Liquidation, Leverage and Optimal Margin in Bitcoin Futures Markets
aa r X i v : . [ q -f i n . T R ] F e b Liquidation, Leverage and Optimal Margin in Bitcoin FuturesMarkets
Zhiyong Cheng a , Jun Deng a ∗ , Tianyi Wang a , Mei Yu a Abstract
Using the generalized extreme value theory to characterize tail distributions, we address liqui-dation, leverage, and optimal margins for bitcoin long and short futures positions. The empiricalanalysis of perpetual bitcoin futures on BitMEX shows that (1) daily forced liquidations to out-standing futures are substantial at 3.51%, and 1.89% for long and short; (2) investors got forcedliquidation do trade aggressively with average leverage of 60X; and (3) exchanges should elevatecurrent 1% margin requirement to 33% (3X leverage) for long and 20% (5X leverage) for short toreduce the daily margin call probability to 1%. Our results further suggest normality assumptionon return significantly underestimates optimal margins. Policy implications are also discussed.
Key words:
Bitcoin futures; Liquidation; Margin; Leverage; Generalized extreme value theory
JEL Classification:
G11, G13, G32 ∗ Corresponding Author. School of Banking and Finance, University of International Business and Economics, Beijing,China, 100029. Email: [email protected] Introduction
The largest cryptocurrency, bitcoin, accounts for more than 70% of the total market capitalizationreported by CoinMarketCap on 14 December 2020. Compared with other traditional assets, bitcoinprice is more volatile: 30-day volatility reaches to 167.24% on 31 March, 2020 . This extraordinarilyhigh price volatility imposes a tremendous risk to various market participants, see Chaim and Laurini(2018), Alexander et al. (2020b), Deng et al. (2020) and Scaillet et al. (2020).Futures contracts are commonly used to hedge spot price risk. We refer this rich field to Figlewski(1984), Daskalaki and Skiadopoulos (2016) and Alexander et al. (2019) for traditional equity, com-modity and currency markets. For bitcoin futures markets, the hedge effectiveness and improvementof portfolio performance are well studied in Alexander et al. (2020a), Sebasti˜ao and Godinho (2020),Deng et al. (2019) and others. By November 2020, bitcoin futures monthly trading volume reachedto $ . Figure 1 plots daily forced liquidation on BitMEX fromJan. 2020 to Feb. 2021, where the average daily liquidation is approximate 20 million USD for longand 10 million USD for short.This article contributes to the literature in several ways. First, by using daily forced liquidationdata of BitMEX, we highlight the speculative activity, aggressiveness and risk preference of market See Forbes report. The highest 30-day volatility of the S&P 500 index so far is 89.53% on 24 October 2008. See Tronweekly and Coingape reports. B i t M EX XB T L i qu i da t i on ( m illi on ) B i t c o i n P r i c e Short Long Spot
Figure 1: Daily Liquidation Volumes of BitMEX Perpetual Futures
Note: The green (blue) bar plots the forced liquidation of long (short) bitcoin perpetual futures on BitMEX from Jan.2020 to Feb. 2021. On “Black Thursday” March 12, 2020, the long position’s daily forced liquidation reached 843 millionUSD. The maximal short liquidation 132 million occurred on June 1, 2020 when the price rise over 10%. The averagedaily liquidations are around 20 million USD and 10 million USD for long and short. The right y-axis corresponds tothe daily bitcoin spot price (red line). The data is manually collected from coinalyze. participants by estimating the speculation index, the average percentage of liquidation and averageleverages used in long and short positions. Second, we show that generalized extreme value theory(GEV) can effectively capture the bitcoin futures price’s fat-tail feature . Finally, we give optimalmargins for long and short positions using GEV and market data that provide guidance for exchangesto design bitcoin futures specifications, especially margin requirements.With bitcoin perpetual futures data from BitMEX, our empirical analysis uncovered several inter-esting aspects of bitcoin futures markets. First, The average speculation index sits at 3.75 which ismuch larger than S&P 500 (0.15), Nikkei (0.21) and DAX (0.45). This shows the extreme speculationactivity in BitMEX perpetual market. Second, the daily average percentage of forced liquidation forlong and short positions are substantial at levels of 3.51% and 1.89%, respectively. Despite those statis-tics, the average leverage of those got forced liquidation is similar (around 60X) for both positions.That suggests some participants do trade aggressively and utilize high leverages regardless of highprice risk. Third, assuming 1% daily margin call probability, the optimal margins are approximate33% (3X leverage) for long position and 20% (5X leverage) for short. At last, the GEV distributionis critical for accurately estimating those margins as the normal distribution assumption on returnsignificantly underestimates margin levels by at least 50%.We organize the rest of the paper as follows. Section 2 introduces bitcoin perpetual futures and GEV distribution is also used for margin settings in Longin (1999), Dewachter and Gielens (1999), Cotter (2001),and more recent work of Gkillas and Katsiampa (2018).
