key: cord-0514043-5ww8zhxo
authors: Matic, Jovanka Lili; Packham, Natalie; Hardle, Wolfgang Karl
title: Hedging Cryptocurrency Options
date: 2021-11-23
journal: nan
DOI: nan
sha: b0b34c2e042888d0b015b4ce7db60a90dc4c8a09
doc_id: 514043
cord_uid: 5ww8zhxo

The cryptocurrency (CC) 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 L'evy processes. First, market data is calibrated to 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) assumption and a close-to-actual-market GARCH-filtered kernel density estimation. In these two markets, 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-date options, good tail risk reduction is consistently achieved with multiple-instrument hedges. This is persistently accomplished with complete market models with stochastic volatility.

but incomplete models. The calibration results reveal a strong indi-

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. Despite growing market volume, cryptocurrency markets continue to exhibit high volatility and frequent jumps, posing challenges to valuation and risk management. From the point of view of market makers and in the interest of financial stability, it is of high priority to understand and monitor risks associated with losses.

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) . 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 the SVCJ model. The second scenario generation method is based on GARCH-filtered Kernel-density estimation (GARCH-KDE) close to actual market dynamics. Under each of the two different market simulation methods, options are hedged where the hedger considers 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) jumpdiffusion 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 (V G) model (Madan et al., 1998) and the CGMY model (Carr et al., 2002) . Options are hedged dynamically with the following hedge strategies: Delta (∆), Delta-Gamma (∆ − Γ), Delta-Vega (∆ − V) and minimum variance strategies. 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 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.

In this section, we introduce the methodology, comprising market scenario generation, option valuation and hedging.

We describe how to generate synthetic market data, which serves as the input for the remainder of the analysis. The principal goal of synthetic scenario generation is to imitate the BTC market behavior, especially retaining its statistical properties. Monte Carlo simulation provides the flexibility 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 Q with a forward looking perspective. The non-parametric simulation relates to the past market behavior performed under the physical measure 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 labeled as the calm period. With an overall standard deviationσ = 756.55, price movements are more stagnant compared to the bullish segment. The last segment is the We now turn to a formal mathematical framework. Let the BTC market to be a continuous-time, frictionless financial market. Borrowing and short-selling are permitted. The constant risk-free interest rate r ≥ 0 and the time horizon T < ∞ are fixed. On a filtered probability space Ω, F, (F t ) t∈[0,T ] , P , the asset price process and the risk-free asset are defined by adapted semimartingales (S t ) t≥0 and (B t ) t≥0 , where B 0 = 1 and B t = e rt , t ≥ 0, respectively. The filtration is assumed to satisfy the usual conditions (e.g. (Protter, 2005) ). To ensure the absence of the arbitrage, we assume the existence of a riskneutral measure Q. We consider an option writer's perspective and short a European call option. The price of the option with strike K and time-tomaturity (TTM) τ = T −t at time t < T is C(t, τ, K). For multiple-instrument hedges, we further assume the existence of a liquidly traded call option suitable for hedging C 2 (t, τ, K 2 ), where K 2 = K. The dynamic hedging strategy ξ = (ξ 0 , ξ 1 ) = (ξ 0 (t), ξ 1 (t))) 0≤t≤T is an F-predictable process, where ξ 0 (t) and ξ 1 (t) denote the amounts in the risk-free security and the asset, respectively. The resulting portfolio process Π = (Π t ) t≥0 is admissible and self-financing. The evolution of the value process Π is reviewed in detail in Appendix A.1, A.2 and A.3.

The finite time horizon T is partitioned into T = {0, δt, 2δt, . . . , mδt = T }, where m ∈ N denotes the m-th trading day and δt = 1 365 . Scenarios are N = 100000 trajectories of the asset price process S(t) = (S t,i ), where i = 1, . . . , N and t = 0, 1, . . . T . The parametric scenario generation approach assumes that the dynamics of the asset price process S t and the volatility process V t 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. Precisely, the model dynamics are

where W s t , W v t are two standard Wiener processes correlated with correlation coefficient ρ. The mean reversion speed is denoted by κ, θ is the mean reversion level and σ v the scale of V t . The model allows for contemporaneous arrivals of jumps in returns and jumps in volatility goverened by the Poisson process N t = N s t = N v t with constant intensity λ = λ s = λ v . Jump sizes in volatility Z v t are exponentially distributed Z v t ∼ ε (µ v ) and jumps sizes in asset prices are conditionally normally distributed

