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## Abstract

The authors demonstrate the construction of an optimal dynamic portfolio of cryptoassets that minimizes either return variance or conditional value at risk. One can view such a portfolio as a minimum-risk index for this asset class. They carefully backtested the dynamic portfolio model and developed a fair valuation model for options based on a dynamic pricing model for the underlying cryptoasset index. They obtain the valuation by passing from the natural world to the equivalent martingale measure via the Esscher transform. The work underscores the need for a cryptoasset index–based exchange-traded fund, the development of derivatives, particularly for cryptoportfolio insurance purposes, and the development of (nearly) riskless rates for this asset class.

**TOPICS:** Currency, VAR and use of alternative risk measures of trading risk, derivatives, options, quantitative methods, statistical methods

**Key Findings**

▪ The study provides a methodology for constructing an optimal dynamic portfolio of major cryptoassets that minimizes either return variance or conditional value at risk.

▪ The authors develop a fair valuation model for options based on a dynamic pricing model for the underlying cryptoindex.

▪ The work emphasizes the need for a cryptoasset index–based exchange-traded fund and the development of derivatives, in particular for cryptoportfolio insurance purposes.

Despite their generic name, cryptocurrencies do not exhibit many of the fundamental properties that define a currency. Thakor (2019) noted that they are not a generally accepted medium of exchange, do not maintain a stable store of value due to their substantial price volatility, and do not serve as a unit of account. A more appropriate description is that they are “cryptoassets,” and we refer to them as such in this article.^{1} We note two exceptions. In February 2018, Venezuela launched the petro, a cryptocurrency backed by the country’s oil reserves (pinned to the price of a Venezuelan barrel of oil).^{2} In November 2019, the petro’s value was anchored instead to the US dollar through the official bolivar–dollar exchange rate. Hyperinflation in Venezuela and uncertainly regarding the petro’s regulatory oversight currently position it in the realm of a cryptoasset instead of a true digital currency. In addition, China announced a pilot run of a national digital currency, the DCEP, built on concepts of peer-to-peer payment, traceability, and tamperproofness without resorting to blockchains.^{3} DCEP is to be backed by the People’s Bank of China and distributed by it and commercial banks. Significantly, it will allow “wallet-to-wallet” transactions that do not require either banks or financial service companies as intermediaries.^{4}

As of September 2020, according to CoinLore, there were 4,987 cryptoassets with a total market capitalization of $352.4 billion and a total trading volume of $62.9 billion. Cryptoassets exhibit extreme price volatility (Thakor 2019). Exhibit 1 compares the mean monthly standard deviation and the maximum drawdown^{5} of the prices for four major cryptoassets with those for the SPDR S&P 500 ETF Trust (SPY) from August 1, 2017–August 31, 2020.^{6} Compared to SPY, the four cryptoassets are very volatile. There are three reasons proffered for the extreme price volatility of digital currencies compared with other assets such as stocks.

First, valuation is difficult. Balcilar et al. (2017) showed that trading volume could not help predict the volatility of bitcoin returns at any point in the conditional distribution of log-returns. Unlike stocks, cryptoasset values are based solely on market sentiment. Although market sentiment partially drives stock values, company fundamentals provide the basis for assessing whether a stock is over- or undervalued. Based on transaction demand, storage demand, and supply, a model for valuing cryptoassets has been proposed by Mitchnick and Athey (2018). The authors note that the application of their eight-parameter model to a given cryptoasset is “an intricate process.”

Second, the market is thin. Although individual investors who have concerns about governments tend to be participants, it is institutional players who are needed to supply liquidity to the cryptoasset market. Without a central bank regulating the cryptomarket, many large institutional investors have a mixed record of participation in this market. As noted in a relatively recent Forbes.com article, “Institutional adoption of bitcoin is here, you just have to know where to look…. While many of the names are well-known mutual funds…the latest [SEC] filings are also rife with relative newbies to the space…. It’s very difficult to have a clean one-to-one signal on who’s entering and exiting the space,” according to Ark Invest crypto analyst Yassine Elmandjra.^{7} Released in 2020, Libra from Facebook is leading a new trend—corporations issuing their own units of exchange linked to baskets of assets.^{8}

Finally, there is no regulatory oversight. Consequently, certain market practices, such as market manipulations prohibited by securities laws in many countries, are not monitored in the digital currency markets. This lack of oversight can result in substantial price movement.

