Hedging with Bitcoin Futures: The Effect of Liquidation Loss Aversion and Aggressive Trading
aa r X i v : . [ q -f i n . R M ] J a n Optimal Hedging with Margin Constraints and Default Aversionand its Application to Bitcoin Perpetual Futures
Carol Alexander a , Jun Deng b , and Bin Zou ca University of Sussex Business School and Peking University HSBC Business School. Jubilee Building,University of Sussex, Falmer, Brighton, BN1 9RH, UK. Email: [email protected] . b School of Banking and Finance, University of International Business and Economics. No.10, HuixinDongjie, Chaoyang District, Beijing, 100029, China. Email: [email protected] . c Department of Mathematics, University of Connecticut. 341 Mansfield Road U1009, Storrs, Connecticut06269-1009, USA. Email: [email protected] . This Version: January 6, 2021
Abstract
We consider a futures hedging problem subject to a budget constraint that limits the ability ofa hedger with default aversion to meet margin requirements. We derive a semi-closed form foran optimal hedging strategy with dual objectives – to minimize both the variance of the hedgedportfolio and the probability of forced liquidations due to margin calls. An empirical analysisof bitcoin shows that the optimal strategy not only achieves superior hedge effectiveness, butalso reduces the probability of forced liquidations to an acceptable level. We also compare howthe hedger’s default aversion impacts the performance of optimal hedging based on minute-leveldata across major bitcoin spot and perpetual futures markets.
Keywords:
Cryptocurrency; Futures Hedging; Leverage; Perpetual Swap; Extreme Value Theo-rem
JEL Classification:
G32; G11 1
Introduction
The cryptocurrency bitcoin is the most volatile of all financial assets held by institutional in-vestors. From 7 to 12 March 2020 its spot price fell by 60%, its 30-day implied volatility indexalmost touched 200% and by 31 March 2020 its 30-day statistical volatility had risen to about170%. This extremely high price uncertainty imposes tremendous risk to various participants inthe cryptocurrency markets and generates a strong hedging demand by investors with long posi-tions in bitcoin. These include, but are certainly not limited to: bitcoin miners such as AntPool; centralised and decentralised cryptocurrency exchanges like Coinbase and Uniswap; businesses andretailers that accept bitcoin as payment, e.g. Microsoft and Wikipedia; and individuals and insti-tutions who hold bitcoin in their asset portfolios. It is common to hedge long (short) spot price risk by buying (selling) futures contracts on thesame asset, or one that is very highly correlated; see Lien and Tse (2002). There is by now ahuge literature on hedging with futures, examining the use of commodities, currency, interest rates,stocks, and indexes futures as effective hedges of spot price risk. We refer to Ederington (1979)and Figlewski (1984a) for classical contributions, and Lien and Yang (2008), Mattos et al. (2008),Acharya et al. (2013), Cifarelli and Paladino (2015), Wang et al. (2015), Billio et al. (2018), andXu and Lien (2020) for more recent works.Bitcoin futures were introduced in December 2017 and by November 2020 their monthly tradingvolume had reached $ Nevertheless, at the time of writing there is little in-depth research into the role that bitcoin futures can play to hedge bitcoin spot risk. The twopapers that study similar contracts to us agree that they are excellent hedging instruments tosegmented bitcoin spot markets: Alexander et al. (2020) show that BitMEX perpetual contractsachieve outstanding hedge effectiveness against the spot risk on Bitstamp, Coinbase, and Kraken.Deng et al. (2020) show that OKEx inverse quarterly contracts are excellent hedging tools for thespot risk on Bitfinex and OKEx, and are superior to CME standard futures in terms of hedgeeffectiveness. However, neither of these studies accounts for the impact of margin constraints anddefault aversion on hedging performance. There are several reasons why the hedging problem is more challenging for bitcoin than it is forother assets: first, many different types of bitcoin futures are traded on multiple exchanges, andthe best choice of product and exchange is a problem not often considered for traditional assets; Details of the bitcoin implied volatility index can be accessed here and the statistical volatility figure is from theForbes report which can be accessed here. Antpool is currently one of the largest of several very large mining cooperatives that aim to optimize bitcoinrevenue for their members. See Alsabah and Capponi (2020) and Cong et al. (2020a) for further analysis on thecentralisation effects of these large pools. For example, Satoshi Nakamoto, Roger Ver, Charlie Shrem, and the institutions listed in this Forbes report. See the CryptoCompare December 2020 Exchange Review here. Deng et al. (2020) claim that margins have little effect on the trading profit and loss and they implicitly assumethat the hedger has an infinite supply of bitcoin to meet margin calls. and thirdly,most exchanges are at most lightly regulated and consequently the risk of default is far higher thanit is on standard derivatives exchanges – so the hedger’s aversion to default needs to be part of themathematical problem.The methodological contribution of this paper is to provide a semi-closed solution to a hedgingproblem that is entirely new to the finance literature, and which incorporates two key features ofbitcoin markets, i.e. margin constraints and default aversion. The margin constraint imposes apositive probability that the hedger’s futures positions will be forced to liquidate during the hedgehorizon, and such forced liquidations are seen as default events. The hedger seeks an optimalhedging strategy with dual objectives to minimize the risk of the hedged portfolio and the defaultprobability. We apply the extreme value theorem to obtain the optimal strategy and show thatboth margin constraint and default aversion play a key role in determining its characteristics.We consider a hedger who has limited resources to meet the margin requirement while tradingbitcoin futures, and hence is subject to an upper margin constraint. A second key novelty in ourmodel is to characterize the hedger as a default averse agent. We define a default event for futurestraders as the circumstance that they cannot raise enough bitcoin (or fiat currency) to meet themargin calls, i.e. a default event happens upon a forced liquidation. A hedger is called defaultaverse if she views default as an undesirable event. There are at least two theoretical reasons toconsider a default averse hedger. First, a default (forced liquidation) is an unexpected and adverseevent to hedgers, causing negative impact on their financial goals. This is certainly the case forthose who use futures as a hedging instrument. An early liquidation means a failure to hedge thespot risk for the chosen horizon (ending with a naked spot position). Second, when the abovedefined default happens, hedgers fail to pay liabilities in full amount to their counterpart in futurestrading. This hurts their creditworthiness immediately and may further cause exchanges to requirehigher collateral or enforce more strict policies in their future trading. Including default aversion isalso a natural choice in our model; because of the already imposed margin constraint, the defaultprobability is strictly positive as long as hedgers trade futures.An empirical analysis considers different bitcoin spot exchanges and perpetual futures contracts.This way, we investigate three important topics in risk management: hedge effectiveness, defaultprobability, and leverage. Since bitcoin markets are highly segmented, we compare results for thethree most liquid bitcoin perpetual futures, BitMEX, Deribit, and OKEx, and three choices forbitcoin spot prices, i.e. the .BXBT index on BitMEX, and the two largest spot markets, Bitstampand Coinbase. Our main empirical findings are threefold: (1) A tighter margin constraint or a Margin mechanisms are essential for ensuring the stability and integrity of futures markets, and there is abundantresearch on this topic for traditional futures like commodities, precious metals, stock indexes, and currencies; seeFiglewski (1984b), Longin (1999), Cotter (2001), Daskalaki and Skiadopoulos (2016), Alexander et al. (2019) andmany others.
There are two distinctive types of bitcoin futures contracts: (1) standard futures that have a fixedexpiry date and a regular schedule for new issues; and (2) ‘perpetual futures’ or just perpetuals ,because they have no expiry date. Perpetual futures are an innovative financial product, so farunique to the cryptocurrency markets. At the time of writing, standard bitcoin futures are tradedon the Chicago Mercantile Exchange (CME), Bakkt and numerous other less-regulated exchangeswhich also trade a variety of direct and inverse perpetual futures. For instance, quarterly contractsare traded on BitMEX and Deribit, and quarterly/weekly are contracts traded on Huobi, Krakenand OKEx.Apart from their term, a key difference between the two types of contracts is that bitcoinstandard futures are denominated and settled in US dollars (domestic currency) with bitcoin as theunderlying asset (foreign currency), while the majority of bitcoin perpetual futures contracts are inverse futures because they take the opposite design, i.e. they set bitcoin as the denomination unit(domestic currency) and the US dollar as the underlying asset (foreign currency). For instance, oneCME bitcoin standard futures contract has a notional value of 5 bitcoin, while one BitMEX bitcoininverse futures contract has a notional value of 1 US dollar. Please see Alexander et al. (2020) forfurther details of the different contract specifications. The official name is [exchange] perpetual contract , e.g. BitMEX perpetual contract. Some exchanges call themperpetual swaps to highlight that they also have features in common with currency swaps, except there is no exchangeof notional. For an example of their terms and conditions, see Deribit. In terms of availability, bitcoin spot is traded continuously – the markets do not even close onreligious holidays – but the CME closes at weekends and holidays. Therefore, we only consider theless-regulated exchanges. Then, comparing hedging costs for standard futures versus perpetuals,the latter are hardly influenced by the swings between backwardation and contango that can inducea high degree of roll-cost uncertainty into hedging with standard futures contracts. This is becausethe price of the perpetual futures is kept very close to the underlying price by changing the fundingrate to be positive (negative) when the perpetual price is greater (less) than the underlying price.Because of this mechanism the basis risk is very small, the perpetual contract price is continuallyre-aligned with the spot price, and these contracts are highly effective hedges.To summarise, we have both theoretical and practical reasons to assume hedgers use bitcoinperpetual futures. First, hedgers can use such contracts to match their preferred horizon, of anylength exactly . Second, an immediate benefit without an expiry is that there is no need to considerrolling the hedge, which can be a highly technical issue in hedging. Third, unregulated exchangesonly offer one inverse perpetual futures contract, but several active regular futures contracts. Ifwe otherwise chose regular futures, we would need to debate on which regular contract(s) to useor even on how to construct a synthetic contract with a constant maturity using several contracts.Lastly, a perpetual contract is likely more cost efficient than a regular contract, since no rollingmeans there is only one entry and one exit in transactions, and the funding payments are usuallyvery small and balance out over the hedge horizon as they regularly fluctuate between positiveand negative cash flows. Finally, the open interest and trading volumes of bitcoin futures variesconsiderably over time, so it may be difficult to arrive at a commonly-accepted preferment ofone contract over another. Therefore, this study considers three of the main perpetuals that aresettled in bitcoin (XBT), i.e. BitMEX, Deribit, and OKEx. For the spot price we consider theBitMEX underlying, i.e. the .BXBT index, and the bitcoin prices on the two largest spot exchanges,Bitstamp and Coinbase. See the CryptoCompare Monthly Exchange Review, available here. When using standard futures in hedging, one often closes the current contract one week prior to its expiry androlls to the next closest contract. For example, Deng et al. (2020) use OKEx quarterly futures and indeed follow suchan ad-hoc one-week rule. See, for example, Deribit perpetual funding rate. We use XBT as the code of bitcoin currency, which is used by the International Standards Organization (ISO),see Kraken.com. Although popular, we do not consider the Binance perpetuals because they are quoted and settledin the stable coin tether instead of XBT. We selected these exchanges because they had the largest trading volumes at the time of writing. However, itshould be noted that the unregulated derivatives exchanges may engage in wash trading practices that artificiallyinflate volumes – see Cong et al. (2020b) – and that new exchanges can quickly rise in volume-based rankings. Forinstance, Binance’s monthly trading volume in November 2020 was up 132% at $ Feb Mar Apr May Jun Jul AugA: BitMEx Perpetuals0200400600800 L i qu i da t i on B i t c o i n P r i c e Short Long Spot Apr May Jun Jul AugB: Deribit Perpetuals02040 L i qu i da t i on B i t c o i n P r i c e Short Long SpotApr May Jun Jul AugC: OKEx Perpetuals01020 L i qu i da t i on B i t c o i n P r i c e Short Long Spot BitMEX: 82% Deribit: 4%OKEx: 14%D: Relative Weight of Liquidations
Note. We plot both the short (in red) and long (in blue) forced liquidations of bitcoin perpetual futures on BitMEXin Panel A (from February to August 2020), Deribit in Panel B (from March to August 2020), and OKEx in Panel C(from April to August 2020). In Panels A-C, the left y -axis is the size (volume) of the forced liquidations in millionUSD, while the right y -axis is the bitcoin spot price in USD, with the black dashed line denoting the spot price. Thepie chart in Panel D plots the relative weight of forced liquidations among the three perpetual futures during theoverlapping period (from April to August 2020). We obtain data from coinalyze.net. Figure 1 depicts daily time series of the USD amounts of long and short forced liquidationsof bitcoin perpetual futures on BitMEX, Deribit, and OKEx during recent months. Upon closerinspection of these data, we find that the average daily volume of short liquidations is around 11million USD, while that of long liquidations is about triple the size, at around 32 million USD. On‘Black Thursday’ (12 March 2020) a total of 800 million USD worth of long positions in BitMEXperpetual futures were forced into liquidation. This is the main reason why 82% of the liquidationamounts are on BitMEX. Also, apart from Black Thursday, the relative amount of forced liquidation largest derivatives exchange by monthly trading volume; see CryptoCompare Exchange Review November 2020.