where µ s is the conditional mean jump size in the asset price given by

In detail, µ s is the unconditional mean, σ s the jump size standard deviation and ρ j is the correlation coefficient between jumps. From an empirical point of view, in most markets, jumps occur seldomly and are difficult to detect, and, as a consequence, the calibration of ρ j is unreliable (Broadie et al., 2007) . We follow the recommendation of Broadie et al. (2007) , Chernov et al. (2003) , Eraker et al. (2003) , Eraker (2004) and Branger et al. (2009) and set ρ j = 0. Furthermore, the risk premium is set to zero, so that µ = r and P = Q. The resulting paths are simulated according to the Euler-Maruyama discretization of (1) suggested in Belaygorod (2005) . The corresponding model parameters are re-calibrated daily according to the methodology described in section 2.3.2. 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 ) denote BTC logreturns and (σ t ) the estimated GARCH(1,1) volatility (Bollerslev, 1986) . The kernel density estimation is performed on "de-garched" residualŝ

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

where K denotes the Gaussian Kernel. The resulting generated paths are

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

This section describes how option prices are derived from the market IV quotes. 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 σ BS (k, τ ) denote the BS IV with log-moneyness k = log (K/S 0 ) and total implied variance w(k, τ ) = σ 2 BS (k, τ )τ . For a fixed τ , the raw SVI parameterization of a total implied variance smile as initially presented in Gatheral (2004) is

In the parameter set χ R = {a, b, ρ, m, σ}, a ∈ R governs the general level of variance, b ≥ 0 regulates the slopes of the wings, ρ ∈ [−1, 1] controls the skew, m ∈ R enables horizontal smile shifts and σ > 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 χ 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 ATM total implied variance θ T = w(0, T ) is interpolated for t 1 < T < t 2 as in Gatheral and Jacquier (2014) . The resulting option price C(T, K) is a convex combination

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.

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 therefore 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 a hedge model. The volatility is constant with V t = σ and there are no discontinuities from jumps N s t = N v t = 0. A slightly more complex model is the JD model. It assumes constant volatility with V t = θ, σ V = 0 and extends the BS model by allowing for jumps in returns, but with N v t = 0. The jump size is log ξ ∼ N µ s , δ 2 s distributed. Evidence for stochastic volatility motivates the choice of the SV model. The jump component is excluded with λ = 0 and N s t = N v 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 this model, the jump component in the variance process V t is dropped while keeping the jump component in the spot process S t , i.e. N v t = 0. 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

with the characteristic function of the VG-process X VG t given by

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 σ; θ governs the symmetry of the distribution and therefore controls the implied volatility skew; ν controls for tails, kurtosis and thus regulates the shape of the volatility surface. An alternative representation of the V G process pleasant for practical interpretation has the characteristic function

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, θ, M and G account for the skewness of the distribution. An increase in C widens the Levy-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 CGM Y t with the characteristic function

The X V G 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 ∈ (1, 2], there is also infinite variation.

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

where c T (k) denotes the α-damped option price c T (k) = e αk C T (k) and ϕ c T (t) its characteristic function. 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

where Γ is a diagonal positive semi-definite matrix. The matrix Γ corresponds to the Tikhonov L 2 -regularization, which gives preference to parameters with smaller norms. The entries in the matrix Γ are chosen individually for each parameter to ensure that they maintain the same reasonable order of magnitude.

The parameter space Θ ⊂ 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. As deep out-of-themoney or deep in-the-money instruments do not provide valuable input for calibration in our case, the ∆ 25 criterion is imposed: all claims whose ∆ BS is smaller than 0.25 or larger than 0.75 in terms of the absolute value, that is 0.25 < |∆ M arket | < 0.75, are disregarded.

To protect against broad market movements, we examine hedging with marketrisk-related sensitivities (∆, Γ, V) = ∂C ∂S , ∂ 2 C ∂ 2 S , ∂C ∂σ . The goal is to protect the position against first-order changes in the underlying S = {S t , t ∈ T }, second-order changes (i.e., first-order changes in ∆) and to changes in σ, respectively. To achieve ∆ − Γ-or ∆ − V-neutrality, an additional liquid option C 2 (S(t), T, K 1 ) with strike K 1 = 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 ∆-and ∆ − Γ-hedging strategies are applied to all hedge models. The ∆ − 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. Models that incorporate jumps are incomplete and difficult to hedge. Jumps and infinite activity Lévy processes are therefore often hedged with quadratic variance-related hedging strategies. Under the assumption of symmetric losses and gains, the aim is to find the strategy ξ * under Q that minimizes the hedging error in terms of the mean-squared error (Föllmer and Sondermann, 1986) (Π(0), ξ * (t)) = argmin 

In a perfect hedge in a complete market we have π rel = 0. However, in practice, due to model incompleteness, discretization and model uncertainty, π rel = 0. We evaluate the hedge performance with the relative hedge error ε hedge as applied in e.g. Poulsen et al. (2009) , defined as ε hedge = 100 Var (π rel ).