While there are a handful of cryptoasset-based ETNs/ETFs trading on European and Canadian markets, as of January 2021, the US Securities and Exchange Commission (SEC) has blocked bitcoin ETF proposals, whether the underlying tracked asset is the cryptoasset or a regulated cryptoasset futures contract. SEC acceptance is often considered a holy grail for institutional investor acceptance. SEC concerns center around the potential for market abuse due to lack of oversight of the cryptoasset trading ecosystem. Improved transparency regarding trading activities and security measures of digital currency exchanges appear to be key to potential SEC acceptance. Our purpose in this work is the following: unlike existing indices that track the performance of cryptoassets, we seek to develop cryptoasset indices that are based on a dynamic asset pricing approach, covering everything from modeling to forecasting and the price of insurance instruments (such as puts), within the setting of rational finance. To this end we have accomplished the following. We identify an appropriate multivariate model, verified by rolling-window VaR/CVaR backtesting, for describing the return distribution of major cryptoassets. We then construct optimal portfolios according to minimum variance or minimum conditional value at risk. Each optimal portfolio can be viewed as a minimum-risk cryptoasset index. We subject these indices to risk-budget and risk-measure analyses. Finally, we determine risk-neutral option pricing where the underlying asset is an optimal portfolio of cryptoassets (i.e., any one of the minimum-risk cryptoasset indices). Prediction of call and put option prices to a six-month maturity date is based on a sample of 10,000 price time series. Each series is generated using a generalized autoregressive conditional heteroskedastic (GARCH)(1,1) model with innovations based on the normal inverse Gaussian distribution.

## MULTIVARIATE MODELS FOR CRYPTOASSETS

Our dataset consists of the top seven cryptoassets’ daily closing prices from July 25, 2017–September 10, 2020.^{9} These cryptoassets are bitcoin (BTC), ethereum (ETH), XRP, litecoin (LTC), bitcoin cash (BCH), EOS, and binance coin (BNB).

As is done in most studies, we measure performance in terms of return rather than price. Compared to a price series, returns offer positive features, including a complete and scale-free summary of the investment opportunity and ease of handling^{10} and desirable statistical properties, such as stationarity.^{11} With denoting the price of the *i*th cryptoasset at time *t*, the log- and simple-return series (either continuous or discrete) are

where *d* denotes the number of cryptoassets and *T* is a terminal time.

It is well known that portfolio optimization based on modern portfolio theory (Markowitz 1952) is unstable, in the sense that small changes in values of return mean, variance, or correlations can lead to large changes in asset weights (Michaud 1989; Tütüncü and Koenig 2004). An examination of each cryptoasset in the dataset reveals that 5% of the time, the differences between log- and simple-returns vary between 0.4% and 2.9%, and 25% of the time, the differences exceed 0.1%. Relevant to this point, care must be taken in optimizations involving log-returns to avoid the use of formulas that are exact for simple-returns and only approximations for log-returns—approximations that improve in accuracy only as . However, if exact formulas^{12} are used, a comparison of optimizations performed using simple- and log-returns could provide informational value, particularly if systematic differences are observed. As our analyses apply to both log- and simple-return time series, for brevity, we simply denote return by . Appendix Exhibit A1 plots the log-return time series for the cryptoassets in our dataset.

We can regard the distribution of returns for an individual cryptoasset as a marginal distribution of the portfolio returns. Hence, we use univariate marginal distributions for each cryptoasset and separately consider multivariate joint distributions to capture the correlations between the cryptoassets.