6s much lower on Deribit than the other two exchanges. These data testify to the utmost importanceof considering margin and liquidation mechanisms when deciding which bitcoin futures to trade.Because the size of forced liquidations on bitcoin futures can be extremely high, the probabilityof margin calls will also be high. To verify this empirically, we use data from BitMEX bitcoinperpetual futures to compute the probability that a hedger receives a margin call during an intervalof five minutes. We do this for different leverages from 5X to 100X, corresponding to 20% to 1%margins, and hedging horizons between 1 and 30 days. Table 1 presents the results. For positionswith the highest leverage (100X), more than 60% of all 1-day hedges end up with facing margincalls; that is a huge risk to hedgers for such a short hedge horizon. The probability quickly gainsas the hedge horizon increases (in particular from 1day to 15 days). For the 30-day horizon, even5X leveraged positions receive margin calls around 40% of the time. The important message fromTable 1 is that failing to account for default aversion likely leads to failing to achieve the hedgingpurpose. Table 1: Historical Probability of Margin Call for BitMEX Perpetual FuturesLeverage 5X 15X 20X 50X 100XShort 1d 0.30% 8.97% 16.05% 44.33% 63.19%5d 3.42% 37.26% 47.15% 75.67% 84.79%15d 19.54% 65.52% 68.97% 85.06% 91.95%30d 41.86% 67.44% 72.09% 83.72% 93.02%Long 1d 1.44% 12.70% 18.40% 43.73% 61.22%5d 9.51% 40.68% 49.81% 67.68% 80.23%15d 24.14% 56.32% 66.67% 80.46% 89.66%30d 37.21% 76.74% 86.05% 93.02% 95.35% Note. Here we define a margin call as an event when losses from trading futures exceed the margin deposit overthe hedge horizon. We use the 5min historical data of bitcoin perpetual futures on BitMEX from January 1, 2017to August 12, 2020 to compute the probability of margin calls. We perform computations under different leveragelevels, ranging from 5X to 100X, and different hedge horizons, ranging from 1d (day) to 30d.
We consider a discrete-time economy indexed by T := { n ∆ t } n =0 , , , ··· , where time steps ∆ t areequal. The time-interval ∆ t can be interpreted as the frequency that the hedged position is mon-itored. We denote the bitcoin spot price by S = ( S t ) t ∈T and the perpetual futures price by The reciprocal of the leverage is the initial margin percentage, e.g. 100X leverage corresponds to 1% initialmargin. Since bitcoin price is highly volatile, in the empirical analysis we choose monitoring frequencies ∆ t = 15 min, 30min and 1hr. = ( F t ) t ∈T . Both S and F are denominated in fiat currency USD. Define the n -period differences:∆ S t ( n ) := S t + n ∆ t − S t and ∆ F t ( n ) := F t + n ∆ t − F t , (1)where t ∈ T and n = 1 , , · · · . If n = 1 in (1), we simply write ∆ S t and ∆ F t and then – when itis possible without ambiguity – we may also suppress the time subscript t : e.g. ∆ S denotes thechange in the bitcoin spot price over the time period ∆ t . Now the nominal value b F of a bitcoininverse perpetual futures contract, and its difference operators are denoted: b F t := 1 F t , ∆ b F t ( n ) := b F t − b F t + n ∆ t , ∆ b F t := b F t − b F t +∆ t , (2)where t ∈ T and n = 1 , , · · · . The n -period difference ∆ b F t ( n ) is the profit and loss (P&L) of aunit long position in bitcoin inverse futures that is opened at t and closed at t + n ∆ t . We commentthat ∆ b F t ( n ) in (2) is defined as ‘opening price − closing price’ which is different from the definitionof ∆ F t ( n ) in (1). Hence, an increase in the inverse futures price leads to a positive P&L for a longposition in inverse futures.For convenience, and without loss of generality, we consider a representative hedger who holds1 bitcoin at time t ∈ T and who seeks to use a position on an inverse perpetual contract to hedgethe spot price volatility from t to t + N ∆ t , for a positive integer N ≥
1. Because this is aninverse contract, the hedger of a long position shorts θ units of bitcoin inverse futures, where thehedging position θ is established at time t and carried over for N time periods. In other words,the hedger follows a static hedge strategy to protect against the volatility of the bitcoin spot price.The (realized) P&L on the hedge is − θ ∆ b F t ( N ), denominated in XBT. Now, the hedged portfolioconsists of one bitcoin spot (long position) and θ bitcoin inverse futures (short position) from t to t + N ∆ t . The P&L of such a portfolio is given by ∆ S t ( N ) − θ ∆ b F t ( N ) · S t + N ∆ t , where we usethe spot price at time t + N ∆ t to convert the trading P&L of inverse futures, − θ ∆ b F t ( N ), fromXBT to USD. This P&L ignores value changes in the margin account, which are minimal becausefunding rate variations are specifically designed to balance any P&L in the margin account arisingfrom changes in the price of bitcoin. This result naturally leads to the first optimization objective: • Under the minimum-variance framework, the hedger wants to choose θ to minimize the vari-ance of the P&L of the hedged portfolio, given by: σ h ( θ ) := V ar (cid:0) ∆ S t ( N ) − θ ∆ b F t ( N ) S t + N ∆ t (cid:1) , (3) Note that b F in (2) is defined in terms of one unit of fiat currency (1 USD), which corresponds to the numerator1 in the definition. If a bitcoin inverse futures contract has a notional value different from 1 (say 100), we simplymultiply b F by a factor (100). V ar( · ) denotes the variance of a random variable. As readily seen from (3), the minimum-variance hedging of inverse futures is already more com-plex than that of standard futures – recall that in the classical minimum-variance framework ofEderington (1979) and Figlewski (1984a) the corresponding variance to be minimized is simply V ar (cid:0) ∆ S t ( N ) − θ ∆ F t ( N )). The margin mechanism is key to the integrity and stability of futures markets. In Deng et al.(2020), the authors study hedging of bitcoin futures without imposing any margin constraint, i.e.they implicitly assume the hedger has an infinite supply of bitcoin to meet margin requirement.However, this is contradicted by the empirical findings presented in Figure 1 and Table 1. To tradebitcoin inverse futures, the hedger needs to deposit a certain amount of bitcoin to meet the initialmargin requirement and maintain the amount at a specified level, both of which are articulatedusing futures contracts. Futures are marked ‘period to period’ and a positive ∆ b F t means a markedloss to the hedger from t to t + ∆ t is θ · ∆ b F t which may result in a margin call or even a forcedliquidation. This call will be triggered if θ · ∆ b F t exceeds the amount of bitcoin in the hedger’smargin account at time t + ∆ t .We model the impact of margin constraint on the hedger’s decisions using the basic approachto margining proposed by Longin (1999). Let m be a positive number, which can be interpreted asan upper constraint on the hedger’s ability to meet margin requirement in trading bitcoin futures.That is, if there is an extreme price movement, such that θ · ∆ b F t + n ∆ t > m , i.e. the total markedloss is greater than the upper constraint m , the hedger will be forced to liquidate the short positionin bitcoin inverse futures, being unable to acquire enough bitcoin to pay the long counterparty. We call such an extreme scenario a default event, since the hedger indeed fails to cover the fulllosses of the short position. This argument motivates us to incorporate the second optimizationobjective: • Under the margin constraint m , the hedger wants to minimize the default probability givenby: P ( m, θ ) = P rob(default events) := P rob (cid:18) θ · max ≤ n ≤ N − ∆ b F t + n ∆ t > m (cid:19) , (4)where P rob( · ) denotes the probability of an event.By definition (4), the default probability P ( m, θ ) is a decreasing function of the constraint m and Regarding the subscript ∆ h of notation σ h ( θ ) in (3), ∆ means the difference in portfolio values (i.e. P&L) and h denotes the h edged portfolio. Minimizing the portfolio variance (risk) is an important criterion in portfolio management, dating back toMarkowitz’s seminar mean-variance portfolio theory, and is recently adopted in robo-advising; see Capponi et al.(2020) and Dai et al. (2020a,b). Recall that ∆ b F t + n ∆ t is the P&L of trading one unit of bitcoin inverse futures for one period from t + n ∆ t to t + ( n + 1)∆ t , and a positive ∆ b F t + n ∆ t means losses to short hedgers.
