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Risk Structure and Optimal Hedging of Bitcoin Inverse Futures

Jun Deng∗Huifeng Pan†Shuyu Zhang‡Bin Zou§

September 8, 2019

Abstract

In the Bitcoin futures markets, the dominating contracts are inverse contracts. Unlike stan-

dard futures, Bitcoin inverse futures have a non-linear payoﬀ structure, are settled in Bitcoin

instead of the ﬁat currency, and require Bitcoins to be deposited into the margin account during

trading. We characterize the unique high-order risk factors, asymmetry eﬀect and (de)leverage

eﬀect of Bitcoin inverse futures, and obtain optimal hedging strategies in closed forms for both

short and long hedges under the minimum-variance framework. We use the market data of Bit-

coin spot and futures to conduct empirical studies. Our ﬁndings show that the optimal hedging

strategies of Bitcoin inverse futures achieve superior hedging performance across exchanges.

Key words: Bitcoin; Inverse futures; Minimum-variance hedging; High-order risk

JEL Classiﬁcation: G110; G320

1 Introduction

Bitcoin (BTC) is the ﬁrst digital currency to operate through a peer-to-peer network without a

central authority (see its blueprint Nakamoto (2008)). Bitcoin is the most prominent and domi-

nating one among thousands of cryptocurrencies, with a market capitalization of 228 billion US

∗School of Banking and Finance, University of International Business and Economics, Beijing, China. Email:

jundeng@uibe.edu.cn.

†School of Banking and Finance, University of International Business and Economics, Beijing, China. Email:

panhf2@126.com.

‡Corresponding Author. 182 Nanhu Avenue, East Lake High-tech Development Zone, Wenlan School of Business,

Zhongnan University of Economics and Law, Wuhan, 430073, China. Email: onething1984@126.com.

§Department of Mathematics, University of Connecticut, Storrs, CT, USA. Email: bin.zou@uconn.edu.

1

dollars (USD, symbol $) by July 10, 2019, followed by Ethereum with only $33 billion.1The price

of Bitcoin climbed from nearly zero at its inception in 2010 to more than $20,000 in January 2018,

but plunged over 50% in the next four months. The year of 2017 witnessed the most dramatic price

movements of Bitcoin, with astonishing annual return of 2,300% and volatility of 98%.

The speculation and high volatility nature of Bitcoin stimulate the launch of Bitcoin futures

by the Chicago Board Options Exchange (CBOE) and the Chicago Mercantile Exchange (CME)

in December 2017. The Bitcoin futures contracts traded on both CBOE and CME are standard

futures, which are not only denominated in USD but also settled in USD, with a linear payoﬀ

structure (see Table A.1 for details of Bitcoin futures contracts on CBOE and CME). These features

of standard futures contradict the essential purposes of Bitcoin as a decentralized currency and a

direct peer-to-peer payment system. The stringent regulations and requirement further diminish

the investors’ interest to trade Bitcoin standard futures on CBOE and CME. Given the above

mentioned factors, it is not surprising that the trading volumes of Bitcoin futures on CBOE and

CME are disappointing. In fact, CBOE had ceased oﬀering Bitcoin futures after June 19, 2019,

leaving CME the only venue for trading standard Bitcoin futures.2

On the other hand, non-standard Bitcoin futures oﬀered by online exchanges (which are less

regulated than CBOE and CME) gain the dominating market shares and are welcomed by traders

around the world (see Table A.2 for trading volumes). Unlike the standard futures on CBOE

and CME, these non-standard Bitcoin futures are denominated and settled in BTC, i.e, the ﬁat

currency (USD) is now seen as the “commodity” or “foreign currency” and the BTC is seen as

the “domestic currency” for quotation and settlement (see Table A.3 for contract details of Bitcoin

futures on OKEx). For instance, let the notional value of such a Bitcoin futures contract be Kand

the reference futures price be F, both quoted in USD, then one such contract is worth of K/F in

BTC, bearing an inverse relation to the futures price F(see Section 2.1 for the unique features

regarding the payoﬀ of inverse futures). As a result, these non-standard Bitcoin futures are called

Bitcoin inverse futures, and we shall adopt this term in this paper as well. To our awareness, the

Russian-based Bitcoin exchange ICBIT was the ﬁrst to list Bitcoin inverse futures back in late

1See market statistics on https://coinmarketcap.com/all/views/all/.

2See news report https://www.wsj.com/articles/cboe-abandons-bitcoin-futures-11552914001.

2

2011 to meet speculation and hedging demand.3Soon after ICBIT, more and more less regulated

cryptocurrency exchanges (e.g., BitMEX, OKEx and bitFlyer) begin to oﬀer Bitcoin inverse futures

contracts, with leverage ranging from 1X to 100X (i.e., margin rate from 100% to 1%), to the public

for trading. Indeed, these inverse contracts attract a broad variety of investors, from arbitrageurs

to speculators, from individual traders to institutional traders.

Bitcoin, as an innovative application of Blockchain technology and the most successful cryp-

tocurrency, is among one of the hottest research topics in academia nowadays. There is already a

large body of literature on Bitcoin (still growing at a fast scale), and we present a brief review on

Bitcoin, with focus on three perspectives: price formulation, diversiﬁcation eﬀect, and market eﬃ-

ciency. There are certainly many other interesting research directions on Bitcoin, e.g., blockchain

and cryptocurrency structure (see Swan (2015)), regulation (see Dwyer (2015)), cybersecurity and

privacy (see Conti et al. (2018)). We refer readers to the survey articles of Tschorsch and Scheuer-

mann (2016) and Corbet et al. (2019) and the book of Narayanan et al. (2016) for comprehensive

studies on Bitcoin and cryptocurrency.