### The Multivariate Time-Series Models

A common model for financial-return time series is , which combines the mixed autoregressive moving-average (ARMA) model of Whittle (1951) with the GARCH model of Bollerslev (1986). We also consider the exponential GARCH (EGARCH) model of Nelson (1991), designed to capture asymmetric effects between positive and negative asset returns.

Describing the realized returns as a linear combination of a drift term and a random process ,

2the drift is modeled using ARMA(1,1)

3and the random process is modeled in terms of the return volatility using GARCH(1,1),

4 5where the sample innovations come from some distribution having zero mean and unit variance. For the EGARCH(1,1) model, Equation 5 becomes^{13}

In order to fit the coefficients of ARMA–GARCH|EGARCH models, an appropriate distribution for the sample innovations must be determined. We have considered Gaussian, Student’s *t*, and normal-inverse Gaussian (NIG) distributions for these innovations.

Our goal is to offer a relatively simple multivariate time-series model with satisfactory out-of-sample performance for the joint distribution of the sample innovations. As the cryptoasset return series’ kurtosis indicates heavy tails, we consider the multivariate *t* and multivariate NIG (MNIG) distributions. The probability density function for a commonly used form of the multivariate *t* distribution *t _{d}*(ν, μ, Σ) with ν degrees of freedom, mean vector μ, and positive-definite dispersion (scatter) matrix Σ is given by

^{14}

In a moving-window scenario (as in the backtesting section), the mean vector μ and dispersion matrix Σ are estimated based on the innovation series computed from the current window. For backtesting, ν is considered a free parameter, and a value, denoted ν_{fit}, is obtained from fitting the multivariate *t* distribution to the innovation matrix.^{15} We compared backtest results obtained from ν_{fit} with results obtained by imposing values ν = 5, 6, and 7.

An MNIG distributed random variable *X* is a mean–variance mixture of a *d*-dimensional Gaussian random variable *Y* with a univariate inverse Gaussian (IG) distributed mixing variable *Z*, where *Y* and *Z* are independent.^{16} For parameters α > 0, β ∈ ℝ^{d}, δ > 0, μ ∈ ℝ^{d}, and Γ ∈ ℝ^{d×d}, the random variable *X* can be constructed as

where *Z* ~ IG(δ^{2}, α^{2} − β^{T}Γβ) with α^{2} > β^{T}Γβ. In the backtesting simulations, all parameters are captured from fitting the MNIG to the innovation matrix.

### Backtesting

For consistent comparison, the ARMA(1,1)–GARCH(1,1)|EGARCH(1,1) models were run on an equal-weighted portfolio of the seven cryptoassets over 1,143 days from July 26, 2017–September 10, 2020, inclusive. An 889-day moving window established in-sample computations covering January 1, 2020–September 10, 2020.^{17} As noted in the previous section, each model was run assuming the individual asset innovation distributions were characterized either by Gaussian, Student’s *t*, or NIG, while the joint probability distributions were assumed to be either multivariate *t* or MNIG. For each model, a total of 10,000 simulated, in-sample, return time series were generated for each asset.^{18}

At each time step, a risk measure was computed from the historic (moving-window) returns for each asset and a single value for that risk measure was computed (by the simple average over assets) to represent the simulated portfolio for that time step. We then compared the time series of this risk measure for the simulated portfolio with the analogous time series computed from the historical, equal-weighted portfolio. Two widely used risk measures, value at risk (VaR) and conditional value at risk (CVaR),^{19} were employed. Let *F*(*x*) = *Pr*{*r* ≤ *x*} denote the cumulative distribution function for return *r*. The VaR and CVaR at confidence level 1 − α are defined as^{20}