9n increasing function of the position θ . The economic meaning is clear: more financial reserves(larger m ) or less risk taking activities (smaller θ ) both reduce the chance of default.From the arguments leading to (3) and (4) the hedger has dual objectives to minimize, and anatural way is to aggregate them into one objective. To balance the magnitude of two objectives wemultiply the default probability P ( m, θ ) by a factor σ S t ( N ) , the variance of the N -period spot pricechanges and to aggregate them we introduce a parameter γ . This way, we consider the followingoptimal hedging problem for the hedger: Problem 3.1.
The hedger, possessing one bitcoin at time t , seeks an optimal static hedging strategy θ ∗ for N periods that solves the following problem: min θ> n σ h ( θ ) + γ σ S t ( N ) P ( m, θ ) o , (5) where σ h ( θ ) is given by (3) , γ > is the aggregation factor, σ S t ( N ) is the variance of the N -periodspot price changes, m > represents the margin constraint, and P ( m, θ ) is given by (4) . One may also interpret γ as a default aversion parameter that captures the extent of the hedger’sdislike of the default events defined in (4). As γ increases, the hedger fears default events moreand will therefore trade in a more conservative way. The limiting case of m = + ∞ (or γ = 0) isstudied in Deng et al. (2020) and corresponds to the scenario where default events have no impacton the hedging decisions. Note that another extreme case is when θ = 0, in which case default willnot occur because the hedger has no position in futures.The main theoretical results of the paper is a semi-closed solution to Problem (5) which is givenin the following theorem and the proof is given in the Appendix A. Theorem 3.2.
The optimal hedging strategy θ ∗ to Problem (5) is given by θ ∗ = F t S t θ , (6) where θ is a positive root to the equation f ( x ) = 0 , where: f ( x ) := γmW t α σ S t ( N ) σ F t ( N ) exp − τ mW t x − βα ! − τ τ mW t x − βα ! − τ − · x + x − σ S t ( N ) , ∆ F t ( N ) σ F t ( N ) . (7) In (7) , W t := S t /F t , constants α , β and τ are estimation parameters of the right tail of ∆ b F based onthe extreme value theorem (see (A.2) in Appendix), σ S t ( N ) (resp. σ F t ( N ) ) denotes the variance ofthe N -period spot (resp. futures) price changes, and σ S t ( N ) , ∆ F t ( N ) denotes the covariance betweenthe two random variables.
10o provide more insight to the rather complex expression in (6), let us take a closer look at twoextreme cases when γ = 0 or m = + ∞ . In these two cases, the first term of f ( x ) in (7) becomeszero and the optimal strategy e θ ∗ (= θ ∗ | γ =0 = θ ∗ | m = ∞ ) is reduced to e θ ∗ = F t S t (cid:18) ρ ∆ S t ( N ) , ∆ F t ( N ) σ ∆ S t ( N ) σ ∆ F t ( N ) (cid:19)| {z } := e θ = F t S t e θ , (8)where ρ ∆ S t ( N ) , ∆ F t ( N ) is the correlation between ∆ S t ( N ) and ∆ F t ( N ). Note that e θ is the optimalhedging strategy when a standard futures contract is used as the hedging instrument. Therefore,even in such simplified cases, the optimal strategy of inverse futures e θ ∗ still differs from that ofstandard futures by a factor F t /S t . Since S t ≈ F t and the current bitcoin price is above 5 digitsUSD, missing this factor could lead to catastrophic consequences when using inverse futures to hedgespot risk. Incorporating a margin constraint m and default aversion γ significantly complicates theanalysis and introduces a non-linear adjustment to correct e θ into θ . At the time of writing, the top three exchanges trading bitcoin perpetual futures by average dailyvolumes are BitMEX, OKEx, and Deribit, with relative market shares given respectively by 77%,17%, and 6% (see the right panel of Figure 2). All three exchanges are online trading platformsthat operate continuously and we refer readers to their websites for full contract specifications. Since bitcoin spot markets are severely segmented, we also consider three different possibilities forthe bitcoin spot price in the empirical analysis: the .BXBT index on BitMEX and the two largestbitcoin spot markets, Bitstamp and Coinbase. We retrieve data at the 5-minute frequency onthese prices using the API (application programming interface) provided by CoinAPI covering theperiods shown in Table 2. Please refer to Table 2 for the information of the bitcoin spot andfutures price data used in the empirical analysis.To calculate 1-period changes in the spot price ∆ S , the futures price ∆ F , and the 1-periodchange in the inverse futures price, ∆ b F , we consider the time step at three different values ∆ t =15 minutes, 30 minutes and 1 hour. Recall that ∆ b F s = b F s − b F s +∆ t = 1 /F s − /F s +∆ t . Table 3 For instance, the full specifications of BitMEX perpetual futures (XBTUSD) can be accessed from link. Bitstamp and Coinbase are two dominant exchanges for trading bitcoin with the latter having the biggest tradingvolumes; see Figure 2 and Table 3 in Alexander and Heck (2020). The .BXBT index is the reference price for theBitMEX perpetual futures (XBTUSD), which tracks the bitcoin price every minute and is calculated as the weightedaverage of the last price of 5 currently active constituent exchanges: Bitstamp (25.73%), Bittrex (2.63%), Coinbase(46.98%), Gemini (6.69%), and Kraken (17.97%); see details on BitMEX link. CoinAPI is a platform providing fast, reliable, and unified data APIs for cryptocurrency markets.
Oct 2019 Jan 2020 Apr 2020 Jul 2020024681012 10 BitMEX: 77% Deribit: 6%OKEx: 17%
Note. The left panel plots daily trading volumes of bitcoin perpetual futures on BitMEX, Deribit, and OKEx, whilethe right panel plots their relative weight of aggregate volumes. The unit of y -axis in the left panel is USD. For bothpanels, data spans from July 2019 to July 2020. Table 2: Data Information of bitcoin Spot and Perpetual FuturesExchange Start Date End Date
Note. This table reports the start/end date and the number of observations for each of the three spot and threefutures price data sources used in the empirical analysis. reports the summary statistics of the price changes ∆ S and ∆ F at different time steps ∆ t . Themean is very close to zero, but the standard deviation is very large. Expressed in daily terms, itranges from 37 to 167. There is a strong negative skewness especially in the BitMEX index andOKEx perpetual futures and the kurtosis is extremely high. Using data at the base frequency of 5minutes data the correlations between ∆ S and ∆ F , for all possible combinations of bitcoin spot andperpetual futures, are displayed in Table 4. As expected, there exists a strong positive correlation.The highest correlation between ∆ S and ∆ F is 0.96 (Coinbase spot and OKEx futures) while thelowest is 0.79 in the cases of BitMEX spot and BitMEX/OKEx futures. To calculate the optimal hedging strategy θ ∗ in (6), we need to estimate the variances σ S ( N ) and σ F ( N ) , the covariance σ S ( N ) , ∆ F ( N ) and the tail distribution parameters α , β and τ . Herewe demonstrate how these quantities are estimated as time series, using a fixed-size window thatrolls over the entire sample. For illustration purpose, we fix the monitoring frequency ∆ t to be 30minutes and the hedge horizon N ∆ t = to be 5 days. We also fix the length of the rolling window12able 3: Summary Statistics of the bitcoin Spot and Futures Price Changes BitMEX Index ∆ S Bitstamp ∆ S Coinbase ∆ S Variables 15min 30min 1h 15min 30min 1h 15min 30min 1hMin -1274.79 -1006.46 -1184.61 -1011.89 -1328.27 -1295.01 -1261.21 -1517.11 -1605.00P25 -8.38 -11.90 -16.32 -8.58 -11.73 -16.15 -8.00 -11.17 -15.81Median 0.15 0.19 0.33 0.16 0.29 0.59 0.06 0.26 0.56P75 8.66 12.36 16.92 9.14 12.62 17.82 8.59 12.04 17.33Max 710.71 645.02 949.39 1045.00 1392.98 1218.00 1116.64 1241.36 1198.12Mean 0.03 0.07 0.14 0.08 0.17 0.34 0.08 0.17 0.33Skewness -2.72 -1.52 -1.14 -0.82 -0.75 -0.74 -0.85 -0.94 -0.67Kurtosis 120.78 59.61 49.75 64.78 53.15 37.47 92.57 61.55 40.65Daily S.D. 30.79 43.33 62.19 41.04 57.08 79.80 41.04 56.66 79.22 F Deribit ∆ F OKEx ∆ F Note. This table reports the summary statistics of spot price changes ∆ S and futures price changes ∆ F acrossdifferent spot and futures markets. P25 and P75 refer to 25% and 75% quantiles, S.D. denotes standard deviation,and Table 4: Correlations of bitcoin Spot and Perpetual Futures Price Changes
Spot Futuresvariables ∆BitMEX ∆Bitstamp ∆Coinbase ∆BitMEX ∆Deribit ∆OKExSpot ∆BitMEX 1.00 0.80 0.81 0.79 0.80 0.79∆Bitstamp 0.80 1.00 0.82 0.87 0.89 0.93∆Coinbase 0.81 0.82 1.00 0.88 0.92 0.96Futures ∆BitMEX 0.79 0.87 0.88 1.00 0.92 0.95∆Deribit 0.80 0.89 0.92 0.92 1.00 0.88∆OKEx 0.79 0.93 0.96 0.95 0.88 1.00
Note. This table reports the correlation coefficient between ∆ S and ∆ F for different choices of bitcoin spot S andperpetual futures F . All the results are computed using the 5-min data over the full available sample period. to be 210 days and use the observations on ∆ S , ∆ F and ∆ b F at ∆ t = 30-min frequency.To estimate the desired variances and covariance we derive a sub-sample from our 210-daywindow of 30-minute returns, which is a sub-sample of 42 observations on each 5-day price dif-ference ∆ S ( N ) and ∆ F ( N ), and apply the standard operators to this sub-sample. Then wecompute another sub-sample which consists of the maximum 30-minute price change ∆ b F oneach day – this is taken as the ‘extreme value’ for that day. That is, we take the maximum of (cid:16) ∆ b F t +∆ t , ∆ b F t +2∆ t , · · · , ∆ b F t +48∆ t (cid:17) as the extreme value on day t . This way, we construct a sub-13ample of 210 extreme values which we use to estimate the right tail distribution parameters α , β and τ for that window (see (A.2)). Please refer to Figure 3 for graphic illustrations of the procedure.Please see Online Supplementary Appendix I for further details.Figure 3: Parameters Estimation ProcedureThe parameter τ measures the heaviness of the right tail, i.e. the losses to hedgers with shortfutures positions, and this parameter has a major impact on the default probability. On the otherhand, the scale parameter α and the location parameter β represent the dispersion and the averageof the extreme value observations, which are less important because we can always change the scaleand location. So let us now discuss our findings on τ , since it is the most important parameter forhedging. Table 5 reports summary statistics of the τ parameter estimates, just for the case ∆ t =30min and N ∆ t = 5 days, and Figure 4 depicts how this estimate evolves as the 210-day windowrolls over the sample.Table 5: Summary Statistics of Estimated Tail Index τ ∆ S Spot Price Difference ∆ F Futures Price Difference τ ∆ S τ ∆ S τ ∆ S τ ∆ F τ ∆ b F τ ∆ F τ ∆ b F τ ∆ F τ ∆ b F Variables BitMEX Bitstamp Coinbase BitMEX BitMEX Deribit Deribit OKEx OKExMin 0.32 0.17 0.15 0.16 0.23 0.30 0.23 0.32 0.46Mean 0.52 0.48 0.49 0.48 0.44 0.54 0.47 0.43 0.61Median 0.45 0.43 0.43 0.45 0.39 0.49 0.49 0.45 0.62Max 0.97 0.98 1.02 0.94 0.71 0.95 0.68 0.52 0.74Daily S.D. 0.16 0.17 0.17 0.16 0.15 0.16 0.14 0.06 0.09Count 631 1111 1111 1111 1111 504 504 183 183