The rationale behind ε hedge is that standard deviation represents a measure of uncertainty. A sophisticated hedge strategy reduces or ideally eliminate uncertainty (Branger et al., 2012) . The tail behavior is evaluated by the expected shortfall Table 4 : Summary statistics of estimated historical densitiesẑ t defined in (3) for a respective scenario. hedging cc significantly with maturity. Therefore, the data is filtered with liquidity cutoffs. All claims without trading volume are disregarded. In addition, the ∆ 25criterion is imposed.

Option prices are obtained at 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 23 . The resulting option prices used in the hedging routine are displayed in Table 3 . 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 σ 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 ρ, varies across market segments. Usually negative values of ρ indicate a preference for OTM puts over OTM calls. In the bullish period, skewness is close to zero across most maturities.

For the GARCH-KDE approach, the estimated residual distributions f h (z) from (4) are displayed in Figure 3 . The empirical moments and quantiles are listed in Table 4 . Figure 9 illustrates the GARCH(1, 1) estimates of BTC returns. As a consequence from de-garching, all three distributions are roughly symmetric and mean-zero. Deviations are direct results from market moves: Table 6 . Selected statistical properties of both scenario generation approaches are given in Table 22 . We observe differences in tails, extreme values and standard deviation. Discrepancies inσ are natural consequences from different methodological assumptions. The SVCJ approach assumes volatility to be stochastic, whereas GARCH-KDE models σ t with GARCH(1,1). Discrepancies in path extremes result from the SVCJ model assumptions on return jump size Ξ SV CJ in (2). In the calibration routine, the 

In each period, calibration is performed daily using instruments satisfying the ∆ 25 -criterion. For an overview, average numbers of options per maturity range used for calibration are summarized in Table 7 . As a consequence of the ∆ 25 -criterion, more longer-dated options are selected. The average parameter values per period are summarized in 

The calibrated parameter σ BS provides meaningful insights into market expectations. Levels vary in the range σ BS ∈ [50 %, 175 %], with summary statistics for this parameter provided in Table 8 . Due to the volatile nature of the CC markets, levels of σ 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 4 shows the dynamics of σ 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 σ BS ∈ [0.61, 0.91]. Figure Figure Appendix 10 . Throughout, yearly jump intensities are low with mostly λ SV (C)J ≤ 2.5. Overall, the conclusion is that jumps are infrequent. We observe contrasting levels of λ SV CJ and λ JD . They are not directly comparable, as the jump intensity λ SV CJ contributes to simultaneous jumps in returns and stochastic volatility, while λ JD and λ SV J corresponds solely to jumps in returns. For example, levels of λ SV CJ in the calm period are high whereas λ SV J is close to zero.

The plausibility of the stochastic volatility assumption is analyzed by the evolution and levels of σ v . In most periods, levels of σ v are higher compared to traditional markets. In the broad picture, the evolution of σ v does not depend on model choice a shown in Figure Appendix 11 . Table 24 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 ≥ 80% and q 50 ≥ 75%, respectively. In the stressed period, levels of σ v SV (C)J remain high for q 50 ≥ 73%.

Empirical evidence suggests that in traditional markets the correlation parameter ρ SV (CJ) is usually negative. Specifically, when prices fall, volatility increases. However, across all three market segments and models, ρ SV (CJ) is mainly positive and close to zero as illustrated in Figure Appendix 12 between the CRIX and the VCRIX under the physical measure P. Pearson's correlation coefficient is ρ pearson = 0.51 in the bullish and ρ pearson = 0.64 in the calm period, respectively. In the stressed segment, correlation is negative with ρ pearson = −0.73.

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 CGM Y portrayed in Figure 6 . In each market segment, as Y > 0 widely, there is strong evidence for infinite activity. In the bullish period, for Y CGM Y ∈ (1, 2] largely, there is also evidence for infinite variation (Carr et al., 2002) . The bullish 

At the beginning of each market period, we short at-the-money options with maturities T = 1 M and T = 3 M at the option prices 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 π rel with regard to the median q 50 , hedge error ε rel , tail measures ES 5% and ES 95% as well as extremes with results in Table 10 to Table 21 . For a concise graphical representation, the best performing hedge strategies across models are compared in boxplots displayed in Figures 7 and Figures 8. For each model , the best performing strategy is selected according to ES 5% .