Price formulation. Urquhart (2018) claims that realized volatility and volume are both signiﬁ-

cant drivers of Bitcoin returns using Google trends data. Liu and Tsyvinski (2018) conclude that

Bitcoin (also Ripple and Ethereum) is almost immune to most common stock/currency/commodity

markets and macroeconomic factors, and its return is largely predictable by factors of the cryp-

tocurrency market alone. Gandal et al. (2018) study the price manipulation in the Bitcoin market.

Zhang et al. (2019a) and Zhang et al. (2019b) ﬁnd that policy related news could signiﬁcantly

impact the Bitcoin returns. Makarov and Schoar (2019) show that the Bitcoin markets exhibit

arbitrage opportunities and a common component explains 80% of the Bitcoin returns.

Diversiﬁcation eﬀect. Two contradictory conclusions coexist in the current literature regarding

the diversiﬁcation eﬀect of Bitcoin (and/or Bitcoin futures). Yermack (2015) shows that Bitcoin

has virtually zero correlation with widely used currencies and gold, making bitcoin useless for risk

management. Bouri et al. (2017) conclude that Bitcoin is suitable for diversiﬁcation, but they also

point out that Bitcoin is not a stable hedging tool. Similarly, the results of Corbet et al. (2018b)

3ICBIT was later acquired by Swedish-based Bitcoin exchange Safello; see https://en.bitcoin.it/wiki/ICBIT.

3

and Feng et al. (2018) also show that cryptocurrencies may oﬀer diversiﬁcation beneﬁts to various

ﬁnancial assets (e.g., bond, gold, stock indices and commodities).

Market eﬃciency. An earlier paper along this direction is Urquhart (2016), and the author

argues that the Bitcoin market is ineﬃcient but may be moving towards an eﬃcient market. How-

ever, Nadarajah and Chu (2017) ﬁnd that, by a simple power transformation, the Bitcoin returns

satisfy the eﬃciency hypothesis. After the introduction of Bitcoin futures on CBOE and CME,

many paper study the price discovery between the Bitcoin spot and futures, and the impact of

futures on market eﬃciency; see, e.g., Corbet et al. (2018a) and Baur and Dimpﬂ (2019). However,

no conclusion is reached yet, as both sides have supporters.

We have better understanding on Bitcoin and other cryptocurrencies by now, thanks to the

quickly accumulated literature in this area. However, the literature on Bitcoin futures is not only

limited but also almost dedicated to the studies of standard contracts traded on CBOE and CME

(see Corbet et al. (2018a), Baur and Dimpﬂ (2019) and the references therein). The territory

of Bitcoin inverse futures is almost unexplored and remains largely mysterious to us. The only

exception, to the best of our knowledge, is our sister paper Deng et al. (2019), in which we study

an optimal trading problem with Bitcoin and Bitcoin inverse futures. Another related article is

Terry (2007), and the author considers hedging with inverse currency futures.

To gain more understanding on the nature of Bitcoin inverse futures, we ask the following two

fundamental questions:

•Question 1. What is the risk structure (mean and variance) of Bitcoin inverse futures?

•Question 2. Are Bitcoin inverse futures an eﬀective hedging tool to Bitcoin?

We answer the above two questions in this paper. To the best of our knowledge, we are the ﬁrst to

study the risk structure and optimal hedging problems of Bitcoin inverse futures. We summarize the

main contributions of this paper as follows. First, we specify the non-linear payoﬀ features of Bitcoin

inverse futures and emphasize the asymmetry eﬀect with respect to the futures price changes.

Second, we obtain the expansions of the returns on Bitcoin inverse futures using the (nominal)

returns on Bitcoin spot and futures reference prices, and their ﬁrst-order approximations, which

4

are proven to be accurate by empirical studies. We ﬁnd that the settlement design of Bitcoin inverse

futures brings non-linear and high-order risk factors into the risk structure. Furthermore, the inverse

structure induces asymmetric leverage eﬀect on volatility. Third, we consider an optimal hedging

problem using Bitcoin inverse futures, and obtain the optimal hedging strategy in closed forms to

both short hedges and long hedges. In empirical studies, we investigate the hedging performance

across diﬀerent exchanges. We ﬁnd that Bitcoin inverse futures of OKEx are an eﬀective hedging

vehicle for the Bitcoin spot markets of OKEx and Bitﬁnex, with hedging eﬃciency over 90%. In

comparison, the standard Bitcoin futures contracts of CBOE and CME achieve moderate hedging

eﬃciency of about 75%. Our ﬁndings suggest that, although Bitcoin inverse futures are not as

straightforward as standard futures and are highly risky, they do provide eﬀective hedging to the

Bitcoin spot markets. The hedging performance obtained in the empirical studies help explain why

investors prefer to trade Bitcoin inverse futures to standard futures.

The rest of the paper is organized as follows. In Section 2, we present our main theoretical results

in three directions of Bitcoin inverse futures: the unique payoﬀ features, the risk structure, and

optimal hedging strategies. In Section 3, we conduct empirical studies. We summarize concluding

remarks in Section 4. Lastly, we provide information on Bitcoin futures contracts in Appendix A

and technical proofs in Appendix B.

2 Main Results

Let us ﬁx a complete probability space (Ω,F,P), where Pis the physical probability measure.

Throughout the paper, E,Var and Cov denote respectively the expectation, variance and covariance

operators under measure P.