Note. This table reports summary statistics for the estimated tail index parameter τ for ∆ S , ∆ F and ∆ b F using arolling 210-day sample from all available data sources. We set ∆ t = 30 min and N ∆ t = 5 days. The most important finding is that τ remains positive over the entire sample period, for allvariables, which immediately implies the limiting distributions are Fr´echet type – see also FigureA.1 in the appendix. Moreover, they have very heavy right tails. This indicates that taking shortpositions can lead to substantial losses. From Table 5, we observe that the mean and median ofthe estimated τ ’s are almost the same except for ∆ b F computed from the OKEx perpetual futuresand possibly the Deribit perpetual futures as well. Panel A of Figure 4 shows that the estimated τ of the spot price changes ∆ S are very close to each other for the three spot exchanges BitMEX,Bitstamp and Coinbase. In comparison, regarding panels B – D, the estimated τ ’s of ∆ F and ∆ b F τ for ∆ S , ∆ F , and ∆ b F Jan 2018 Jan 2019 Jan 2020
A: BitMEX, Bitstamp and Coinbase Spot T a il I nde x Jan 2018 Jan 2019 Jan 2020
B: BitMEX Futures T a il I nde x Jul 2019 Jan 2020 Jul 2020
C: Deribit Futures T a il I nde x Mar Apr May Jun Jul Aug
D: OKEx Futures T a il I nde x Note. This figure plots the rolling estimation of the tail index parameter τ for ∆ S , ∆ F , and ∆ b F using a rollingwindow of 210 days. The full sample data are specified in Table 2. For illustration we set the monitoring frequencyof ∆ t = 30 min and a hedge horizon N ∆ t = 5 days. Panel A plots the estimated τ of ∆ S for BitMEX, Bitstampand Coinbase. This shows that the different τ estimates are very close to each other for the three exchanges. PanelsB to D plot the estimated τ of ∆ F and ∆ b F for the bitcoin perpetual futures traded on BitMEX, Deribit and OKEx,respectively. These exhibit very different features across the three exchanges. All we can say is that, in most cases, τ ∆ b F is larger than τ ∆ F . vary more significantly from exchange to exchange, and for a fixed exchange, ∆ b F produces a higher τ than ∆ F in most cases.Now at each time t we use the rolling window of the preceding 210 days to estimate all theparameters and we apply (6) to compute the current optimal strategy θ ∗ t , which will be followedfrom t to t + N ∆ t . We denote by ∆ V ∗ the P&L of the hedged portfolio for N periods underthe optimal strategy θ ∗ . For instance, the P&L of strategy θ ∗ t from t to t + N ∆ t is given by∆ V ∗ t = ∆ S t ( N ) − θ ∗ t · ∆ b F t ( N ) S t + N ∆ t . Once we have the realized price data at time t + N ∆ t , wecan compute ∆ V ∗ t and store it. Then, we roll the 210-day sample of 30-minute data forward by30-minutes and repeat all parameter estimates, compute the optimal strategy and calculate ∆ V ∗ t .We continue this process until we arrive at the last available time point, i.e. exactly N ∆ t periodsprior to the end of the full sample. This constitutes our realized time series of ∆ V ∗ , the P&L of theoptimally hedged portfolio, that will be used to investigate hedge effectiveness in the next section.15 Empirical Analysis
In this section, we conduct empirical analysis to investigate the economic consequences of the opti-mal hedging strategy θ ∗ , derived in (6), for the representative hedger. We begin with a sensitivityanalysis for θ ∗ in Section 5.1. Next, we study two important topics related the hedger’s dual objec-tives: hedge effectiveness in Section 5.2 and default probability in Section 5.3. We close the sectionwith investigations on the implied leverage under θ ∗ in Section 5.4. To understand the results obtained in this sub-section, recall that the representative hedger seeksthe optimal strategy θ ∗ with dual objectives to minimize the portfolio variance (risk) σ h in (3) andthe default probability P ( m, θ ) in (4) and that the hedger’s optimal strategy θ ∗ can be computedefficiently once all the parameters in (6) have been estimated or assigned. Since we are mostlyinterested in the qualitative behavior of the optimal strategy θ ∗ with respect to various parameters,we assign reasonable base values and conduct sensitivity analysis focusing on four parameters: thedefault aversion γ and margin constraint m , which are specific to the hedger’s own profile, and thetail index τ and correlation coefficient ρ ∆ S, ∆ F which are purely market-specific. The results arepresented in Figure 5.We now discuss the findings from Figure 5. The first impression is that both margin constraintand default aversion have a major effect on the optimal strategy θ ∗ , by carefully examining therelative scales along the y-axis. This provides further justifications for incorporating them inthe problem setup (5). The impact of the margin constraint m and the default aversion γ onthe optimal strategy θ ∗ is only through their influence on the default probability. When defaultaversion γ increases, minimizing the default probability becomes a more important task to thehedger and a natural solution is to reduce the size of the futures position – because, by definition(4), a decrease of θ leads to a smaller P ( m, θ ). This observation amounts to the optimal strategy θ ∗ being a decreasing function of γ , as shown in the upper right panel. When m increases, thehedger’s financial reserves improve, naturally reducing the likelihood of default and thus alleviatingthe constraint on trading. In consequence, we expect the hedger to increase their futures position,and this is confirmed by the lower left panel in Figure 5.Next we analyse the effect of the other two parameters, the tail index τ and the correlationcoefficient ρ ∆ S, ∆ F , which are market-related and independent of the hedger’s particular profile.From the standard hedging theory and also the result in (8), the correlation coefficient ρ ∆ S, ∆ F appears as a positive multiplier in the optimal strategy and a near-linear positive relation between ρ ∆ S, ∆ F and θ ∗ is anticipated, which is verified by the lower right panel in Figure 5. According Recall from (8) that, without margin constraint and default aversion, the optimal strategy e θ ∗ is indeed a linearfunction of ρ ∆ S, ∆ F . Once both features are included, a non-linear adjustment is needed; however, the leading term θ ∗ Note. We conduct sensitivity analysis of the optimal strategy θ ∗ on four parameters: tail index τ , default aversion γ ,margin constraint m , and correlation coefficient ρ ∆ S, ∆ F . Here, we express m in percentage since the hedger’s initialholding in bitcoin is normalized to one. In each of the four panels, we study the impact of one target parameter(shown as the x -axis label) on the optimal strategy, while fixing other three parameters as shown on top. We set theremaining parameters as: σ S = σ F = 1, S t = F t = 1, α = β = 0 . to the extreme value theorem, tail index τ measures the heaviness of the tails of ∆ b F ; a larger τ means heavier tails or more extreme price changes (see Figure A.1). The impact of τ on the optimalstrategy is rather complex and highly non-linear, as seen from the definition of f in (7). The upperleft panel in Figure 5 indicates a positive relationship between τ and θ ∗ . In this sub-section, we study the hedge effectiveness of bitcoin perpetual futures in hedging bitcoinspot risk. To that end, we consider two strategies (portfolios) for the hedger: the first one is an unhedged portfolio holding one bitcoin, and the second one is an optimally hedged portfolio thatconsists of one bitcoin and a short position of θ ∗ perpetual futures contracts, where the optimalstrategy θ ∗ is given by (6). Since the tail index parameter τ is rather stable with respect to thechoice of time step ∆ t (see Table I.1), we fix the monitoring frequency ∆ t = 30 minutes in thefollowing analysis.First, we focus on the P&L of these two (unhedged and hedged) portfolios. We apply the rolling is still determined by e θ ∗ . This positive relationship can also be verified mathematically by showing that the first non-linear term in (7)increases as τ increases. N ∆ t =5 days, 15 days, and 1 month. The margin constraint m is taken to be 10% for the moment. Thisis equivalent to assuming that the hedger who is long 1 bitcoin is able to acquire an additional 0.1bitcoin to top up the margin requirement when trading bitcoin futures. Figures 6 and 7 draw thehistograms of the value changes of these two portfolios, along with the optimal strategy θ ∗ , in thecases that γ = 20 and γ = 100 respectively. For illustration we choose the spot price to be theBitMEX .BXBT index and the futures price to be that of BitMEX perpetuals in Figures 6 and 7.Recall that γ is the default aversion parameter (or aggregation factor) in the main problem (5).A larger γ means the hedger is more risk averse