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. The calm and COVID periods in the GARCH-KDE approach are exceptions 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. In the GARCH-KDE approach, paths are simulated from return residualsẐ t . By definition, returns are correlated with S t . Hence, under this market simulation, an important risk factor is the underlying itself. Especially on short time intervals, the option is most sensitive to the underlying itself.

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 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 model, 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 ∆ SV − 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 λ SV J and λ SV CJ are low, the SVJ or SVCJ are often similar to the SV leading to comparable hedge results. For illustrative purposes π rel is truncated at q 5% and q 95% . The vertical axis portrays ∆ BS hedge results compared to the best performing strategy of a given hedge model. The best performing strategy is selected for the minimal ES 5% . hedging cc Table 10 : Hedge performance for T = 1 M under GARCH-KDE simulation in the bullish segment with the best and the worst in worst performing strategy. For illustrative purposes π rel is truncated at q 5% and q 95% . The vertical axis portrays ∆ BS hedge results compared to the best performing strategy of a given hedge model. The best performing strategy is selected for the minimal ES 5% . hedging cc 

-14.67 -11.58 -11.57 -11.51 -11.55 -9.30 -11.6 -11.6 ES 5% -1. 

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 tradedand 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 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 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 arbitragefree 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 SCVJ, 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. 

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

At maturity T , the final P nL distribution vector is

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

At time t, the value of portfolio V (t) is

The changes evolve through

A.3 Dynamic Delta-Gamma-hedging

We will explain the ∆ − V hedge in detail. The ∆ − Γ-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 ∆ in the asset and Λ in the second contingent claim. At time t, the value of the portfolio is

with the change in the portfolio V (t)

That is

For the choice of ∆ = ∂C ∂S − Λ ∂C 2 ∂S Λ = ∂C/∂v ∂C 2 /∂v the portfolio is ∆ − V hedged. Analogously, for the choice of

this is a ∆ − Γ hedge. For comparison, these hedges are applied to all models in the class of affine jump diffusion models.

The alternative representation of the VG process has the characteristic function

where C, G, M > 0 with

An increase in G increases the size of upward jumps, while an increase in M increases the size of downward jumps. Accordingly, θ, 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. 

Hedging with bitcoin futures: The effect of liquidation loss aversion and aggressive trading

Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options

Solving continuous time affine jump-diffusion models for econometric inference

The pricing of options and corporate liabilities

Generalized autoregressive conditional heteroskedasticity

On the hedge and safe haven properties of bitcoin: Is it really more than a diversifier?

Expected option returns and the structure of jump risk premia

Hedging under model misspecification: All risk factors are equal, but some are more equal than others

Model specification and risk premia: Evidence from futures options

Option valuation using the fast fourier transform

The fine structure of asset returns: An empirical investigation

Alternative models for stock price dynamics

Model risk in incomplete markets with jumps

Transform analysis and asset pricing for affine jump diffusions

Hedging capabilities of bitcoin. is it the virtual gold?

Do stock prices and volatility jump? reconciling evidence from spot and option prices

The impact of jumps in volatility and returns

Hedging of non-redundant contingent claims

A parsimonious arbitrage-free implied volatility parameterization with application to the valuation of volatility derivatives. Presentation at Global Derivatives & Risk Management

Arbitrage-free svi volatility surfaces

A closed-form solution for options with stochastic volatility with applications to bond and currency options

Pricing cryptocurrency options: the case of crix and bitcoin

Vcrix -a volatility index for cryptocurrencies. International Review of Financial Analysis p 101915

A software package for sequential quadratic programming

Option hedging with stochastic volatility

The variance gamma process and option pricing

Advanced model calibration on bitcoin options

Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach

Option pricing when underlying stock returns are discontinuous

Hedging bitcoin with conventional assets

Risk minimization in stochastic volatility models: model risk and empirical performance

Stochastic Integration and Differential Equations. Stochastic Modelling and Applied Probability

High-frequency jump analysis of the bitcoin market*

Bitcoin futures: An effective tool for hedging cryptocurrencies

Is bitcoin a hedge, a safe haven or a diversifier for oil price movements? a comparison with gold

Nonlinear ill-posed problems

Crix an index for cryptocurrencies

Acknowledgments. Financial support from the Deutsche Forschungsge-