 | 54.43 | 51.02 | 64.12
Backtest | 2 M | -0.5 | $\varepsilon^{rel}$ | 32.61 | 10.06 | 10.51 | 46.97

Table 20: Comparison of SVCJ hedge performance for different values of
$\rho^{j}$ during the calm period. We observe consistently worse hedge
performances for $\rho^{j}\in\\{-0.5,0.5\\}$.

##  Appendix C Additional plots

###  C.1 GARCH(1,1) model

Figure 12: Estimated GARCH(1,1) volatility $\hat{\sigma}_{t}$ during bullish
market behavior, calm period and stressed scenario and the 7-day close-to-
close historical volatility.  hedging_cc

###  C.2 Calibration

Figure 13: Daily calibrated jump intensity $\lambda^{JD}$, $\lambda^{SVJ}$ and
$\lambda^{SVCJ}$ segregated chronologically by market segment. In each market
segments, annual jump intensity is generally $\lambda\leq 2$.  hedging_cc
Figure 14: Daily calibrated volatility of volatility $\sigma^{v}$ (SV),
$\sigma^{v}$ (SVJ) and $\sigma^{v}$ (SVCJ) plotted in chronological order by
market segment. For illustrative purposes, extremes are disregarded.
Information about extremes is provided in Table 16. Regardless of the model
choice, levels of $\sigma^{v}$ are high. This provides strong indication for
stochastic volatility.  hedging_cc Figure 15: Daily calibrated correlation
parameter $\rho^{SV}$, $\rho^{SVJ}$ and $\rho^{SVCJ}$ plotted in chronological
order by market segment. For illustrative purposes, extremes are disregarded.
As generally $\rho>0$, there is an indication for an inverse leverage effect
as reported in Hou et al. (2020).  hedging_cc Figure 16: Daily calibration of
$\sigma^{VG}$ plotted against $\sigma^{BS}$. Both models capture comparable
volatility levels.

(a) downward jumps M

(b) upward jumps G

Figure 17: (a) Evolution of $G_{CGMY}$ and (b) $M_{CGMY}$ segregated by market
segment. High magnitudes for both parameter values are observed during the
bullish and stressed scenario. For illustrative purposes, extremes are
excluded from this graph.

###  C.3 RMSE

(a) BS

(b) JD

(c) SV

Figure 18: RMSE with 95 %-confidence interval of the (a) BS, (b) JD and (c) SV
model.  hedging_cc

(a) SVJ

(b) SVCJ

Figure 19: RMSE with 95 %-confidence interval of the (a) SVJ, (b) SVCJ, (c) VG
and (d) CGMY model.  hedging_cc

(a) VG

(b) CGMY

Figure 20: RMSE with 95 %-confidence interval of the (a) VG and (b) CGMY
model.  hedging_cc