to ‘default events’ and concerns more about forcedliquidations. When the hedger’s default aversion is low (e.g. γ = 20 in Figure 6), the volatility ofthe hedged portfolio is much smaller than that of the unhedged portfolio, indicating the optimalhedging strategy works as hoped in reducing risk. However, if the hedger has a very high defaultaversion (e.g. γ = 100 as in Figure 7), the degree of volatility reduction from hedging is limited.The first economic explanation is that the non-linear payoff of bitcoin inverse futures inducesan asymmetry effect, whereby a decline in the bitcoin spot price imposes a stronger effect onthe P&L of inverse futures than an increase in spot price of the same size. This effect makesinverse futures considerably more risky than bitcoin spot, as already noted by Deng et al. (2020).Secondly, a more risk averse hedger acts more conservatively by shorting less futures contracts thanthe ‘global’ optimal level (i.e. θ ∗ under γ = 0, shown in red in the low panels of Figures 6 and 7),which naturally leads to a diminished effect on volatility reduction. In addition, we observe fromboth figures that the volatility of both portfolios increases with the hedge horizon N ∆ t , althoughthe optimal strategy θ ∗ remains relatively stable.To investigate the hedging performance of the optimal strategy θ ∗ , we define hedge effectiveness(HE) as the percentage reduction in portfolio variance. Let ∆ V ( θ ) denote the P&L of a portfoliowith one bitcoin and θ short positions in bitcoin futures. It is clear that θ = 0 corresponds to theunhedged portfolio. Mathematically, we define HE byHE( θ ) = 1 − V ar(∆ V ( θ )) V ar(∆ V (0)) , θ > , (9)where V ar( · ) denotes the variance of a random variable. We calculate the hedge effectiveness of theoptimal portfolio (i.e. when θ = θ ∗ given by (6)) for different pairs of bitcoin spot and perpetualfutures under different margin constraints m , default aversion γ , and hedge horizon N ∆ t . Wereport the results in Table 6 under three margin constraint levels ( m = 10% , , γ = 20 , N ∆ t = 5d, 30d). In each ofTables II.1-II.9 in Online Supplementary Appendix II, we calculate HE defined in (9) for one pair ofbitcoin spot (from BitMEX, Bitstamp, or Coinbase) and perpetual futures (from BitMEX, Deribit,or OKEx) under four margin constraint levels ( m = 10% , , , θ ∗ under γ = 20 Note. The top three panels plot the histogram of the P&L of the unhedged and optimally hedged portfolios underthree different hedge horizons 5d, 15d and 1m. The lower three panels plot the optimal hedging strategy θ ∗ under γ = 20 for the three hedge horizons, along with θ ∗ under γ = 0 for comparison purpose. We observe that the hedgertrades less bitcoin futures when she is default averse, captured by a strictly positive γ . We set monitoring frequency∆ t = 30min, m = 10% and γ = 20. For illustration purpose, we only focus on BitMEX index and BitMEX perpetualfutures. Figure 7: P&L of Unhedged and Hedged Portfolios, and Optimal Strategy θ ∗ under γ = 100 Note. All the explanations in note of Figure 6 apply here as well, except now we set the default aversion parameter γ = 100. γ = 20 , , , N ∆ t = 5d, 15d, 30d, 60d).Table 6: Hedge Effectiveness of bitcoin Perpetual FuturesSpot MarketsBitMEX Bitstamp Coinbase BitMEX Bitstamp CoinbaseFutures m γ
5d 30dBitMEX m = 10% γ = 20 93.61% 74.98% 75.08% 94.45% 78.53% 76.82% γ = 100 68.92% 51.38% 55.65% 67.31% 52.28% 51.78% m = 20% γ = 20 98.75% 92.31% 92.28% 99.14% 94.68% 94.35% γ = 100 89.89% 76.00% 75.50% 90.07% 79.38% 78.67% m = 50% γ = 20 99.54% 99.20% 99.39% 99.87% 99.68% 99.71% γ = 100 98.69% 96.12% 96.29% 99.00% 97.69% 97.57%Deribit m = 10% γ = 20 93.99% 93.96% 93.92% 94.74% 94.72% 94.71% γ = 100 70.18% 70.01% 69.93% 69.11% 69.06% 69.05% m = 20% γ = 20 99.03% 99.08% 99.06% 99.26% 99.27% 99.26% γ = 100 90.72% 90.65% 90.59% 91.11% 91.08% 91.07% m = 50% γ = 20 99.77% 99.85% 99.84% 99.92% 99.94% 99.94% γ = 100 99.03% 99.08% 99.06% 99.22% 99.23% 99.22%OKEx m = 10% γ = 20 97.68% 97.77% 97.79% 97.83% 97.84% 97.86% γ = 100 76.71% 76.74% 76.77% 76.12% 76.14% 76.17% m = 20% γ = 20 99.60% 99.71% 99.71% 99.74% 99.75% 99.75% γ = 100 94.67% 94.80% 94.81% 95.35% 95.37% 95.39% m = 50% γ = 20 99.86% 99.96% 99.96% 99.97% 99.98% 99.98% γ = 100 99.42% 99.55% 99.55% 99.64% 99.66% 99.66% Note. This table reports the hedge effectiveness, defined in (9), of the optimal strategy θ ∗ at different hedge horizon N ∆ t , margin constraint m , and default aversion γ . We consider three bitcoin spot markets BitMEX (.BXBT index),Bitstamp and Coinbase, and three bitcoin perpetual futures on BitMEX, Deribit and OKEx. We set ∆ t = 30min. We summarize the key findings on hedge effectiveness as follows: • The margin constraint has a major impact on both the hedge effectiveness and the optimalstrategy θ ∗ in bitcoin futures markets. In all cases, once the hedger has sufficient capital tomeet margin requirement (e.g. m = 50%), the optimal hedging strategy achieves superiorhedge effectiveness (with HE close to 100%). This finding offers strong support to the inclusionof margin constraint in the hedging analysis of bitcoin futures. The hedging performance ofthe optimal strategy may be over-exaggerated if the analysis does not consider the possibilityof margin constraint (such as Alexander et al. (2020) and Deng et al. (2020)). • The effect of margin constraint is more profound for hedgers with high default aversion (large γ ). To see this, we consider the case of using BitMEX perpetual futures to hedge the spotrisk on Bitstamp given γ = 100. By comparing the hedge effectiveness under m = 10% and20 = 50%, we observe a very significant improvement: HE increases sharply from around 40%to 95% for both 5d and 30d hedge horizon. This extreme example also serves as anotherevidence to the above conclusion that it is critical to include margin constraint in the hedgingstudy of bitcoin inverse futures. In addition, for a fixed margin constraint level m , hedgerswith lower default aversion short more futures and are able to reduce the portfolio risk in amore significant amount than those with higher default aversion. • As margin constraint parameter m increases (i.e. the hedger has a ‘deeper’ pocket), theoptimal strategy’s hedge effectiveness improves to almost 100% rather quickly across all pairsof spot and futures markets. This result testifies that bitcoin perpetual futures (offeredby BitMEX, Deribit, and OKEx) provide an effective instrument to hedge against the pricevolatility of bitcoin spot price, which is consistent with the findings in Alexander et al. (2020).On the other hand, there is no clear winner among the three perpetual futures consideredin the analysis, as they deliver similar hedging performance across different spot markets.However, upon close examination, we find that (1) OKEx perpetual futures is marginallybetter than the other two perpetual futures in terms of HE, and (2) BitMEX perpetualfutures is not a good choice for high default averse hedgers who want to hedge the spot riskon Bitstamp and Coinbase. • As the hedge horizon ( N ∆ t ) increases, the hedge effectiveness improves, although at a tinyscale. To see this, we refer to Tables II.1-II.9 in Online Supplementary Appendix II where weconsider four different choices for the hedge horizon at N ∆ t = 5d, 15d, 30d, and 60d. Thisresult is due to the fact that a longer horizon naturally smooths the price time series. We regard a default event for futures traders as the circumstance that they cannot meet a margincalls – see (4) – and is forced to liquidate their futures positions. Clearly, as readily seen fromFigure 1 and Table 1, both the historical forced liquidation volumes and the simulated margincall probabilities (using BitMEX perpetual futures data) are very high, imposing enormous riskto futures traders. Our approach to alleviating this risk is to take account of the impact of thedefault probability P ( m, θ ), defined in (4), on the hedger’s hedging strategy θ . To be precise, weassume the representative hedger is default averse and aims to minimize the default probability P ( m, θ ), as formulated in Problem (5). With that in mind, we naturally hypothesize that followingthe optimal strategy θ ∗ should reduce the default probability to a reasonable level. But how muchof a reduction is possible? So in this sub-section we ask, to what degree does the optimal hedgingstrategy reduce the default probability?To investigate this, we obtain the default probability P ( m, θ ∗ ) in (10) under the optimal strategy θ ∗ , which we call the ‘optimal default probability’ for short. We derive this using the extreme value21heorem (A.2) in Appendix A as: P ( m, θ ∗ ) ≃ − exp " − (cid:18) τ (cid:18) mθ ∗ − βα (cid:19)(cid:19) − /τ , (10)where α and β are the scale and location parameters respectively, and τ is the tail index parameter.All the parameters α , β , and τ have been estimated and summarized in Table 5 and the optimalstrategy θ ∗ has been obtained as well – see, e.g. the lower panels in Figures 6 and 7. Hence, wecan calculate the right hand side of (10), and use it as an approximation to the optimal defaultprobability P ( m, θ ∗ ). We emphasis that the default probabilities reported in Table 7 are all at themonitoring frequency of ∆ t =30min.Table 7: Default Probability Under the Optimal Strategy θ ∗ Spot MarketsBitMEX Bitstamp Coinbase BitMEX Bitstamp CoinbaseFutures m γ N ∆ t = 5d N ∆ t = 30dBitMEX m = 10% γ = 20 0.55% 0.52% 0.52% 0.55% 0.52% 0.52% γ = 100 0.23% 0.19% 0.19% 0.23% 0.19% 0.19% m = 20% γ = 20 0.20% 0.19% 0.20% 0.20% 0.20% 0.20% γ = 100 0.14% 0.12% 0.12% 0.14% 0.12% 0.12% m = 50% γ = 20 0.05% 0.04% 0.04% 0.05% 0.04% 0.04% γ = 100 0.04% 0.03% 0.03% 0.04% 0.03% 0.03%Deribit m = 10% γ = 20 0.53% 0.54% 0.54% 0.54% 0.54% 0.54% γ = 100 0.22% 0.22% 0.22% 0.22% 0.22% 0.22% m = 20% γ = 20 0.19% 0.19% 0.19% 0.19% 0.19% 0.19% γ = 100 0.13% 0.13% 0.13% 0.13% 0.13% 0.13% m = 50% γ = 20 0.04% 0.04% 0.04% 0.04% 0.04% 0.04% γ = 100 0.04% 0.04% 0.04% 0.04% 0.04% 0.04%OKEx m = 10% γ = 20 0.57% 0.58% 0.57% 0.57% 0.57% 0.57% γ = 100 0.26% 0.26% 0.26% 0.26% 0.26% 0.26% m = 20% γ = 20 0.20% 0.21% 0.21% 0.21% 0.21% 0.21% γ = 100 0.15% 0.15% 0.15% 0.16% 0.16% 0.16% m = 50% γ = 20 0.05% 0.05% 0.05% 0.05% 0.05% 0.05% γ = 100 0.04% 0.04% 0.04% 0.04% 0.04% 0.04% Note. This table reports the default probability P ( m, θ ∗ ) under the optimal strategy θ ∗ , which is computed using(10). We perform calculations for all pairs of bitcoin spot (BitMEX, Bitstamp, and Coinbase) and bitcoin perpetualfutures (BitMEX, Deribit, and OKEx) under different margin constraint m ( m = 10% , , γ ( γ = 20 , N ∆ t ( N ∆ t = 5d, 30d). Default probabilities reportedhere are all at the monitoring frequency of ∆ t =30min. Table 7 reports the results and we outline the main findings in the following: • The optimal default probability P ( m, θ ∗ ) is very small, close to or less than 1%, in all scenar-22os. Therefore, this result strongly supports our hypothesis that the hedger is able to reducethe default probability to a desirable level by following the optimal strategy θ ∗ . We alsoemphasize that such a finding is robust, in the sense that it holds in regardless of the marginconstraint level m , default aversion γ , hedge horizon N ∆ t , and the choice of bitcoin spot andperpetual futures. • The margin constraint m has a major impact on the optimal default probability. A larger m leads to a smaller optimal default probability. For a hedger with relatively low defaultaversion ( γ = 20), increasing the margin constraint from 10% to 20% reduces the optimaldefault probability by about 50%. To achieve the similar 50% of reduction, a hedger withrelatively high default aversion ( γ = 100) needs to increase m from 10% to 50% or more.With sufficient resource to meet margin requirement (say m ≥ • As expected, the optimal default probability decreases as the default aversion γ increases.Hedgers with higher γ are more default averse and hence act more conservatively by takingless shorts positions in bitcoin perpetual futures when hedging spot risk (see the lower panelsin Figures 6 and 7 for evidence). • In the computations leading to Table 7, we consider three markets for bitcoin spot (BitMEX,Bitstamp, and Coinbase) and three bitcoin perpetual futures (BitMEX, Deribit, and OKEx).We do not see any significant difference on the optimal default probability among differentpairs of bitcoin spot and perpetual futures, although Deribit perpetual futures is the mostpreferred choice. This provides a positive signal to apply our approach to the cases when adifferent bitcoin spot market or/and a different perpetual futures contract is taken. • Hedge horizon only has a marginal effect on the optimal default probability, with a weak posi-tive correlation. That is, as the hedge horizon N ∆ t increases, the optimal default probability P ( m, θ ∗ ) increases, at a negligible scale though. A distinguish feature of various bitcoin derivatives is high leverage, e.g. up to 100X for BitMEXperpetual futures. In the last part of empirical analysis, we study the implied leverage under theoptimal strategy θ ∗ taken by the representative hedger.Recall that in our setup, the margin constraint m is applied to the entire short position of θ ∗ futures contracts. Hence, the margin constraint per contract is m/θ ∗ in XBT. Since the nominalvalue per futures contract is equal to b F = 1 /F also in XBT, the implied leverage in futures under23he optimal strategy θ ∗ is given by θ ∗ / ( F · m ). We then compute the implied leverage under theoptimal strategy θ ∗ and present the results in Table 8. We summarize our observations from Table8 as follows: Table 8: Implied Leverage Under the Optimal Strategy θ ∗ Spot MarketsBitMEX Bitstamp Coinbase BitMEX Bitstamp CoinbaseFutures m γ N ∆ t = 5d N ∆ t = 30dBitMEX m = 10% γ = 20 8.75 8.32 8.32 8.81 8.34 8.36 γ = 100 6.02 5.86 5.86 6.06 5.88 5.88 m = 20% γ = 20 4.76 4.70 4.71 4.80 4.72 4.74 γ = 100 4.17 4.11 4.12 4.19 4.13 4.14 m = 50% γ = 20 1.96 1.96 1.97 1.98 1.97 1.98 γ = 100 1.90 1.91 1.91 1.92 1.92 1.93Deribit m = 10% γ = 20 8.73 8.75 8.76 8.77 8.77 8.77 γ = 100 6.03 6.04 6.04 6.03 6.03 6.03 m = 20% γ = 20 4.77 4.79 4.79 4.80 4.80 4.80 γ = 100 4.19 4.20 4.20 4.20 4.20 4.20 m = 50% γ = 20 1.97 1.97 1.97 1.98 1.98 1.98 γ = 100 1.91 1.92 1.92 1.92 1.92 1.92OKEx m = 10% γ = 20 8.81 8.83 8.83 8.87 8.87 8.87 γ = 100 5.89 5.89 5.89 5.94 5.94 5.94 m = 20% γ = 20 4.79 4.80 4.80 4.81 4.81 4.81 γ = 100 4.16 4.17 4.17 4.18 4.18 4.18 m = 50% γ = 20 1.97 1.98 1.98 1.98 1.98 1.98 γ = 100 1.91 1.92 1.92 1.92 1.92 1.92 Note. This table reports the implied leverage under the optimal strategy θ ∗ , which is defined as θ ∗ / ( F · m ). Weperform calculations for all pairs of bitcoin spot (BitMEX, Bitstamp, and Coinbase) and bitcoin perpetual futures(BitMEX, Deribit, and OKEx) under different margin constraint m ( m = 10% , , γ ( γ = 20 , N ∆ t ( N ∆ t = 5d, 30d). All Leverages reported here are atthe monitoring frequency of ∆ t =30min. • The implied leverage the optimal strategy θ ∗ varies from 1X to 9X, which is at a reasonablelevel for bitcoin futures. For instance, BitMEX perpetual futures allow up to 100X leverage,while the implied leverage in the case of BitMEX perpetual futures is below 5X for a hedgerwith a moderate margin constraint (say m ≥ m = 10%, and γ = 20.We thus conclude that the hedger reduces leverage considerably in trading perpetual futureswhen she follows the optimal strategy θ ∗ derived under the margin constraint. • Similar to the analysis of hedge effectiveness and default probability, both the margin con-24traint and default aversion play a key role in determining the implied leverage. Given asufficiently large margin constraint (say m = 50%), the implied leverage is less than 2X in allcases, and remains stable for different default aversion γ and pairs of bitcoin spot and per-petual futures. On the other hand, the impact of hedge horizon N ∆ t on the implied leverageis negligible. Motivated by both empirical findings and practical needs, we study how margin constraints anddefault aversion affect the optimal hedging of bitcoin using futures. A representative investor is as-sumed to hold one bitcoin and hedges against its price volatility by trading bitcoin perpetual inversefutures, with a limited ability to acquire bitcoin to meet margin calls. The representative investoris averse to both market volatility and default risks, where default is defined as the circumstancewhen marked trading losses exceeding the margin constraint. The main theoretical contributionof this paper is to formulate the optimal hedging problem under dual objectives: minimizing boththe volatility of the hedged portfolio and the probability of default. An application of the extremevalue theorem yields the optimal hedging strategy in semi-closed form.In the empirical part of this paper we investigate three important topics in risk management:hedge effectiveness, default probability, and implied leverage. Based on three popular bitcoin per-petual futures (BitMEX, Deribit, and OKEx) and three bitcoin spot markets (BitMEX, Bitstamp,and Coinbase) we conduct a sensitivity analysis of the optimal strategy, to examine how it changeswith various parameters controlling the characteristics of both the contract and the hedger.The primary goal of using futures in hedging is to reduce the risk exposure to the spot mar-kets; see for instance Ederington (1979) and Figlewski (1984a). As such, we analyze the hedgeeffectiveness of the optimal hedging strategy, i.e. whether and to what degree the optimal strategyhelps reduce the variance. However, of particular interest here, given the novelty of our theoreticalresults, is the effect of margin constraint and default aversion on the hedger’s decisions. We findthat the optimal hedging strategy achieves superior hedge effectiveness, in terms of reducing port-folio volatility, as long as the hedger is not overly default averse or the margin constraint is not tootight.As is well known, bitcoin futures are among the most leveraged financial products in the markets(e.g. BitMEX perpetual futures allow up 100X leverage). But the high leverage feature of bitcoinfutures is a double-edged sword: on the positive side, this feature helps attract a large amount ofnoise traders and speculators, and thus improves the overall market depth and liquidity; while onthe negative side, it could easily lead to forced automatic liquidations (default events). Therefore,in the last part, we investigate the implied leverage under the optimal strategy. By following ourstrategy the hedger is able to reduce the default probability significantly (less than 1% in most25ases) and control the leverage in futures trading to a reasonable level (mostly below 5X).
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In this section, we provide detailed derivations leading to our main results in Theorem 3.2.
Step 1.
Approximation of the variance term σ h ( θ ) defined in (3).Recall that σ h ( θ ) is the variance of the P&L of the hedged portfolio consisting of one long positionin bitcoin spot and θ short positions in bitcoin inverse futures. By following the same argumentsin Deng et al. (2020) (see Eq.(3.1) therein), we obtain σ h ( θ ) ≃ σ S t ( N ) + (cid:18) S t θF t (cid:19) σ F t ( N ) − S t θF t σ S t ( N ) , ∆ F t ( N ) , (A.1)where σ S t ( N ) (resp. σ F t ( N ) ) denotes the variance of the N -period spot (resp. futures) pricechanges, and σ S t ( N ) , ∆ F t ( N ) denotes the covariance between the two random variables. The ap-proximation result in (A.1) depends on one condition, S t /F t ≃ S t + N ∆ t /F t + N ∆ t . Since this ratio isalmost unchanging over time, being very close to one for all the exchanges considered, we concludethat the approximation to σ h ( θ ) in (A.1) is very accurate. Step 2.