Thus, our backtesting is based on different combinations^{21} of two return types (logarithm and simple); two time-series models (ARMA-GARCH and ARMA-EGARCH); three choices of innovation distribution (Gaussian, Student’s *t*, and NIG) for the individual asset returns; two multivariate distributions (multivariate *t* and MNIG) for their joint distribution; and two risk measures (VaR and CVaR) at two choices of confidence level 1 − α, (α = 0.01 and 0.05). The backtesting results for these combinations are summarized in Exhibits A2–A5 in the appendix. Preferred models were identified based on the following backtesting measures: smallest ratio of observed to expected^{22} failures in the VaR-based backtests; the smallest ratio of observed to expected severity^{23} in the CVaR-based backtests; and success in the traffic light^{24} and binomial tests.^{25} Four models, listed in Exhibit 2, emerge as “competitively best.”

To identify a single, most appropriate model, we augmented the backtesting results with a scoring rule—the continuous-ranked probability score (CRPS). We evaluated the CRPS in a closed form (Gneiting and Raftery 2007; Baringhaus and Franz 2004; Székely and Rizzo 2005),

10where *X* and *X*′ are independent copies of a random variable with cumulative distribution function *F* and finite first moment. CRPS measures the difference between the cumulative distribution functions of the forecasts and the observations; a model with perfect forecasting power results in a CRPS value of zero. CRPS values for the four competitively best models are shown in Exhibit 2. Noticeably, the values of the CRPS are of magnitude 10^{−3}. Based on the additional information provided by CRPS, we consider EGARCH/Student’s *t*/multivariate *t* − ν_{fit} to be the single most appropriate model. We note that values of ν_{fit} for the seven-dimensional multivariate *t* joint distribution fall in the range [2.2, 2.8]. In the remainder of this article, we use this model to build optimum cryptoasset portfolios and derive option pricing.

## MINIMUM-RISK PORTFOLIOS, CRYPTOASSET INDICES

In this section, we optimize asset weights in the portfolio by using an objective function that minimizes risk. The resulting optimal portfolio can then be used as a minimum-risk cryptoasset index.^{26} Specifically, we considered the mean–variance minimized portfolio of Markowitz (1952) and the CVaR minimized portfolio of Krokhmal et al. (2002). We minimized CVaR at the 99% confidence level. We refer to these two optimized portfolios as the *min Var* and *min CVaR* portfolios, respectively. We used the dataset described in the section on backtesting with the 889-day moving window so that each optimized portfolio consisted of 254 daily returns. We assumed no transaction costs, so weights were optimized purely to minimize risk. As the cryptomarket runs 24/7, to benchmark the cryptoasset indices against the SPY, we adjusted SPY data by interpolating returns for weekends and holidays.^{27} The parameters of our ARMA(1,1)-EGARCH(1,1)/Student’s *t* innovation model were calibrated using the data in each estimation window. The multivariate *t* − ν_{fit} distribution was used to generate 10,000 scenarios for each time-step forecast. 10,000 scenarios provided a sufficiently dense set of portfolio values to determine the efficient frontier (Cornuejols and Tütüncü 2007). Denoting the weights for each cryptoasset in the portfolio as ; *i* = 1, …, *d*; *j* = minV, minC; *k* = simple, log; and *t* = 0,…,*T*, where *j* = minV represents the *min Var* portfolio and *j* = minC represents the *min CVaR* portfolio, then the return of each optimized portfolio at time *t* is^{28}

Exhibit 3 compares the price time series computed for the four optimized portfolios and the benchmark SPY. Among other variables, the cumulative return is a function of investment timing. Our portfolio investments on January 1, 2020, occur during a period of strong, prepandemic growth in cryptoasset prices. With the pandemic’s onset, cryptoasset prices (and hence our four index values) plunged sharply, even more dramatically than the benchmark SPY. However, during the pandemic, cryptoasset prices and our index values rebounded more strongly, although with greater volatility, than SPY—presumably representing the sentiment of crytpoasset buyers regarding a safe-haven investment. There was a noticeable burst of optimism in cryptoasset investment during the third quarter of 2020.