We denote S= (St)t≥0and F= (Ft)t≥0the Bitcoin spot and futures reference prices. Note

that both Sand Fare denominated in USD. We consider Bitcoin inverse futures contracts, with

notional value K(USD) and margin rate qper contract. Here, constants K > 0 and q∈(0,1]. We

5

deﬁne FB= (FB

t)t≥0by

FB

t=K

Ft

,

and interpret FB

tas the nominal value of the Bitcoin inverse futures per contract, denominated in

BTC, at time t. Hereinafter, we use superscript ·Bon a random variable (process) to emphasize

that such a random variable (process) is denominated in BTC, instead of USD. We use ∆Sand

∆Fto denote the changes of the spot price Sand the futures price F, respectively. For instance,

let ∆tbe a time increment, ∆St=St+∆t−Stand ∆Ft=Ft+∆t−Ftfor all t≥0.

Remark 2.1. In the above model setup, we do not specify whether the underlying ﬁnancial market

is a continuous-time market or a discrete-time market. In what follows, the default choice is the

discrete-time market. To be speciﬁc, we may think ∆tof a time unit (e.g., ∆t= 1 day or 1 minute)

and time ttakes values in {0,∆t, 2∆t, 3∆t, ···}. A discrete setup has at least two advantages.

First, all the analytical results are directly applicable to empirical studies, without the need for

discretization. Second, the (nominal) return of an asset is better deﬁned over discrete times, and

one key objective of this paper is to analyze the risk structure of the returns of Bitcoin inverse

futures. However, we note that all our results hold under a continuous-time model as well (one

simply takes the limit ∆t→0in deﬁning returns). In fact, our results are model-free since no price

model for either Sor Fis assumed.

2.1 Payoﬀ Features of Bitcoin Inverse Futures

The Proﬁt & Loss (P&L) of trading Bitcoin inverse futures comes from two sources: (1) the direct

changes of the inverse contract value, denoted by P&L1, and (2) the changes of the margin account,

denoted by P&L2. Both P&L1and P&L2are denominated in USD. In the following, we analyze

P&L1and P&L2separately.

•Let ∆FBdenote the changes of contract value of Bitcoin inverse futures (per contract). By

inverse contract covenants, ∆FBis deﬁned as “enter value” minus “exit value”.4Namely, for

4Such a choice on inverse futures settlement is beneﬁcial to the understanding of P&L. As can be seen from

6

one long position opened at tand closed at t+ ∆t, ∆FB

tis given by

∆FB

t=FB

t−FB

t+∆t=K

Ft−K

Ft+∆t

,(2.1)

which is diﬀerent from the deﬁnitions of ∆Sand ∆F. We easily see from (2.1) that ∆FBis

non-linear with respect to ∆F, as conﬁrmed by Figure 1.

To convert the value changes ∆FBfrom BTC into USD, we multiply it by the spot price

when the position is closed (i.e., the liquidation value of ∆FBBitcoins in the spot market).

We obtain P&L1for a trader with one unit long position in the inverse futures by

P&L1= ∆FB

t×St+∆t=K

Ft−K

Ft+∆t×St+∆t.(2.2)

•Suppose an investor enters into δcontracts of Bitcoin inverse futures at time t(either long

or short positions). With notional value Kand margin rate qper contract, the investor

is required to deposit |δ|qK/Ftnumber of Bitcoins into the margin account. Suppose the

investor purchases all required amount at the unit price Sfrom the Bitcoin spot market. The

change of values from the margin account over [t, t + ∆t] is then given by

P&L2=|δ|qK

Ft×(St+∆t−St).

To gain a better understanding of the P&Lfrom trading Bitcoin inverse futures, we present a

toy example (Example 2.2). This example delivers an important message that the P&L of trading

Bitcoin inverse futures is not only aﬀected by the futures price Fbut also the Bitcoin spot price S

in a non-linear fashion.

Example 2.2. Consider a Bitcoin inverse futures contract with notional value $100 and leverage

10X. Equivalently, we set K= 100 and q= 1/10 = 10%. Suppose a trader enters into one

long position of such an inverse contract, when the current quoted Bitcoin spot price and futures

(2.1), an increase in the futures price leads to trading gains in long positions, which is consistent with common

knowledge on standard futures. More details can be found on https://support.okex.com/hc/en-us/articles/

360000104591-Futures-Account-Profit- Loss.

7

Figure 1: Payoﬀ of Bitcoin Inverse Futures

Note. We choose the notional value K= 1 and initial futures price F0= 1. We plot the payoﬀ (1/F0−1/Ft) for one

long position and (1/Ft−1/F0) for one short position, where the x-axis is the futures price Ft.

price are $3000 and $4000, respectively. As a result, the nominal value per inverse contract is

100/4000 = 0.025 BTC. Opening one long (or short) Bitcoin futures contract would require the

trader to buy 0.1×100/4000 = 0.0025 BTC (equivalently 0.0025 ×$3000 = $7.5) on the Bitcoin

spot market and deposit them into the margin account. Assume, by tomorrow, the Bitcoin spot

and futures prices rise to $3500 and $5000, and the trader closes her futures position. Notice the

nominal value of the futures contract becomes 100/5000 = 0.02 BTC. Using (2.1), we obtain the

value changes of Bitcoin futures ∆FBby

100

4000 −100

5000 = 0.005 BTC.