Two types of futures contracts are traded across exchanges: standard futures on CME and Bakkt andperpetual futures (perpetuals) without expiry date on BitMEX, Binance, and Huobi, to name a few.Perpetual futures depart from standard futures in two aspects. Firstly, bitcoin perpetuals are quotedin US dollars (USD) and settled in Bitcoin (XBT); while bitcoin standard futures are both quotedand settled in USD . For instance, each CME bitcoin standard futures contract has a notional valueof 5 XBT, while one BitMEX bitcoin perpetual futures has a notional value of 1 USD. Secondly, thebitcoin margin deposit is required for trading perpetuals; while the USD margin is used for standardfutures. Since perpetuals account for around 90% trading volume, we only focus on perpetuals here .Specifically, suppose an investor who enters into one long position of bitcoin perpetual futures witha fixed notional amount of Π USD at time t , and closes her position at later time t . The perpetualsprices, denominated in USD, changed from F t at time t to F t at time t . This is equivalent to thechange from Π F t XBT to Π F t XBT for each perpetuals. Then, the realized pay-off in XBT is given bythe difference between the “enter value” and “exit value”:
Long Pay-off = Π F t − Π F t . (2.1)As a result, the pay-off is inversely related to quotes and perpetuals are also called “inverse” futures.The gain increases when futures price increases as standard futures. However, the magnitude issignificantly different from the latter . Similarly, for the short position, the pay-off in XBT is Short Pay-off = Π F t − Π F t . (2.2)Implied by equations (2.1) and (2.2), long positions incur greater loss than short positions for thesame magnitude of price change ∆ F . As a result, margins for long and short would be asymmetriceven the perpetuals price ( F t ) fluctuates symmetrically. Empirical results in Section 3 confirm the It resembles quanto-options that are settled in domestic currency but quoted in foreign currency. For more discussion on perpetuals in hedge, we refer to Alexander et al. (2020b). For standard futures, the long gain equals F t − F t . Let ∆ F t , ∆ F t , · · · , ∆ F t n be quote price changes observed on discrete time t i = 1 , , · · · , n . Themaximum and minimum changes over n periods are Max ( n ) = max(∆ F t , ∆ F t , · · · , ∆ F t n ) and Min ( n ) = min(∆ F t , ∆ F t , · · · , ∆ F t n ), respectively. For a wide range of possible distributions ofprice changes (∆ F t i ), the limit distributions of Max ( n ) and Min ( n ) follow the generalized extremevalue distribution (GEV), see Jenkinson (1955). The extreme distribution is characterized by locationparameter µ , scale parameter σ , and tail parameter τ . Due to possible asymmetry in left and righttails, GEV parameters could be different for Max ( n ) and Min ( n ) in equations (2.3) and (2.4). G max ( x ) = exp " − (cid:18) − x − µ max σ max · τ max (cid:19) /τ max , and (2.3) G min ( x ) = 1 − exp " − (cid:18) x − µ min σ min · τ min (cid:19) /τ min . (2.4)The introduction of GEV to study optimal margin can be traced back to Longin (1999), which fo-cuses on silver futures traded on COMEX. Recently, Gkillas and Katsiampa (2018) also used GEV toestimate Value-at-Risk and expected shortfall of several cryptocurrencies.Depending on the tail parameter τ , extreme distributions fall into three categories: Gumbel ( τ =0), Weibull ( τ < e chet ( τ >
0) distributions, which are illustrated in Figure 2. < Insert Figure 2 here. > Over n periods, for a long position, the margin call is triggered once the minimum price change Min ( n ) is lower than margin deposit MD long . Once triggered, the long position would be automat-ically liquidated if the investor could not pull up enough bitcoin deposit. The corresponding margincall probability, p long , is given by p long = P rob ( Min ( n ) < − MD long ) ≃ G min ( − MD long ) . In parallel, for the short position, we have p short = P rob ( Max ( n ) > MD short ) ≃ − G max ( MD short ) . Using limiting distributions in (2.3) and (2.4), given acceptable margin call probabilities p long and5 short , minimal margins MD long and MD short can be analytically expressed as MD long = − µ min + σ min τ min (cid:20) − (cid:16) − ln (cid:16) − p long (cid:17)(cid:17) τ min (cid:21) , and (2.5) MD short = µ max + σ max τ max (cid:20) − (cid:16) − ln (cid:16) − p short (cid:17)(cid:17) τ max (cid:21) . (2.6)In Section 3 below, we conduct empirical analysis to estimate left and right GEV distribution param-eters ( τ min , µ min , σ min ) and ( τ max , µ max , σ max ) and related long and short margin levels MD long and MD short . The BitMEX exchange, founded in 2014, is the largest online platform for trading bitcoin futures,which runs continuously 24/7 a week and offers various cryptocurrency futures contracts, such asBitcoin and Ethereum. On BitMEX, both perpetuals and fixed expiration futures are listed. Theperpetual futures contracts dominate the market by contributing 90% of the total trading volume.Perpetuals traded on BitMEX have notional value Π = 1 USD and the maximum leverage allowed is100X (margin requirement is 1%).To investigate the average leverage used by market participants, we manually collect the dailyvolume, forced liquidation, open interest and daily OHLC prices (open, high, low, close) data fromCoinalyze. The data spans from January 29, 2020 to February 3, 2021 and contains 372 entries . Be-sides, we also download 5min BitMEX perpetual futures price data using the API provided by BitMEXto investigate optimal margins. The BitMEX perpetual data-set consists of 431, 346 observations fromJanuary 1, 2017 to February 6, 2021.Following Longin (1999), price changes for perpetual futures are defined in percentage changes∆ b F t = Ft − − Ft Ft − = 1 − F t − F t . This definition has the advantage of being a stationary time series andnum´eraire independent. Margin level is in terms of percentage. Its left (right) tail is associated with theloss of a long (short) position. We also define an alternative price change as ∆ F t = F t − F t − F t − = F t F t − − < Insert Table 1 here. > As BitMEX runs 24/7, the open (close) price for each day is defined as the first (last) trade at 0:00 UTC and 24:00UTC. Since liquidation data is seldomly released, we manually collect the earliest of BitMEX from Coinalyze. F and ∆ b F , sampled at different frequen-cies. On daily frequency, both price changes are negatively skewed and leptokurtic, and the magnitudefor perpetuals is higher, evidenced by a more negative skewness (-3.67 vs. -0.24). The price changedistribution for perpetuals favors the short position than the long position as ∆ b F has a lower minimum(-82.67% vs. -45.26%), a lower maximum (22.00% vs. 28.20%) and, as a result, a lower mean (0.14%vs. 0.35%) as well as a thicker tail . These facts are also confirmed by the Q-Q plot in Figure 3. < Insert Figure 3 here. > As mentioned above, the left/right tail is directly linked to the loss of long/short position. Two setsof parameters ( µ, σ, τ ) are estimated separately.As there is only one extreme value for the full sample, estimating GEV distribution parametersoften relies on the block extreme technique. Here, we follow the estimation procedure of Longin (1999).The technique divides the whole sample into non-overlapping sub-samples containing n observations,and each sub-sample provides us a maximum change and a minimum change. For the i -th block { ∆ X ( i − n +1 , ∆ X ( i − n +2 , · · · , ∆ X in } ( X = b F and F , respectively), we denote the block maximumand minimum as min i and max i . If we have a total of N observations, the block extreme sets { min , ..., min [ N/n ] } and { max , ..., max [ N/n ] } are used to estimate GEV parameters . For differentsampling frequencies, the block size n varies. In particular, the block spans 8/24/48 hours for 5/30/60min sampling and 5/10 days for 8/24 hours sampling frequencies , and we point out the samplingfrequency could be treated as the screening-frequency that