Before April 2020, the prices of both *min Var* portfolios, optimized using log- and simple-returns, are very close. Subsequently, a price gap between these two optimizations appears, remaining relatively constant over time, with the simple-return portfolio producing higher prices. For the *min CVaR* portfolios, prices based on the log-return portfolio tend to outperform those based on the simple-return portfolio, most noticeably during periods of high prices.

To further quantify the performance of these four indices, we subjected them to risk-budget and risk-measure analyses.

### Risk-Budget Analysis

We consider two approaches to risk-budget analysis based on the homogeneous risk measures of portfolio volatility and CVaR, both of which are coherent risk measures (Artzner et al. 1999). We first consider portfolio volatility. Define as the vector of weights for portfolio *j* at time *t*. The risk contribution of a portfolio is considered under the condition of equally weighted assets; . We consider the volatility risk measure , where Σ is the covariance matrix of the return series. Then the risk contribution of the *i*th cryptoasset in portfolio *j* at time *t* is , where is the marginal risk of the *i*th cryptoasset. The second risk measure we consider for budgeting analysis is CVaR_{α}, as defined in Equation 9, at the 99% confidence level. Thus, each asset will have a portfolio volatility risk contribution that we will refer to as the asset’s RC^{Vol}, as well as a CVaR risk contribution that we will refer to as its RC^{CVaR}.

These two risk measures were computed using the same 889-day, rolling-window strategy as in the section on backtesting. Panels A and B of Exhibit A6 in the appendix plot the asset risk contributions RC^{Vol} and RC^{CVaR} for the two portfolios based on log-returns, while Panels C and D present the asset risk contributions for the simple-return portfolios. Exhibit 4 summarizes the percentage of total risk contribution of each asset over the period 01/01/2020 through 09/10/2020. Comparing the log- and simple-return portfolios, RCVol measures for each asset agree to within 3.8%. The interquartile distance and total spread (i.e., maximum − minimum) values shown in Exhibit 4 indicate that the risk contributions, as measured by volatility, for the assets are narrowly spread. RC^{CVaR} measures for each asset have much larger differences between the log- and simple-return portfolios. The interquartile distance and total spread values indicate that the risk contributions measured by CVaR for the assets are spread more widely. These observations are consistent with the fact that volatility measures differences in the central portion of a distribution while CVaR measures differences in the tails. BNB and BTC have consistently low RC values, especially for RC^{CVaR}, implying they play the largest risk diversifying role in the portfolio. BCH has consistently high RC values, indicating it is the greatest risk contributing asset.

### Measures of Risk-Adjusted Return

As indicated by the large volatility (relative to SPY) of their price time series in Exhibit 3, portfolios composed solely of cryptoassets carry high risk. We utilized four measures to characterize the risk of our four cryptoasset indices.

**1.**The maximum drawdown (MDD),which characterizes the maximum loss incurred from peak to bottom during a specified period of time [0,

*T*].**2.**The Sharpe ratio (Sharpe 1994),where

*r*is the portfolio return,_{p}*r*is a risk-free rate, and σ_{f}_{p}is the standard deviation of the portfolio’s excess return,*r*−_{p}*r*_{f}.**3.**The M2 ratio (Modigliani 1997),where σ

_{M}is the standard deviation (volatility) of a specified market benchmark.**4.**The Rachev ratio (Rachev et al. 2008),which represents the reward potential for positive returns compared to the risk potential for negative returns at confidence levels defined by the user.

^{29}In our analysis, we consider the cases α = β = 0.01 (denoted R-ratio_{0.01}) and α = β = 0.05 (denoted R-ratio_{0.05}).

Daily rates based on the US 10-year Treasury rates were used as the risk-free rate. SPY provided the benchmark standard deviation used in the computation of M2.