The trader can either keep the gain of 0.005 BTC in her account or sells on the spot market to

receive 0.005 ×$3500 = $17.5(in USD). Furthermore, the trader’s margin account, with 0.0025

BTC, is now worth 0.0025 ×$3500 = $8.75 (in USD), that is a gain of $1.25 to the trader. The

P&Lof such a trade is $18.75 USD; see Table 1for summary.

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Table 1: Value Changes in Example 2.2

S(USD) F(USD) FB(BTC) Margin (USD)

Today $3000 $4000 0.025 $7.50

Tomorrow $3500 $5000 0.020 $8.75

Changes ∆ +$500 +$1000 +0.005 +$1.75

Total P&L = 0.005 ×3500 + 1.75 = $18.75

Notes. The changes ∆ are computed as “Tomorrow” - “Today”, except for the column FB(see Footnote 4).

Another unique feature of Bitcoin inverse futures is that the payoﬀ of both long and short

positions are more sensitive to the decline of futures price than to the increase of futures price (see

Figure 1). We call such a feature “asymmetry eﬀect”. In comparison, the payoﬀ of standard

futures is a linear function ∆F, so the impact of ∆Fon payoﬀ is symmetric. Ignoring or misun-

derstanding asymmetry eﬀect could lead to catastrophic consequences in trading Bitcoin inverse

futures. We further explore the details of asymmetry eﬀect in Example 2.3.

Example 2.3. Suppose the notional value and leverage of a Bitcoin inverse futures contract is $1

and 4X, implying K= 1 and q= 1/4 = 25%. An investor longs one inverse contract at time 0

when F0= $10. We consider two scenarios at time 1:20% increase on F(F1= $12) and 20%

decrease on F(F1= $8). We report results in Table 2, where the return in the last column is

computed as ∆FB/F B

0. We observe that 20% increase (resp. decrease) on the futures price leads

to a return of 16.67% (resp. −25%), which conﬁrms the asymmetry eﬀect. More importantly, 20%

decrease will trigger a margin call (or force the investor to liquid her long position). To see this, we

tract the total value of the investor’s aggregate positions in BTC. If the scenario is 20% increase,

the investor will gain 0.0167 BTC from trading futures, plus 0.025 BTC in the margin account,

and thus end up with 0.0417 BTC. However, if the scenario is 20% decrease, the investor will lose

0.025 BTC from trading futures, which completely washes out 0.025 BTC in the margin account,

and thus end up with 0BTC. In fact, even without any leverage (1X), 50% decline of the futures

price would cause the trader losing all margin deposit.

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Table 2: Value Changes in Example 2.3

F(USD) FB(BTC) Margin (BTC) ∆FB(BTC) Return (%)

t= 0 $10 0.1 0.025 NA NA

t= 1 (20% up) $12 0.083 0.025 0.0167 16.67%

t= 1 (20% down) $8 0.125 0.025 −0.025 -25%

2.2 Risk Structure of Bitcoin Inverse Futures

Having studied the payoﬀ features of Bitcoin inverse futures in Section 2.1, we now turn our atten-

tion to the inverse contract’s risk structure (expectation and variance) and their approximations.

We summarize key results in Theorem 2.5.

Let us consider a unit long position in Bitcoin inverse futures, which is initiated at time tand

closed at time t+ ∆t. Following the setup from Section 2.1, the direct gains from the long futures

position in BTC and in USD are given respectively by (2.1) and (2.2). We deﬁne (nominal) returns

for assets as follows:

RF:= Ft+∆t−Ft

Ft

=∆Ft

Ft

,(2.3)

e

R:= St+∆t−St

Ft

=∆St

Ft

,(2.4)

RB:= FB

t−FB

t+∆t

K=∆FB

t

K=1

Ft−1

Ft+∆t

,(2.5)

R:= FB

t−FB

t+∆tSt+∆t

K=1

Ft−1

Ft+∆tSt+∆t.(2.6)

RFdeﬁned in (2.3) is the return on standard futures; see Figlewski (1984) for similar deﬁnition.

We interpret e

Rin (2.4) as the mixed spot-futures return, which will be proven to be useful in the

analysis. RBdeﬁned in (2.5) (resp. Rdeﬁned in (2.6)) is the return on Bitcoin inverse futures

denominated in BTC (resp. USD).

Remark 2.4. To ease notations, we decide not to include time argument tin the above return

deﬁnitions (2.3)-(2.6). It is clear that all of them are random variables well deﬁned at time t, and

as tevolves, they become time series (stochastic processes).

Our main focus is to investigate the time tconditional expectation (notation Et) and variance

10

(notation Vart) of returns RBand Rof Bitcoin inverse futures. The non-linear payoﬀ feature of

Bitcoin futures brings high-order risk factors to both RBand R. Theorem 2.5 below delivers this

message.

Theorem 2.5. We obtain the time tconditional expectation and variance of RBand Rby

Et(RB) = 1

Ft

∞

X

i=0

(−1)iEt(R1+i

F),(2.7)

Et(R) = St

Ft

∞

X

i=0

(−1)iEt(R1+i

F) +

∞

X

i=0

(−1)iEthe

R·R1+i

Fi,(2.8)

Vart(RB) = 1

F2

t

∞

X

i,j=0

(−1)i+jCovtR1+i

F, R1+j

F,(2.9)

Vart(R) = S2

t

F2

t

∞

X

i,j=0

(−1)i+jCovtR1+i

F, R1+j

F

+St

Ft

∞

X

i,j=0

(−1)i+jCovtR1+i

F,e

R·R1+j

F

+

∞

X

i,j=0

(−1)i+jCovte

R·R1+i

F,e

R·R1+j

F,(2.10)

where Covtdenotes conditional covariance.