Approximation of the default probability P ( m, θ ) defined in (4).Due to the margin constraint m , the representative hedger may not be able to meet margin calls ifthe futures price encounters a sharp movement, which is referred to as a default event. The proba-bility of default is then given by (4). Since the hedger holds short positions in the futures, the righttail of the value changes ∆ b F (defined in (2)) is the risk part to the hedger. This argument naturallyinspires us to apply the extreme value theorem to study the (right) tail risk of ∆ b F . We refer toEmbrechts et al. (1997) for general theory and Longin (1999) for applications in optimal marginproblem in futures. By applying the extreme value theorem, we obtain the following convergenceresult P (cid:18) max ≤ n ≤ N − ∆ b F t + n ∆ t ≤ x (cid:19) −→ exp " − (cid:18) τ (cid:18) x − βα (cid:19)(cid:19) − /τ , (A.2)where α , β , and τ are respectively the scale parameter, location parameter, and right tail index ofthe value changes of the inverse futures ∆ b F , and all of them can be easily estimated from bitcoinfutures prices. The most important parameter among the three is the right tail index τ , whichmeasures (right) tail fatness of the data. To be precise, the sign of τ determines the type of theextreme value distribution as follows: the limiting case τ = 0 corresponds to the double exponential Note that the limiting result in (A.2) is not achieved as N → ∞ , but under that the number of observations over N periods goes to infinity. For instance, let us take ∆ t = 1h (hour) and N = 8, meaning the hedger wants to hedgethe position for 8 hours. Then the result in (A.2) holds as long as we have enough observations of the maximumprice changes over a 8h ( N ∆ t ) window. τ > τ < τ , the fatter the right tail. The limiting result in (A.2) is model-free , i.e. it holds forany distribution of ∆ b F under mild conditions.Figure A.1: Generalized Extreme Value Distribution -10 -5 0 5 10 15 2000.050.10.150.20.25 E x t r e m e V a l ue D i s t r i bu t i on pd f Note. We set the scale and location parameters α = 2 and β = 0 . τ at τ = 0 . , , − . Using (A.2), we approximate the default probability P ( m, θ ) by P ( m, θ ) ≃ − exp " − (cid:18) τ (cid:18) mθ − βα (cid:19)(cid:19) − /τ . (A.3)We comment that the right hand side in (A.3) provides an accurate approximation to P ( m, θ ),which is empirically verified in the Online Supplementary Appendix I. Step 3.
Approximation of the main problem defined in (5).Using (A.1) and (A.3), the hedger’s optimization problem in (5) can be approximated bymin θ> σ S t ( N ) + (cid:18) S t θF t (cid:19) σ F t ( N ) − S t θF t σ S t ( N ) , ∆ F t ( N ) + γ σ S t ( N ) " − exp " − (cid:18) τ (cid:18) mθ − βα (cid:19)(cid:19) − /τ . (A.4)Since the approximations in Steps 1 and 2 are accurate enough, Problem (A.4) serves as a goodapproximation to Problem (5), and from now on we focus on Problem (A.4). To solve this, weapply a change of variable e θ = S t θ/F t . We then obtain the first-order condition of Problem (A.4)by 2 σ F t ( N ) f ( e θ ) = 0, where f is defined in (7). Hence, θ ∗ given in (6) is a necessary solution toProblem (A.4). Step 4.
Verification of the optimal solution θ ∗ given by (6).29ince S t ≈ F t and the bitcoin price S t is in the scale of thousands (even tens of thousands), thelinear term in function f has the dominating role in its second derivative. As a result, the secondderivative of the objective functional in Problem (A.4) is positive, and thus the necessary conditionof optimality is also sufficient. Therefore, θ ∗ in (6) is indeed an optimal strategy to Problem (A.4). Step 5.
Showing that the equation f ( x ) = 0 has a positive solution if τ > f ( x ) = 0. As the timeseries of ∆ b F has very fat tails, we expect τ > τ >
0, we havelim x → f ( x ) < x →∞ f ( x ) > , where we have used the fact that S and F are positively correlated (which is always the case forspot and futures, but see also Table 4). Together with the continuity property of function f , weapply the mean value theorem to conclude that there always exists a positive root to the equation f ( x ) = 0.Since the function f is highly non-linear, the uniqueness result regarding its root is not availablein general. However, in all scenarios considered during our empirical analysis, we always find aunique solution to f ( x ) = 0. 30 nline Supplementary AppendixI Approximation Accuracy of Tail Distribution In this section, we investigate the approximation accuracy to the tail distribution based on theextreme value theorem, which is derived in (A.2). The ultimate purpose here is to verify that theright hand side of (A.3) provides an accurate approximation to the default probability P ( m, θ ).In the empirical investigation, we calculate the left hand side of (A.2) directly from the historicaldata. To calculate the right hand side, we estimate the parameters α , β and τ from data. Werefer to Hosking et al. (1985) and Longin (1999) for parameter estimation details. Once we havesuccessfully obtained both sides of (A.2), we can easily examine the approximation accuracy. Inthe analysis, we consider three variables (time series): 1-period spot price difference ∆ S , 1-periodfutures price difference ∆ F , and 1-period inverse futures’ value changes ∆ b F , although (A.2) onlyconcerns ∆ b F . Please refer to (1) and (2) for the definition of ∆ S , ∆ F , and ∆ b F . We include theresults of ∆ S and ∆ F to showcase the model-free feature and powerful applications of the extremevalue theorem.We take the BitMEX .BXBT Index to be the spot price S and the BitMEX perpetual XBTUSDfutures price to be the futures price F , sampled at 5min frequency from Jan. 1, 2017 to Aug. 12,2020. We consider three possible time steps ∆ t = 15min, 30min, and 1h, and obtain the full sample at the frequency of ∆ t from raw data. Since the hedger in our analysis shorts bitcoin inversefutures, the right tail of the futures price changes ∆ F (or ∆ b F ) is the risk (losses) to the hedger. Assuch, we use the maximum price change of each day sampled at monitoring frequency ∆ t = 15min,30min, and 1h to form a sample of ‘right tail’, which we use to estimate the parameters ( α , β , and τ ). Table I.1: Estimated Parameters α , β , and τ Using BitMEX Full SampleVariable ∆ t τ α β S F b F Note. This table reports estimated parameters α , β , and τ for ∆ S , ∆ F , and ∆ b F using the full sample of BitMEXdata. We set time step ∆ t = 15min, 30min and 1h. Since the values of ∆ b F are quite small, we multiply ∆ b F by 1000to alleviate numerical errors. This does not affect the tail index τ of ∆ b F , even through it does change α and β . iecall from (A.2) that, the scale parameter α and location parameter β represent the dispersionand the average of the extremes, while the right tail index τ determines the type of the extremevalue distribution. We report the estimation results in Table I.1. We find that τ is positive in allscenarios, and as a result, the limiting distributions are all Fr´echet type (see Figure A.1). Althoughthe scale and location parameters α and β depart from each other, the estimated values of the tailindex τ are rather stable for all variables (∆ S , ∆ F , and ∆ b F ) and for all choices of ∆ t (15min,30min, and 1h). The right tail indices τ of inverse futures ∆ b F are larger than those of ∆ S and∆ F . That means inverse futures has more extreme values and fatter right tail.Next, we use the estimated parameters in Table I.1 to examine the accuracy of the limitingresult in (A.2). We plot the empirical and fitted cumulative distribution function (CDF) for ∆ S ,∆ F , and ∆ b F in Figure I.1. The fitted CDF’s show great accuracy to their empirical counterpartfor all three variables ∆ S , ∆ F , and ∆ b F under all three choices of ∆ t = 15min, 30min and 1h. Thatgives us confidence to use the simplified problem in (A.4) to approximate the original problem in(5), and claim that the optimal strategy θ ∗ to Problem (A.4) is near optimal to Problem (5).Figure I.1: Empirical and Fitted Tail CDF of ∆ S , ∆ F , and ∆ b F Using BitMEX Full Sample
Fitted CDFEmpirical CDF
Fitted CDFEmpirical CDF
Fitted CDFEmpirical CDF
Fitted CDFEmpirical CDF
Fitted CDFEmpirical CDF
Fitted CDFEmpirical CDF
Fitted CDFEmpirical CDF
Fitted CDFEmpirical CDF
Fitted CDFEmpirical CDF
Note. All the empirical CDF’s are obtained using the maximum price change of each day sampled at monitoringfrequency ∆ t = 15min, 30min, and 1h to form a sample of ‘right tail’. All the fitted tail CDF’s are computed by theright hand side of (A.2) under the estimated parameters in Table I.1. We use the full sample data from BitMEX. ii I Tables of Hedge Effectiveness