## References

  *   * Alexander and Imeraj (2022) Alexander C, Imeraj A (2022) Alexander, carol and imeraj, arben, delta hedging bitcoin options with a smile. SSRN Working Paper 10.2139/ssrn.4097909
  * Alexander et al. (2021) Alexander C, Deng J, Zou B (2021) Hedging with bitcoin futures: The effect of liquidation loss aversion and aggressive trading 10.13140/RG.2.2.25471.23200/2
  * Athey et al. (2016) Athey S, Parashkevov I, Sarukkai V, et al. (2016) Bitcoin pricing, adoption, and usage: Theory and evidence. Stanford Graduate School of Business Research Paper Series No 16-42
  * Bates (1996) Bates DS (1996) Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options. The Review of Financial Studies 9(1):69–107. 10.1093/rfs/9.1.69
  * Belaygorod (2005) Belaygorod A (2005) Solving continuous time affine jump-diffusion models for econometric inference. John M Olin School of Business working paper
  * Biais et al. (2022) Biais B, Bisiere C, Bouvard M, et al. (2022) Equilibrium bitcoin pricing. Journal of Finance, forthcoming
  * Black and Scholes (1973) Black F, Scholes M (1973) The pricing of options and corporate liabilities. Journal of Political Economy 81(3):637–54. 10.1086/260062
  * Bollerslev (1986) Bollerslev T (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31(3):307 – 327. 10.1016/0304-4076(86)90063-1
  * Bouri et al. (2017) Bouri E, Molnár P, Azzi G, et al. (2017) On the hedge and safe haven properties of bitcoin: Is it really more than a diversifier? Finance Research Letters 20:192–198. 10.1016/j.frl.2016.09.025
  * Branger et al. (2009) Branger N, Hansis A, Schlag C (2009) Expected option returns and the structure of jump risk premia. AFA 2010 Atlanta Meetings Paper 10.2139/ssrn.1340575
  * Branger et al. (2012) Branger N, Krautheim E, Schlag C, et al. (2012) Hedging under model misspecification: All risk factors are equal, but some are more equal than others …. Journal of Futures Markets 32:397 – 430. 10.1002/fut.20530
  * Broadie and Kaya (2005) Broadie M, Kaya O (2005) Exact simulation of option greeks under stochastic volatility and jump diffusion models. In: Proceedings of the 36th Conference on Winter Simulation, pp 1607–1615, 10.1109/WSC.2004.1371506
  * Broadie et al. (2007) Broadie M, Chernov M, Johannes M (2007) Model specification and risk premia: Evidence from futures options. The Journal of Finance 62(3):1453–1490. 10.1111/j.1540-6261.2007.01241.x
  * Carr and Madan (1999) Carr P, Madan D (1999) Option valuation using the fast fourier transform. Journal of Computational Finance 2:61–73. 10.21314/JCF.1999.043
  * Carr et al. (2002) Carr P, Geman H, Madan D, et al. (2002) The fine structure of asset returns: An empirical investigation. The Journal of Business 75(2):305–332. 10.1086/338705
  * Chernov et al. (2003) Chernov M, Gallant A, Ghysels E, et al. (2003) Alternative models for stock price dynamics. Journal of Econometrics 116(1):225–257. 10.1016/S0304-4076(03)00108-8
  * Clark (2011) Clark I (2011) Foreign Exchange Option Pricing: A Practitioner’s Guide. The Wiley Finance Series
  * Detering and Packham (2015) Detering N, Packham N (2015) Model risk in incomplete markets with jumps. Springer Proceedings in Mathematics and Statistics 99:39–59. 10.1007/978-3-319-09114-3_3
  * Detering and Packham (2016) Detering N, Packham N (2016) Model risk of contingent claims. Quantitative Finance 16(9):1357–1374. 10.1080/14697688.2016.1142671
  * Duffie et al. (2000) Duffie D, Pan J, Singleton K (2000) Transform analysis and asset pricing for affine jump diffusions. Econometrica 68(6):1343–1376. 10.1111/1468-0262.00164
  * Dyhrberg (2016) Dyhrberg AH (2016) Hedging capabilities of bitcoin. is it the virtual gold? Finance Research Letters 16:139–144. 10.1016/j.frl.2015.10.025
  * Eraker (2004) Eraker B (2004) Do stock prices and volatility jump? reconciling evidence from spot and option prices. The Journal of Finance 59(3):1367–1403. 10.1111/j.1540-6261.2004.00666.x
  * Eraker et al. (2003) Eraker B, Johannes M, Polson N (2003) The impact of jumps in volatility and returns. The Journal of Finance 58(3):1269–1300. 10.1111/1540-6261.00566