In the empirical studies, we take the terms with index i= 0 in (2.7)-(2.10) and denote them

the ﬁrst-order approximations to Et(RB), Et(R), Vart(RB) and Vart(R). We have:

Et(RB)'1

Ft

Et(RF),(2.11)

Et(R)'St

Ft

Et(RF) + Ethe

R·RFi,

Vart(RB)'1

F2

tVart(RF)−2Covt(RF, R2

F):= d

Vart(RB),(2.12)

Vart(R)'S2

t

F2

t

Vart(RF)+2St

Ft

Covt(RF,e

R·RF) + Vart(e

R·RF) := d

Vart(R).(2.13)

The results of Theorem 2.5 shed light on the complexity of risk factors of Bitcoin inverse

futures’ return RBand R. First, the variance of RFand higher-order covariance related to RFand

e

Rboth contribute to the risk (variance) of inverse futures’ return, and these factors are intertwined.

11

This observation explains why perfect hedging of Bitcoin futures is nearly impossible in practice.

Second, the skewness and kurtosis of standard futures’ return RFplay an important role in the

risk of inverse futures’ intrinsic return R. Third, there is a leverage eﬀect on the variance Vart(R).

Here, leverage eﬀect means the magnitude of the inverse futures’ intrinsic risk Vart(R) comparing

to the quoted prices variance Vart(RF). When the market is in contango, i.e., when futures price

F > spot price S(resp. backwardation, i.e., F < S), the variance Vart(R) is deleveraged (resp.

leveraged) by the ratio S/F . To summarize, the non-linear payoﬀ of Bitcoin inverse futures enriches

the risk structure of returns RBand R, and at the same time makes hedging with inverse futures

a complicate task. In the empirical studies, we show that the ﬁrst-order approximations in (2.11)-

(2.13) provide accurate estimations to the intrinsic risk of returns RBand R, and the (de)leverage

eﬀect.

2.3 Optimal Hedging with Bitcoin Inverse Futures

One essential use of futures is to provide a hedging tool to the underlying asset. In this section,

following the minimum-variance hedging framework of Ederington (1979) and Figlewski (1984), we

study optimal hedging problems with Bitcoin inverse futures. We refer to Lien and Tse (2002) and

references therein for literature on futures hedging.

To begin, we deﬁne two types of hedging strategies: short hedge and long hedge. We need to

distinguish between short and long hedges, since Bitcoin margin requirement sets them apart from

traditional hedges. Without loss of generality, we assume the absolute position in Bitcoin spot is

1 in the studies. Using Deﬁnition 2.6, we establish the positions in Bitcoin (spot), inverse futures

and margin account in Table 3, where columns “Spot” and “Margin Account” are the number of

Bitcoins in possession.

Deﬁnition 2.6. A short hedge (S-hedge) is a hedging strategy that longs one Bitcoin and shorts

δBitcoin inverse futures contracts. A long hedge (L-hedge) is a hedging strategy that shorts one

Bitcoin and longs δBitcoin inverse futures contracts. Here, δ > 0is the number of inverse contracts.

Let us denote by VSand VLthe portfolio value of an S-hedge and a L-hedge, and by ∆VSand

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Table 3: Positions of S-hedge and L-hedge (unit in BTC)

Spot Futures Margin Account

S-hedge 1 −δ δ qK

F

L-hedge -1 δ δ qK

F

∆VLthe corresponding P&L, denominated in USD. We derive ∆VSby

∆VS=1 + δqK

Ft∆St−δK

Ft−K

Ft+∆tSt+∆t

'1 + δqK

Ft∆St−δK St

F2

t

∆Ft,(2.14)

where we have used the approximation St+∆t/Ft+∆t'St/Ftto derive the second (approximation)

equality.5Similarly, we obtain ∆VLby

∆VL=δqK

Ft−1∆St+δK

Ft−K

Ft+∆tSt+∆t

'δqK

Ft−1∆St+δK St

F2

t

∆Ft.(2.15)

We proceed to formulate our optimal hedging problems as follows.

Problem 2.7. The investor seeks an optimal hedging strategy δi(i∈ {S, L}) that minimizes the

variance of trading proﬁt and loss ∆Vi, i.e.,

δi= arg min

δ>0

Vart(∆Vi), i ∈ {S, L},

where ∆VSand ∆VLare given by (2.14)and (2.15). We call δS(δL) an optimal S-hedge (L-hedge)

strategy or ratio.

We are able to solve the above optimization problem (Problem 2.7) and obtain optimal strategies

δSand δLin closed forms in Theorem 2.8. For notational simplicity, we suppress time tsubscript

5The exact form of ∆VScan be derived, but will lead to much complicate expression due to the inﬁnite series

of Vart(R) in (2.10). Numeric studies show that such an approximation is accurate from the perspective of hedging

eﬃciency.

13

in the rest of this section.

Theorem 2.8. We obtain optimal strategies δSand δLto Problem 2.7 by

δS=δ0+δS

qand δL=δ0+δL

q,(2.16)

where δ0,δS

qand δL

qare deﬁned by

δ0=ρ√k, δS

q=n2δ2

0−nkδ0−k

1−2nδ0+n2k, δL

q=n−2δ2

0−nkδ0+k

1+2nδ0+n2k,(2.17)

with parameters given by

ρ=Cov(∆S, ∆F)

pVar(∆S)Var(∆F), k =Var(∆S)

m2Var(∆F), m =KS

F2, n =qK

F.(2.18)

Three important implications are due, thanks to Theorem 2.8. First, if margin deposit is not

required when establishing futures positions (q= 0 ⇒n= 0), the optimal S-hedge strategy δS

is equal to the optimal L-hedge strategy δL. To be precise, given q= 0, we have δS=δL=δ0.