Two considerations affected our computations of these risk measures:

**1.**It is common to provide the risk-measure values as annual averages. As noted,^{30}our data period was reduced to just over eight months (January 1, 2020–September 10, 2020)—a period that is affected by the current global Covid-19 pandemic. As our period is less than one year, we instead quote daily and monthly averages, based on 254 daily and eight monthly returns, for the Sharpe and M2 ratios. Values for MDD and the Rachev ratio are based on the entire 254-day period.**2.**Cryptoasset prices change daily, while SPY prices are available on market trading days and Treasury rates are available for federal government workdays. This required interpolation of the SPY and risk-free rate data in order to provide daily values to match that of the cryptoassets. Interpolation of financial datasets is a complex issue. As our goal is to provide values for benchmarking and risk measures that are characteristic for the time period, we interpolated the SPY prices and Treasury rates using a moving median window centered on the date to be interpolated.^{31}

The results, reported in Exhibit 5, are based on the out-of-sample dataset covering the period from January 1, 2020–September 10, 2020. MDD values for the four optimal portfolios are essentially identical and are twice as large as those for the benchmark. Based on daily and monthly Sharpe and M2 ratio values, the cryptoasset indices outperform the SPY. Rachev ratio values are less conclusive; relative to SPY, the reward-to-risk potential for cryptoasset investing is less clear. The risk measures generally indicate that the *min Var* indices outperform *min CVaR* indices.

## OPTION PRICING

If options on a cryptoasset index existed, standard methodologies for hedging would exist (e.g., protective purchases of put options to create a nonlinear payoff profile). However, in the absence of options where the underlying asset is a cryptoasset index, hedging is still possible if there is a cryptoasset index. The dynamic trading strategy to do so, called “portfolio insurance,” was formulated in the 1980s by the advisory firm of Leland O’Brien Rubinstein Associates. We apply this strategy to calculate the fair value of cryptooptions in which the underlying asset is a cryptoasset index. The option valuation model is applied to the log-return-based *min Var* and *min CVaR* portfolios derived in the previous section (which we treat as cryptoasset indices).

We start by finding an appropriate distribution for the innovations of the log-return time series for each index. Since the ARMA effect is irrelevant for risk-neutral option pricing, we utilize the GARCH(1,1) Model 5 for returns. We fit the GARCH model using the generalized hyperbolic distribution introduced by Barndorff-Nielson (1977) to find an appropriate distribution for the innovation ε. After extensive backtesting, the results indicate that the innovations of the log-returns for both the *min Var* and *min CVaR* indices follow the NIG(α, β, δ, μ) distribution with the moment-generating function

where μ sets location; α, the tail heaviness; and the parameters β and δ, asymmetry and scale, respectively. We obtain the valuation of a cryptoindex-based option by passing from the natural world to the unique equivalent martingale measure via the Esscher transform (Gerber and Shiu 1994). Following Chorro (2012), we apply the following methodology to incorporate the NIG innovation:

**1.**We select a data set of 254 daily index returns (e.g., from the*min CVaR*portfolio). For our purposes below, we label this set with dates*t*_{−253}to*t*_{0}, with*t*_{0}being the last day of the dataset.**2.**Using this dataset, fit the Models 2, 4, and 5 with NIG innovations. This fit produces the NIG parameters α, β, δ, and μ, as well as a one-step forecast for the conditional variance .**3.**Repeat the following Steps (a)–(d) for*t*= 1, …,*T*, where*T*is the maturity date for the option.^{32}**(a)**Solve the following function for the stochastic discount factor θ_{t}: 13**(b)**Update the value of β to β + σ_{t}θ_{t}.**(c)**Generate a sample ε_{t+1}from the NIG(α, β, δ, μ).**(d)**Compute*r*_{t+1}and σ_{t+2}.