  * Föllmer and Sondermann (1986) Föllmer H, Sondermann D (1986) Hedging of non-redundant contingent claims. Contributions to Mathematical Economics, in Honour of Gérard Debreu North-Holland, Amsterdam
  * Gatheral (2004) Gatheral J (2004) A parsimonious arbitrage-free implied volatility parameterization with application to the valuation of volatility derivatives. Presentation at Global Derivatives & Risk Management, Madrid
  * Gatheral and Jacquier (2014) Gatheral J, Jacquier A (2014) Arbitrage-free svi volatility surfaces. Quantitative Finance 14(1):59–71. 10.1080/14697688.2013.819986
  * Heston (1993) Heston SL (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6(2):327–343. 10.1093/rfs/6.2.327
  * Hou et al. (2020) Hou A, Wang W, Chen K, et al. (2020) Pricing cryptocurrency options: the case of crix and bitcoin. Journal of Financial Econometrics 10.1007/s42521-019-00002-1
  * Kim et al. (2021) Kim A, Trimborn S, Härdle WK (2021) Vcrix — a volatility index for crypto-currencies. International Review of Financial Analysis p 101915. 10.1016/j.irfa.2021.101915
  * Kraft (1988) Kraft D (1988) A software package for sequential quadratic programming. Deutsche Forschungs- und Versuchsanstalt für Luft- und Raumfahrt Köln: Forschungsbericht, Wiss. Berichtswesen d. DFVLR
  * Kurpiel and Roncalli (1999) Kurpiel A, Roncalli T (1999) Option hedging with stochastic volatility 10.2139/ssrn.1031927
  * Madan et al. (1998) Madan DB, Carr PP, Chang EC (1998) The variance gamma process and option pricing. Review of Finance 2(1):79–105. https://doi.org/10.1023/A:1009703431535
  * Madan et al. (2019) Madan DB, Reyners S, Schoutens W (2019) Advanced model calibration on bitcoin options. Digital Finance 1(1):117–137. 10.1007/s42521-019-00002-1
  * Makarov (2020) Makarov AIgor Schoar (2020) Trading and arbitrage in cryptocurrency markets. Journal of Financial Economics 135(2):293–319. https://doi.org/10.1016/j.jfineco.2019.07.001
  * McNeil and Frey (2000) McNeil AJ, Frey R (2000) Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach. Journal of Empirical Finance 7(3):271–300. 10.1016/S0927-5398(00)00012-8
  * Merton (1976) Merton RC (1976) Option pricing when underlying stock returns are discontinuous. Journal of financial economics 3(1):125–144. 10.1016/0304-405X(76)90022-2
  * Nekhili and Sultan (2021) Nekhili R, Sultan J (2021) Hedging bitcoin with conventional assets. Borsa Istanbul Review 10.1016/j.bir.2021.09.003
  * Poulsen et al. (2009) Poulsen R, Schenk-Hoppé K, Ewald CO (2009) Risk minimization in stochastic volatility models: model risk and empirical performance. Quantitative Finance 9(6):693–704. 10.1080/14697680902852738
  * Protter (2005) Protter PE (2005) Stochastic Integration and Differential Equations. Stochastic Modelling and Applied Probability, Springer Berlin Heidelberg
  * Scaillet et al. (2018) Scaillet O, Treccani A, Trevisan C (2018) High-frequency jump analysis of the bitcoin market*. Journal of Financial Econometrics 18(2):209–232. 10.1093/jjfinec/nby013
  * Schwartz and Trolle (2009) Schwartz E, Trolle A (2009) Unspanned stochastic volatility and the pricing of commodity derivatives. Review of Financial Studies 22:4423–4461. 10.2139/ssrn.1108546
  * Sebastião and Godinho (2020) Sebastião H, Godinho P (2020) Bitcoin futures: An effective tool for hedging cryptocurrencies. Finance Research Letters 33:101,230. 10.1016/j.frl.2019.07.003
  * Selmi et al. (2018) Selmi R, Mensi W, Hammoudeh S, et al. (2018) Is bitcoin a hedge, a safe haven or a diversifier for oil price movements? a comparison with gold. Energy Economics 74(C):787–801. 10.1016/j.eneco.2018.07.007
  * Tikhonov et al. (2011) Tikhonov A, Leonov A, Yagola A (2011) Nonlinear ill-posed problems. De Gruyter, doi:10.1515/9783110883237.505
  * Ting and Ewald (2013) Ting SHM, Ewald CO (2013) On the performance of asymptotic locally risk minimising hedges in the heston stochastic volatility model. Quantitative Finance 13(6):939–954. 10.1080/14697688.2012.691987
  * Trimborn and Härdle (2018) Trimborn S, Härdle WK (2018) Crix an index for cryptocurrencies. Journal of Empirical Finance 49:107–122. 10.1016/j.jempfin.2018.08.004