Second, the unique design of Bitcoin inverse futures brings intricate components into the expression

of optimal hedging strategies. Both δSand δLconsist of two terms. The ﬁrst term is δ0, induced

by the payoﬀ structure of Bitcoin inverse futures. The second term is δS

q(or δL

q), induced by the

margin requirement on Bitcoin inverse futures. Notice that various factors, including correlation

coeﬃcient ρ, variance ratio of ∆Sand ∆F, leverage of contract (1/q), are entangled in the second

term δS

qand δL

q. Third, optimal hedging strategies δSand δLare both non-linear functions of

the correlation coeﬃcient ρ. This result means that, unlike traditional futures hedging, the co-

movement of spot price and futures price is not the only factor which determines the optimal

hedging strategy. Hedging with Bitcoin inverse futures is a lot more complicated than that with

standard futures.

In addition, using (2.17), we can easily establish conditions for δS

q>0 and δL

q<0. For instance,

if ρ > 1/√2=0.707, we have δL

q<0. However, as ρis usually high (close to 1) in the real Bitcoin

14

markets, we expect that

δS

q>0 and δL

q<0⇒δL≤δ0≤δS.(2.19)

Recall δ0is the optimal hedging strategy for both S-hedge and L-hedge problems without Bitcoin

margin requirement (i.e., q= 0). The above relation reveals that introducing margin on inverse

futures will cause hedgers to short more contracts in S-hedge but long less in L-hedge.

In the remaining part of this section, we study the hedging eﬃciency of optimal strategies δS

and δLin (2.16). Following Ederington (1979), we deﬁne hedging eﬃciency as the percentage of

variance reduced by hedging. To emphasize the dependence of P&L ∆VSand ∆VLon hedging

strategy δ, we write them as ∆Vi= ∆Vi(δ), where i=S, F . Given an S-hedge or a L-hedge

strategy δ, the hedging eﬃciency of such a strategy is deﬁned by

HE(δ)=1−Var(∆Vi(δ))

Var(∆Vi(0)), δ > 0, i =Sor F, (2.20)

where ∆VSand ∆VLare given by (2.14) and (2.15). We have the following result regarding the

hedging eﬃciency of optimal strategies δSand δL.

Theorem 2.9. We obtain the hedging eﬃciency of the optimal S-hedge strategy δSand optimal

L-hedge strategy δLby

HE(δS) = 1

k(1 −2nδ0+n2k)(δS)2and HE(δL) = 1

k(1 + 2nδ0+n2k)(δL)2,(2.21)

where δ0is given by (2.17)and nand kare given by (2.18).

Furthermore, we have

HE(δS)≤HE(δL)if ρ≥0.(2.22)

We oﬀer several explanations to Theorem 2.9. If margins are not required on trading futures

(q= 0), we have n= 0 and δS=δL=δ0, and in consequence, the hedging eﬃciency of δSand

15

δLare the same HE(δS) = HE(δL) = ρ2. Such an equality immediately implies that if ρ=±1

(spot price change ∆Sand futures price change ∆Fare perfectly correlated), perfect hedging

(HE(δS) = HE(δL) = 1) is achieved if the investor follows the optimal strategy. Since ρ > 0 holds

almost surely in the real markets, the relation (2.22) reveals that a L-hedge is more eﬀective than

an S-hedge, again caused by the asymmetry eﬀect of Bitcoin inverse futures.

3 Empirical Studies

In this section, we conduct empirical studies to investigate the risk structure and hedging perfor-

mance of Bitcoin inverse futures.

3.1 Data Description

BitMEX and OKEx are the two largest exchanges ranked by the trading volumes of Bitcoin futures,

with daily volumes of $9.13 billion and $3.71 billion respectively (see Table A.2). Although BitMEX

is the largest Bitcoin futures exchange, around 97% of the Bitcoin futures contracts traded on

BitMEX are perpetual contracts. On the other hand, nearly 85% of the Bitcoin futures contracts

traded on OKEx are inverse contracts with a ﬁxed term (ﬁnite expiry). In particular, quarterly

inverse contracts account for about 80% of the total trading volumes on OKEx, and they are among

the most liquid Bitcoin futures contracts in the markets. Therefore, we choose the quarterly inverse

futures of OKEx as the Bitcoin inverse futures in the following studies. As a comparison, we also

consider standard Bitcoin futures contracts from CME and CBOE. Please refer to Tables A.1 and

A.3 for more information on Bitcoin futures of CME, CBOE and OKEx. Since Bitcoin market is

severely segmented, we choose Bitcoin spot prices from OKEx and Bitﬁnex exchanges to verify the

hedging performance of OKEx futures across diﬀerent exchanges. All data are in daily frequency

and sampled at UTC time. We summarize the basic information regarding the sources and time

range of the data used in our empirical studies in Table 4.