exchanges/investors monitor the futurespositions. We report the corresponding results in Table 2.The tail parameter τ is consistently positive, indicating extreme distributions are all Fr´ e chet types.The magnitude of τ increases as the sampling frequency increases, which means the tail is thickerfor high-frequency returns. On the other hand, the location and scale parameters are higher for low-frequency returns as extreme price changes increase for longer horizons. Slight differences are observedfor left and right tail parameters of standard futures ∆ F . However, for perpetual futures ∆ b F t , thetail parameter τ ’s differ significantly for left and right tails, especially for low-frequency cases such as8h (8 hours) and 1d (1 day). For daily return, the left tail parameter τ is more than double of theright, and this indicates a fatter left tail (higher risk for the long position). The kurtosis 65.36 for ∆ b F and 14.10 for ∆ F . [ N/n ] is the integer part of
N/n Our empirical results are robust to the block size. The 8 hours are selected as BitMEX charges funding rate forinvestors holding positions every 8 hours. Insert Table 2 here. > Finally, we plot the empirical and fitted cumulative distribution functions (CDFs) for both ∆ F t and ∆ b F t in Figure 4. Results suggest that the GEV method is suitable for capturing tail behaviorsand investigating optimal margins for bitcoin futures. < Insert Figure 4 here. > In this section, using estimated parameters in Table 2 of Section 3.2, we calculate optimal marginsvia (2.5) and (2.6) and present empirical results in Table 3. Four margin call probabilities and fiveholding periods are discussed. Both perpetual futures and standard futures are studied to illustrate theimpact of the inverse feature on margin requirements. The results for standard futures also provideguidelines for bitcoin futures traded on CME and Bakkt. Moreover, to match the current marginrequirements set by exchanges, we also calculate the one common optimal margin level for bothpositions by simultaneously encompassing the left and right tails. For each case, we also provide theoptimal margin based on the normal distribution assumption (in parentheses) to show the importanceof GEV distribution. < Insert Table 3 here. > For each holding period, as expected, the optimal margin increases with the decreasing margin calltolerance. For standard futures, the margin increases from 7.86% (7.66%) to 32.95% (35.93%) for short(long) position when margin call probability varies from p = 0 . p = 0 . asymmetric effect is the direct result of thedifference in tail parameters listed in Table 2.To match the exchanges’ practice of setting equal margins for both positions, we pool maximumsand minimums together as {− min , ..., − min [ N/n ] , max , ..., max [ N/n ] } and estimate one set of param-eters and the corresponding margin levels in Panel C of Table 3. As expected, the common margin8evel lies somewhere between the long and short cases. For example, for 8 hours holding period withcall probability p = 0 . Previously, we estimated optimal margins for both long and short positions under different monitoringfrequencies by using perpetual price data. However, to get a sense of investors’ trading activity and riskpreference in the bitcoin futures market, calculating speculation percentage and precise leverages usedby investors requires traders’ account-level data, which is unavailable from the exchange. Fortunately,Coinalyze reports the forced liquidation of both long and short. Combined with the daily OHLCprices (open, high, low, close) of bitcoin perpetual futures and trading volume and open interest, wecan roughly estimate speculation activity and the leverage used by investors who experienced forcedliquidation within a particular day.First, following Garcia et al. (1986), Lucia and Pardo (2010), and Kim (2015), we define the spec-ulation index as the ratio of daily trading volume to daily open interest, i.e. SI = trading volumeopen interest . Intuitively, total trading