With

*S*_{0}as the initial capital, the estimated price of the index at maturity*T*is 14where

*S*_{t}is the current price based on the initial capital.**4.**To price call and put options, we repeat Steps 4 and 5*N*times,^{33}generating samples*S*_{T,i},*i*= 1, …,*N*, for the maturity price of the underlying index.**5.**For a strike price*K*, the call option price and put option price are estimated using sample averages:

Using this Monte Carlo method, we calculated the fair price of call and put options for the two cryptoasset indices, *min Var* and *min CVaR*, based on log-returns. The short-term, *T* = 180 days, behavior of the call and put options based on the *min CVaR* portfolio is shown in Exhibit A7 in the appendix. For the options based on the *min Var* index log-returns, we have similar figures.

Note that our method for the valuation of cryptooptions can be applied to all European style options. Using the put option pricing given by Equation 16, an investor can hedge the risk for the minimum-risk cryptoportfolios (cryptoasset indices) with a specified strike price *K*. We cannot compare our results with option prices in the cryptomarket at the current time because such market prices do not exist.

## CONCLUSION

In this article, we have conducted an empirical analysis of minimum-risk portfolios composed solely of the highest-market-capitalization cryptoassets. Using VaR/CVaR backtesting augmented by a continuous-ranked probability score, we first evaluated the goodness-of-fit of ARMA(1,1)-GARCH(1,1)|EGARCH(1,1) models to the log- and simple-return time series of the individual assets. Three univariate distributions were tested for the individual asset innovations, and two multivariate distributions were tested for their joint distributions. We have concluded that the optimal fitting model is ARMA(1,1)-EGARCH(1,1) (Student’s *t* innovation) with a multivariate *t* distribution for the joint distribution. The degrees of freedom for the multivariate *t* distribution were best determined from the fit. The observation that the fitted degrees of freedom fall in the range [2.2, 2.8] for the multivariate *t* joint distribution is a strong warning signal that investing in cryptoasset portfolios is subject to extreme tail risk. The optimal fitting model was then used to develop dynamically optimized portfolios that minimized either mean–variance (the *min Var* portfolio) or CVaR at the 99% confidence level (the *min CVaR* portfolio). The optimizations employed both log- and simple-returns. The result produced four optimized portfolios, which we consider as cryptoasset indices.

With the cryptoasset market being new and the number of assets rapidly expanding, our dataset for testing and development covered three calendar years (July 21, 2018–September 10, 2020). With in-sample datasets produced by an 889-day moving window, our results are based on a 254-day dataset, which covered the current period of the Covid-19 pandemic, including a few prior months. A risk-budget analysis, based on portfolio volatility and CVaR, performed on the four indices showed that the assets bitcoin and binance coin acted as risk-diversifier assets. In contrast, bitcoin cash was the major risk contributor over this time period. Analyses, based on four risk-adjusted return measures, quantify the outperformance of the four cryptoasset indices relative to our chosen market index, SPDR S&P 500 ETF Trust, during this period. Finally, we offer a novel solution for calculating cryptoasset options’ fair value based on any of the underlying cryptoasset indices. Specific evaluations of European-style call and put options for one of the cryptoasset indices are presented.

This work points to the following needs. As for the S&P 500 Index, there should be a cryptoasset-based index that, given the volatility of the cryptomarket, should be based on elements of modern portfolio theory. At least one exchange-traded fund should be tracking this cryptoindex. There should be traded derivatives (particularly futures and put options) on both the index and the crypto-ETF. To properly compute the Sharpe ratio, there is a need for an appropriate risk-free rate for this asset class. Future work should be directed to defining a near-riskless rate for the cryptomarket, as (currently) no central bank is backing the cryptomarket.