In this section, we report full details on the hedge effectiveness (HE) of different pairs of bitcoinspot and perpetual futures. Each of the tables below considers a pair of bitcoin spot (from BitMEX.BXBT index, Bitstamp, or Coinbase) and perpetual futures (from BitMEX, Deribit, or OKEx)under four different default aversion levels ( γ = 20 , , , N ∆ t = 5 , , , m = 10% , , , m γ
5d 15d 30d 60d m = 10% γ = 20 93.61% 93.97% 94.45% 95.79% γ = 50 82.05% 81.63% 82.32% 83.61% γ = 100 68.92% 67.18% 67.31% 64.51% γ = 200 55.70% 51.99% 51.40% 42.31% m = 20% γ = 20 98.75% 99.03% 99.14% 99.45% γ = 50 95.77% 95.98% 96.24% 97.24% γ = 100 89.89% 89.68% 90.07% 91.53% γ = 200 79.96% 78.69% 78.87% 78.37% m = 50% γ = 20 99.54% 99.82% 99.87% 99.93% γ = 50 99.34% 99.60% 99.66% 99.78% γ = 100 98.69% 98.92% 99.00% 99.30% γ = 200 96.73% 96.81% 96.93% 97.64% m = 80% γ = 20 99.57% 99.86% 99.90% 99.95% γ = 50 99.53% 99.81% 99.85% 99.92% γ = 100 99.38% 99.64% 99.69% 99.80% γ = 200 98.84% 99.05% 99.12% 99.35% Note. This table reports the hedge effectiveness defined in (9) for the optimal hedging strategy θ ∗ given by (6) underfour different default aversion levels ( γ = 20 , , , N ∆ t = 5 , , , m = 10% , , , iiiable II.2: Hedge Effectiveness for BitMEX Spot and Deribit Futures m γ
5d 15d 30d 60d m = 10% γ = 20 93.99% 94.12% 94.74% 96.28% γ = 50 82.83% 82.19% 83.33% 85.76% γ = 100 70.18% 68.31% 69.11% 69.17% γ = 200 56.46% 53.66% 53.56% 48.54% m = 20% γ = 20 99.03% 99.13% 99.26% 99.56% γ = 50 96.24% 96.24% 96.65% 97.81% γ = 100 90.72% 90.29% 91.11% 93.29% γ = 200 81.36% 79.91% 80.88% 82.75% m = 50% γ = 20 99.77% 99.89% 99.92% 99.96% γ = 50 99.59% 99.69% 99.76% 99.85% γ = 100 99.03% 99.08% 99.22% 99.52% γ = 200 97.29% 97.19% 97.52% 98.35% m = 80% γ = 20 99.79% 99.92% 99.95% 99.97% γ = 50 99.76% 99.88% 99.92% 99.95% γ = 100 99.63% 99.74% 99.79% 99.87% γ = 200 99.18% 99.23% 99.35% 99.58% Note. This table reports the hedge effectiveness defined in (9) for the optimal hedging strategy θ ∗ given by (6) underfour different default aversion levels ( γ = 20 , , , N ∆ t = 5 , , , m = 10% , , , ivable II.3: Hedge Effectiveness for BitMEX Spot and OKEx Futures m γ
5d 15d 30d 60d m = 10% γ = 20 97.68% 97.91% 97.83% 97.34% γ = 50 90.08% 90.76% 90.30% 87.80% γ = 100 76.71% 77.48% 76.12% 69.17% γ = 200 58.80% 58.75% 56.05% 42.73% m = 20% γ = 20 99.60% 99.72% 99.74% 99.61% γ = 50 98.28% 98.60% 98.60% 97.94% γ = 100 94.67% 95.46% 95.35% 93.14% γ = 200 86.00% 87.42% 86.87% 80.42% m = 50% γ = 20 99.86% 99.95% 99.97% 99.95% γ = 50 99.76% 99.87% 99.90% 99.82% γ = 100 99.42% 99.61% 99.64% 99.40% γ = 200 98.27% 98.71% 98.74% 97.86% m = 80% γ = 20 99.88% 99.96% 99.99% 99.96% γ = 50 99.85% 99.94% 99.97% 99.93% γ = 100 99.75% 99.87% 99.90% 99.82% γ = 200 99.41% 99.62% 99.65% 99.39% Note. This table reports the hedge effectiveness defined in (9) for the optimal hedging strategy θ ∗ given by (6) underfour different default aversion levels ( γ = 20 , , , N ∆ t = 5 , , , m = 10% , , , vable II.4: Hedge Effectiveness for Bitstamp Spot and BitMEX Futures m γ
5d 15d 30d 60d m = 10% γ = 20 74.98% 76.33% 78.53% 86.28% γ = 50 64.77% 63.61% 63.29% 73.30% γ = 100 51.38% 52.22% 52.28% 60.29% γ = 200 43.56% 41.35% 39.25% 45.51% m = 20% γ = 20 92.31% 94.11% 94.68% 97.16% γ = 50 84.35% 86.31% 87.65% 92.85% γ = 100 76.00% 77.54% 79.38% 86.91% γ = 200 65.97% 66.72% 68.71% 77.38% m = 50% γ = 20 99.20% 99.66% 99.68% 99.72% γ = 50 98.12% 98.97% 99.03% 99.43% γ = 100 96.12% 97.46% 97.69% 98.67% γ = 200 92.41% 94.26% 94.90% 96.85% m = 80% γ = 20 99.54% 99.81% 99.86% 99.76% γ = 50 99.34% 99.73% 99.76% 99.74% γ = 100 98.87% 99.45% 99.49% 99.62% γ = 200 97.70% 98.65% 98.74% 99.20% Note. This table reports the hedge effectiveness defined in (9) for the optimal hedging strategy θ ∗ given by (6) underfour different default aversion levels ( γ = 20 , , , N ∆ t = 5 , , , m = 10% , , , viable II.5: Hedge Effectiveness for Bitstamp Spot and Deribit Futures m γ
5d 15d 30d 60d m = 10% γ = 20 93.96% 94.14% 94.72% 96.28% γ = 50 82.70% 82.19% 83.29% 85.75% γ = 100 70.01% 68.28% 69.06% 69.16% γ = 200 56.29% 53.62% 53.51% 48.47% m = 20% γ = 20 99.08% 99.16% 99.27% 99.57% γ = 50 96.23% 96.26% 96.64% 97.81% γ = 100 90.65% 90.30% 91.08% 93.29% γ = 200 81.24% 79.89% 80.84% 82.74% m = 50% γ = 20 99.85% 99.92% 99.94% 99.96% γ = 50 99.66% 99.72% 99.77% 99.86% γ = 100 99.08% 99.11% 99.23% 99.52% γ = 200 97.30% 97.21% 97.52% 98.36% m = 80% γ = 20 99.88% 99.95% 99.97% 99.98% γ = 50 99.84% 99.91% 99.93% 99.96% γ = 100 99.71% 99.76% 99.81% 99.88% γ = 200 99.23% 99.25% 99.36% 99.58% Note. This table reports the hedge effectiveness defined in (9) for the optimal hedging strategy θ ∗ given by (6) underfour different default aversion levels ( γ = 20 , , , N ∆ t = 5 , , , m = 10% , , , viiable II.6: Hedge Effectiveness for Bitstamp Spot and OKEx Futures m γ
5d 15d 30d 60d m = 10% γ = 20 97.77% 97.95% 97.84% 97.33% γ = 50 90.15% 90.82% 90.32% 87.78% γ = 100 76.74% 77.54% 76.14% 69.14% γ = 200 58.79% 58.81% 56.08% 42.67% m = 20% γ = 20 99.71% 99.75% 99.75% 99.62% γ = 50 98.40% 98.65% 98.61% 97.94% γ = 100 94.80% 95.52% 95.37% 93.12% γ = 200 86.11% 87.49% 86.90% 80.39% m = 50% γ = 20 99.96% 99.98% 99.98% 99.96% γ = 50 99.87% 99.91% 99.91% 99.83% γ = 100 99.55% 99.66% 99.66% 99.40% γ = 200 98.42% 98.76% 98.76% 97.86% m = 80% γ = 20 99.97% 99.99% 99.99% 99.98% γ = 50 99.95% 99.97% 99.97% 99.95% γ = 100 99.87% 99.91% 99.91% 99.83% γ = 200 99.55% 99.66% 99.67% 99.40% Note. This table reports the hedge effectiveness defined in (9) for the optimal hedging strategy θ ∗ given by (6) underfour different default aversion levels ( γ = 20 , , , N ∆ t = 5 , , , m = 10% , , , viiiable II.7: Hedge Effectiveness for Coinbase Spot and BitMEX Futures m γ
5d 15d 30d 60d m = 10% γ = 20 75.08% 75.06% 76.82% 86.16% γ = 50 62.33% 62.19% 63.68% 72.68% γ = 100 55.65% 50.14% 51.78% 60.43% γ = 200 41.25% 39.77% 38.79% 45.62% m = 20% γ = 20 92.28% 93.43% 94.35% 96.89% γ = 50 84.02% 85.20% 87.07% 92.51% γ = 100 75.50% 76.22% 78.67% 86.57% γ = 200 65.99% 65.49% 68.13% 77.08% m = 50% γ = 20 99.39% 99.62% 99.71% 99.72% γ = 50 98.35% 98.79% 99.00% 99.34% γ = 100 96.29% 97.11% 97.57% 98.50% γ = 200 92.44% 93.67% 94.66% 96.59% m = 80% γ = 20 99.63% 99.85% 99.91% 99.81% γ = 50 99.51% 99.72% 99.80% 99.76% γ = 100 99.09% 99.37% 99.50% 99.59% γ = 200 97.93% 98.45% 98.71% 99.11% Note. This table reports the hedge effectiveness defined in (9) for the optimal hedging strategy θ ∗ given by (6) underfour different default aversion levels ( γ = 20 , , , N ∆ t = 5 , , , m = 10% , , , ixable II.8: Hedge Effectiveness for Coinbase Spot and Deribit Futures m γ
5d 15d 30d 60d m = 10% γ = 20 93.92% 94.12% 94.71% 96.28% γ = 50 82.63% 82.15% 83.28% 85.76% γ = 100 69.93% 68.24% 69.05% 69.17% γ = 200 56.22% 53.64% 53.51% 48.54% m = 20% γ = 20 99.06% 99.15% 99.26% 99.57% γ = 50 96.20% 96.24% 96.63% 97.82% γ = 100 90.59% 90.27% 91.07% 93.30% γ = 200 81.16% 79.85% 80.83% 82.75% m = 50% γ = 20 99.84% 99.91% 99.94% 99.96% γ = 50 99.65% 99.71% 99.76% 99.86% γ = 100 99.06% 99.10% 99.22% 99.52% γ = 200 97.26% 97.20% 97.51% 98.36% m = 80% γ = 20 99.87% 99.95% 99.96% 99.98% γ = 50 99.83% 99.90% 99.93% 99.96% γ = 100 99.69% 99.76% 99.80% 99.88% γ = 200 99.21% 99.24% 99.35% 99.59% Note. This table reports the hedge effectiveness defined in (9) for the optimal hedging strategy θ ∗ given by (6) underfour different default aversion levels ( γ = 20 , , , N ∆ t = 5 , , , m = 10% , , , xable II.9: Hedge Effectiveness for Coinbase Spot and OKEx Futures m γ
5d 15d 30d 60d m = 10% γ = 20 97.79% 97.96% 97.86% 97.34% γ = 50 90.18% 90.84% 90.34% 87.79% γ = 100 76.77% 77.58% 76.17% 69.16% γ = 200 58.83% 58.85% 56.10% 42.70% m = 20% γ = 20 99.71% 99.76% 99.75% 99.62% γ = 50 98.41% 98.65% 98.62% 97.94% γ = 100 94.81% 95.53% 95.39% 93.13% γ = 200 86.13% 87.51% 86.93% 80.41% m = 50% γ = 20 99.96% 99.98% 99.98% 99.96% γ = 50 99.87% 99.90% 99.91% 99.83% γ = 100 99.55% 99.66% 99.66% 99.40% γ = 200 98.42% 98.76% 98.77% 97.86% m = 80% γ = 20 99.97% 99.99% 99.99% 99.98% γ = 50 99.95% 99.97% 99.98% 99.95% γ = 100 99.86% 99.91% 99.91% 99.83% γ = 200 99.54% 99.66% 99.67% 99.40% Note. This table reports the hedge effectiveness defined in (9) for the optimal hedging strategy θ ∗ given by (6) underfour different default aversion levels ( γ = 20 , , , N ∆ t = 5 , , , m = 10% , , ,80%). The hedger shorts the OKEx perpetual futuresto hedge the spot risk on Coinbase.