volume is a proxy for speculation activity and hedging demand is representedby open interest. A lower ratio implies lower speculative activity relative to hedging demand, or viseversa.Second, we estimate leverage using liquidation data. At time t , suppose the futures price is F t (USD) and an investor enters one long (short) position with leverage L long ( L short ). The marginrequired in XBT is L long F t for long position and L short F t for short position. According to long andshort pay-offs in (2.1) and (2.2), at a later time s , margin call is triggered once the trading loss wipesout margin deposit, i.e., 1 F s − F t = 1 L long F t , and 1 F t − F s = 1 L short F t . (4.1)9s Coinaylze does not report the exact time when each forced liquidation happens, we madefurther assumption that investors open their positions at the opening of day d and the possible forcedliquidation happens when the perpetuals price hits the daily low (high) for the long (short) positionon day d . The corresponding returns are: r maxd = F highd − F opend F opend , and r mind = F lowd − F opend F opend . (4.2)Here, F opend , F highd and F lowd stand for the open, high, and low prices. Once got forced liquidation, wecould infer traders’ minimal long and short leverages via (4.1) as L long = − r mind r mind , and L short = 1 + r maxd r maxd . (4.3)The long (short) liquidation percentage p long ( p short ) is estimated by the proportion of long (short)liquidation to open interest. < Insert Table 4 here. >< Insert Figure 5 here. > The summary statistics for maximal and minimal futures returns, speculation index, liquidationpercentage, and leverages are presented in Table 4. The average of speculation index sits at 3.75which is much larger than S&P 500 (0.15), Nikkei (0.21) and DAX (0.45) in Lucia and Pardo (2010).Figure 5 plots its time series in the sample period and shows more speculative trades in both bullishand bearish markets. This reflects extreme speculative activity in bitcoin futures market. The dailyaverage percentages of long and short liquidation to open interest are 3.51% and 1.89%, and the averageleverage used by those got forced liquidation is at least 58.13X for long and 59.94X for short. Thisshows liquidation is substantial due to high spot price risk and some investors’ aggressive leverage.
It is crucial and of self-interest for exchanges keeping futures margins at a proper level: high enough topreserve market integrity yet low enough to attract broad participants and maintain market liquidity.This paper employed the generalized extreme value theory to address optimal margins for bitcoinfutures. Using the BitMEX perpetual futures, we found: first, average leverages used by forcedliquidation traders are 58.13X for long and 59.94X for short for the short position. Second, highleverage leads to substantial forced liquidation, where the daily liquidation percentage is 3.51% for10ong and 1.89% for short. Third, investors should be more rational and cautious to use leverage inthe concurrent high volatile market. We suggest 5X leverage for short and 3X leverage for long ifone accepts 1% daily margin call probability. Furthermore, the normality assumption of return willsignificantly underestimate margin levels by at least half.The policy implications of our results are: (1) exchanges should give more clear documentation ofthe asymmetric risk character of perpetuals to market participants; (2) exchanges should also considerperpetuals’ inverse-payoff structure and set higher margin for long positions than shorts; (3) it isnecessary and important for exchanges to limit leverage allowed and impose more margin deposit tolower forced liquidation, maintain market integrity, and improve the functioning of futures markets.11 eferences
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Finance Research Letters , 33:101230. 13 ppendixFigures -6 -4 -2 0 2 4 600.10.20.30.40.5
Figure 2: Gumbel, Weibull, and Fr´ e chet Distributions Note: We consider three tail parameters τ = 0 . , , − .