## APPENDIX

## ENDNOTES

↵

^{1}The Bank of Israel, for example, has stated that bitcoin and similar virtual currencies are assets and not currencies (https://www.coindesk.com/bank-of-israel-digital-currencies-are-an-asset-not-a-currency).↵

^{2}See https://en.wikipedia.org/wiki/Petro_(cryptocurrency).↵

^{3}See https://www.wsj.com/articles/china-rolls-out-pilot-test-of-digital-currency-11587385339.↵

^{4}See https://decrypt.co/33866/dcep-an-inside-look-at-chinas-digital-currency.↵

^{5}Maximum drawdown is precisely defined in the section on risk-adjusted return measures.↵

^{6}To compare performance in the form of a classical portfolio horse race, the benchmark must be tradable. We chose the most representative ETF for the S&P 500 index.↵

^{7}See “Editor’s Pick,” August 6, 2020, https://www.forbes.com/sites/michaeldelcastillo/2020/08/06/valuable-sec-data-on-20-institutional-bitcoin-investors-could-soon-disappear/?sh=127aacc71de2.↵

^{8}See https://www.ipe.com/home/are-cryptocurrencies-an-asset-class-for-institutional-investors/10043517.article.↵

^{9}As cryptoasset data are relatively recent, the choice of assets limited the time period of available data.↵

^{11}See Chapter 2 of Tsay (2010). Any unit-root test can be applied to check stationarity. Prices usually have a unit root, while returns can be assumed to be stationary.↵

^{12}See, for example, Equations 11a and 11b later in this article.↵

^{13}See Tsay (2010 3.28).↵

^{14}See Demarta and McNeil (2005). The covariance matrix is given by ν/(ν − 2)Σ for ν > 2.↵

^{15}Note that the value obtained for ν_{fit}generally changes at each time step.↵

^{16}See Øigård et al. (2004).↵

^{17}The choice of 889 days was guided by the desire to capture an adequate in-sample time period of historical volatility while ensuring that the out-of-sample time period covered the Covid-19 pandemic, including a few months prior to its onset.↵

^{18}This represented a compromise between run-time and accuracy.↵

^{19}CVaR is also known as the expected shortfall (ES).↵

^{20}Note that the “units” of VaR and CVaR are those of a return. Definition (9b) holds under the assumption that the distribution for*r*is continuous.↵

^{21}The multivariate*t*joint distribution was used in combination with the Gaussian and Student’s*t*innovation distributions. The MNIG joint distribution was used only with the NIG innovation distribution.↵

^{22}As determined by α for VaR_{α}.↵

^{23}Severity is the average ratio of CVaR to VaR over periods with VaR failures.↵

^{24}Basel Committee on Banking Supervision, January 1996, green, yellow and red zones (see https://www.bis.org/publ/bcbs22.htm).↵

^{25}The binomial test is based on the normal approximation to a binomial distribution (see Jorion 2011). It assesses whether the number of failures is consistent with the VaR(CVaR) confidence level.↵

^{26}Although there are currently capitalization-weighted cryptocurrency indices, the index we create here is one that minimizes variance or CVaR. Note the parallel to the development of smart beta indices in the equity space (i.e., rule-based indices that are not based on market capitalization). For example, there are popular, minimum-variance, smart beta indices that are not based on expected returns.↵

^{27}See the section below on risk-adjusted returns for a full discussion of this interpolation.↵

^{28}Equation 11b is exact and is written in the form convenient for portfolio optimization done in terms of log-returns.↵

^{29}The Rachev ratio can be negative—for example, when the entire sample of returns consists of positive values. In such cases, the Rachev utility function,should be used as a performance measure. The constant

*c*≥ 0 is a risk-aversion parameter; in practice, a value of*c*= 1 can be used. With*c*varying over [0, ∞), the*Rachev effcient frontier*(Proposition 10 in Stoyanov et al. 2007) can be depicted in the plane having*x*-axis values given by CVaR_{β}(*r*−_{p}*r*) and_{f}*y*-axis values given by CVaR_{α}(*r*−_{p}*r*_{f}).↵

^{30}See footnotes 10 and 18.↵

^{31}The moving window used data from the five days on each side of the date to be interpolated. Interpolated values were stored separately and never used in providing interpolations for other dates.↵

^{32}We set maturity*T*at 180 days and use the 10-year Treasury yield-curve rate on September 10, 2020, as the risk-free rate*r*_{f}.↵

^{33}We chose*N*= 10,000, as suggested by Chorro (2012).

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