We plot the Bitcoin spot price Sand futures price Ffor all four exchanges considered (OKEx,

Bitﬁnex, CME and CBOE) in Figure 2. We observe from Figure 2that Sand Fare highly positively

16

Table 4: Information of Data Used in Empirical Studies

Exchange Data Source Time Range

Inverse Futures OKEx www.okex.com From 2018/10/07 to 2019/07/26

Standard Futures CBOE www.investing.com From 2017/12/12 to 2019/06/19

Standard Futures CME www.investing.com From 2017/12/18 to 2019/07/03

Bitcoin Spot OKEx & Bitﬁnex www.investing.com From 2018/10/07 to 2019/07/26

Notes. CBOE discontinued its Bitcoin futures contracts on 2019/06/19.

correlated across all four exchanges, which suggest ρ, deﬁned in (2.18), is close to 1. We report the

summary statistics of the Bitcoin spot price changes ∆Sand futures price changes ∆Fin Table 5.

The results show that both Bitcoin spot and futures prices are extremely volatile, with standard

deviations of daily price changes exceeding 300 in all cases. We also ﬁnd both ∆Sand ∆Fexhibit

spike kurtosis and negative skewness across all four exchanges.

Table 5: Summary Statistics

Variables ∆S∆F

Exchanges OKEx Bitﬁnex OKEx CBOE CME

Min -1776.30 -2503.00 -1921.48 -2745.00 -3030.00

P25% -60.05 -119.60 -63.33 -140.00 -125.00

Median 8.70 5.50 9.36 0.00 0.00

Mean 10.99 -11.56 11.34 -23.48 -20.29

P75% 90.70 118.10 91.24 117.00 130.00

Max 1267.90 2056.00 1162.35 1850.00 2505.00

Skewness -0.57 -0.69 -0.73 -0.63 -0.89

Kurtosis 11.42 10.54 11.16 10.59 13.31

Standard Deviation 307.07 425.32 330.40 438.73 477.99

Count 292 292 292 378 381

Notes. ∆Sand ∆Fare daily price changes of Bitcoin spot and futures. P25% and P75% refer to the 25% quantile

and 75% quantile. The settlement reference price of Bitcoin futures on OKEx is the weighted average of the last

Bitcoin prices in USD on top exchanges, such as Bitﬁnex, Gemini, Coinbase, Bitstamp, OKCoin and Kraken. The

Bitcoin Reference Rate (BRR) and Gemini auction price are used as the daily Bitcoin reference rate by CME and

CBOE, respectively.

3.2 Risk Structure of Bitcoin Inverse Futures

We take variance (volatility) as a risk measure, and study the risk of Bitcoin futures in this sub-

section. In particular, we focus on the risk of return variables RBand R, which are the returns of

17

Bitcoin inverse futures and deﬁned in (2.5) and (2.6). Recall Vart(RB) and Vart(R) are given by

(2.9) and (2.10), and their ﬁrst-order approximations d

Vart(RB) and d

Vart(R) by (2.12) and (2.13).

In what follows, we denote σthe standard deviation (volatility) of a random variable and bσthe

ﬁrst-order approximation of σ, e.g., σ(R) = pVart(R) and bσ(R) = qd

Vart(R). We report the

summary statistics of RBand R, along with returns RFand ˜

R(see (2.3) and (2.4) for deﬁnitions),

in Table 6, which are calculated using the data from OKEx only over the full sample period.

Table 6: Full Sample Returns Summary Statistics on OKEx

Variables RFe

R RBR

Min -0.14 -0.13 -3.17E-05 -0.15

P25% -0.01 -0.01 -1.93E-06 -0.01

Median 0.00 0.00 3.80E-07 0.00

Mean 0.0023 0.0020 1.76E-07 0.0021

P75% 0.02 0.02 3.26E-06 0.02

Max 0.19 0.17 3.97E-05 0.19

Skewness 0.15 0.16 -2.17E-02 0.09

Kurtosis 6.31 6.15 7.59 6.34

Daily S.D. 4.28% 3.97% 0.00081% 4.30%

Count 292 292 292 292

We ﬁrst look at the accuracy of the ﬁrst-order approximations bσ(RB) and bσ(R) to σ(RB) and

σ(R). To calculate the standard deviations of RFand RB, and the approximation bσ(RB), we

only need the Bitcoin futures prices from OKEx. However, the standard deviation of R, and the

approximation bσ(R) rely on the choice of both spot prices and futures prices. Here we use Bitcoin

spot prices from OKEx and Bitﬁnex, and report the corresponding results for comparisons. To this

purpose, we calculate them on a 60-day rolling window using the full sample data of OKEx and

Bitﬁnex exchanges. The computation details are given as follows.

•We ﬁrst use the raw data of Sand Fto compute RF, RBand Rusing deﬁnitions in (2.3),

(2.5) and (2.6). Next, we use the sample standard deviations of RBand Rover a 60-day

rolling window as estimates for σ(RF), σ(RB) and σ(R).

•To obtain bσ(RB) and bσ(R), we repeat the same process as above by using the approximations

in (2.12) and (2.13).

18

We plot the graphs of σ(RB) vs bσ(RB) and σ(R) vs bσ(R) for exchanges OKEx and Bitﬁnex

in panels 1-4 of Figure 3. In addition, we also include the comparisons between σ(RF) (return

on standard futures) and σ(R) (return on inverse futures) in panels 5-6 of Figure 3. We observe

that bσ(RB) and bσ(R), given by (2.12) and (2.13), are as good as the commonly used sample

standard deviations. It is well known that the latter is an unbiased and consistent estimator to

the true volatility. Regarding the volatilities σ(R) and σ(RF), they are not identical, although the

diﬀerence is not signiﬁcant either. Recall that RFdeﬁned in (2.3) is the return on Bitcoin futures

prices F(standard futures), while Ris the return on inverse futures, and both are denominated in

USD.