5. The Fr´ e chet distribution (green line) is fat-tailed as its tail isslowly decreasing; the Gumbel distribution (red line) is thin-tailed as its tail is rapidly decreasing; and the Weibull (blueline) has no tail after a certain point. Figure 3: Q-Q Plot of Price Changes ∆ F and ∆ b F on BitMEX15
10 20 3000.20.40.60.81 0 10 20 3000.20.40.60.81 0 5 10 1500.20.40.60.81 0 10 2000.20.40.60.81 -10 0 10 20 3000.20.40.60.810 10 2000.20.40.60.81 0 10 2000.20.40.60.81 0 5 1000.20.40.60.81 -5 0 5 10 1500.20.40.60.81 -10 0 10 2000.20.40.60.81
Figure 4: Empirical and Fitted Right Tail CDF of ∆ F t and ∆ b F t Note: This figure shows the empirical and fitted CDF of ∆ F t and ∆ b F t under 5min, 30min, 1h, 8h, and 1d observationfrequencies. All the fitted right tail CDF’s are computed by (2.3) with parameters in Panel A of Table 2. L i qu i da t i on R a t i o t o O pen I n t e r e s t B i t c o i n P r i c e Short Long Spot 01-Jul-2020 01-Jan-202105101520 S pe c u l a t i on I nde x B i t c o i n P r i c e Speculation Index Spot
Figure 5: Liquidation Ratio and Speculation Index
Note. The speculation index is defined as the ratio of trading volume to open interest, SI = trading volumeopen interest . A lowerratio implies lower speculative activity relative to hedging demand. ables Variable Standard Futures ∆ F Perpetuals ∆ b Ft − − − − − − − − − − − − − − − − − − − − − − − − − − − − Table 1: Summary Statistics of Price Changes ∆ F and ∆ b F on BitMEX Note. It reports summary statistics of price changes ∆ F and ∆ b F using BitMEX perpetual futures from January 1, 2017,to February 6, 2021, sampled at 5min, 30min, 1h, 8h, and 1d frequencies. P25 and P75 refer to 25% and 75% quantiles,and S.D. is the standard deviation. ariable Standard Futures ∆ F Perpetuals ∆ b Ft τ σ µ τ σ µ τ σ µ Table 2: GEV Parameter Estimations for ∆ F and ∆ b F Note: This table reports left and right tail parameter estimations for ∆ F and ∆ b F under 5min, 30min, 1h, 8h, and 1dobservation frequencies. For comparison purposes, we take the absolute value of the left tail to estimate correspondingextreme parameters and report a common tail that simultaneously encompasses absolute left tail and right tail. Sincevariables ∆ F t and ∆ b F t are quite small, we multiply them by 100. The corresponding standard errors are included inparentheses. Note: This table reports optimal margins for Bitcoin standard futures and perpetual futures for short (Panel A) andlong (Panel B) positions. The optimal margins for long and short positions are calculated via (2.5) and (2.6). Panel Cgives the margin level for the common position. The corresponding optimal margin levels obtained under the assumptionof normality are also presented in parentheses, which are calculated via p long = √ πσ R − ML long −∞ e − ( x − µ ) / σ dx and p short = √ πσ R + ∞ ML short e − ( x − µ ) / σ dx . Here, µ and σ are the mean and standard deviation of price changes ∆ F t and∆ b F t . ML long and ML short denote optimal margins of long and short positions. r mind r maxd long liq.(M) short liq.(M) SI p long (%) p short (%) L long L short min -0.46 0.00 0.00 0.00 0.66 0.00 0.00 1.16 3.84median -0.02 0.02 6.93 4.60 3.12 1.22 0.86 55.97 56.34mean -0.03 0.03 20.14 10.17 3.75 3.51 1.89 58.13 59.94max 0.00 0.35 843.39 132.70 18.42 169.39 24.32 100.00 100.00Nobs 372 372 372 372 372 372 372 372 372Table 4: Summary Statistics for Speculation, Liquidation and Leverage Note. The maximal and minimal return r mind and r maxd are estimated via (4.2). The speculation index is defined as theratio of trading volume to open interest, SI = trading volumeopen interest . The long and short liquidation percentage p long and p short are estimated by the proportion of long/short liquidation to open interest. The long and short leverage L long and L short are estimated via (4.3). All the data is daily frequency from January 29, 2020 to February 3, 2021. M denotesthe Million USD.are estimated via (4.3). All the data is daily frequency from January 29, 2020 to February 3, 2021. M denotesthe Million USD.