One interesting ﬁnding of Theorem 2.5 is the (de)leverage eﬀect. Here, leverage eﬀect means

the magnitude of the inverse futures’ intrinsic risk Vart(R) comparing to the quoted prices variance

Vart(RF). For standard futures, there is no (de)leverage eﬀect and the ratio Vart(R)/Vart(RF)

always equals to one. Precisely, we deﬁne the (de)leverage eﬀect as

Leverage Eﬀect (LE) = max(Vart(R)−Vart(RF),0)

Vart(RF),(3.1)

Deleverage Eﬀect (DE) = max(Vart(RF)−Vart(R),0)

Vart(RF).(3.2)

The (de)leverage eﬀect measures the evolution of (de)inﬂation rate of the inverse futures’ intrinsic

variance Vart(R) compared with Vart(RF) through time. We calculate LE and DE using a 60-day

rolling window similar to σ(R) and σ(RF), and plot them in Panels 7-8 of Figure 3. As shown by

Figure 3, when the market is in contango (the ratio S/F < 1), the volatility σ(R) is deleveraged

and reduced by the ratio S/F , and vice versa. The economic insight is that, in a contango (resp.

backwardation) market, inverse futures contracts are deleveraged (resp. leveraged) and less (resp.

more) risky than standard contracts on the same quoted futures price. When the market is neutral

(S/F '1), two volatilities are almost indistinguishable. To further explore the (de)leverage eﬀect,

inherited from (3.1) and (3.2), we deﬁne

Max Leverage Eﬀect (MLE) = max(LE),Max Deleverage Eﬀect (MDE) = max(DE),(3.3)

19

where LE and ME are deﬁned in (3.1) and (3.2). We calculate MLE and MDE and report the

results in Table 7. The MLE and MDE of Bitcoin futures traded on OKEx (Bitﬁnex) are 8.68%

(12.19%) and 4.51% (4.62%) respectively. These numbers show that the leverage eﬀect is more

signiﬁcant and Bitcoin inverse futures are more risky than the quoted futures prices, although

deleverage eﬀect coexists in the market.

Table 7: 60-day Rolling Daily Volatility and Leverage Eﬀect

Variables OKEx Bitﬁnex

Average σ(RB) 8.25E-06 8.25E-06

Average bσ(RB) 7.79E-06 7.79E-06

Accuracy 91.16% 91.16%

Average σ(R) 4.07% 4.12%

Average bσ(R) 4.16% 4.22%

Accuracy 97.43% 97.20%

Average σ(RF) 4.03% 4.03%

Max Leverage Eﬀect 8.68% 12.19%

Max Deleverage Eﬀect 4.51% 4.62%

Notes. σ(RB), σ(R) and σ(RF) are empirical (sample) standard deviations, while bσ(RB) and bσ(R) are their ﬁrst-

order approximations given by (2.12) and (2.13). To calculate the standard deviations σ(RB), σ(RF), and bσ(RB),

we only need the Bitcoin futures prices from OKEx; therefore, the corresponding entries for OKEx and Bitﬁnex are

identical in the table. Max (De)Leverage Eﬀect is calculated using the deﬁnitions in (3.3). Accuracy is calculated

as |(Average bσ(RB)−Average σ(RB))/Average σ(RB)|for RBand |(Average bσ(R)−Average σ(R))/Average σ(R)|

for R.

3.3 Hedging Performance

In this subsection, we study the hedging performance of various strategies, including optimal hedg-

ing strategies δSand δLin (2.16), where the spot market is either OKEx or Bitﬁnex. In the

analysis, we consider three strategies for both S-hedge (short hedge) and L-hedge (long hedge); see

their deﬁnitions in Deﬁnition 2.6.

1. Optimal hedging strategy (δ=δSin S-hedge and δ=δLin L-hedge);

2. No-margin hedging strategy (δ=δ0);

3. Na¨ıve hedge (δ=F

K:= ˆ

δ).

20

We include the na¨ıve hedge in the studies because it is a popular strategy used in practice and the

performance is acceptable in most scenarios (see Kroner and Sultan (1993), Alexander and Barbosa

(2008), and Wang et al. (2015)). Since we normalize the size of the spot position to be always one,

the na¨ıve strategy simply enters F/K futures contracts to match exactly one Bitcoin. We balance

all three hedging strategies δS, δL, and δ0on a daily basis, as our data come in daily frequency.

We use the data over the ﬁrst 100 days in the full sample (from 2018/10/07 to 2019/07/26 with

292 observations of ∆Sand ∆F) to compute the initial values of δ0,δS

qand δL

qin (2.17). Recall

from (2.16) that δS=δ0+δS

qand δL=δ0+δL

q. At any given day t(t≥100), we use all historical

data up to tas the in-sample data to estimate the latest values of δS(or δL), δ0and ˆ

δ, which

are the hedging strategies adopted from day tto day t+ 1. The P&L ∆VS(δ) (or ∆VL(δ)) are

then recorded for each of the three strategies. We ﬁnally use the deﬁnition (2.20) to calculate the

hedging eﬃciency (HE) of each strategy. We present the hedging eﬃciency of all three strategies

for OKEx and Bitﬁnex spot markets under 5% and 10% margin rates in Table 8. In Theorem

2.9, we have derived the exact expressions of HE(δS) and HE(δL) in closed forms in (2.21). We

report the results of HE(δS) and HE(δL) from (2.21) over the full sample under the “Average Model

Prediction” row in Table 8, and illustrate its evolution in Figure 4.

Table 8: Hedging Eﬃciency of Bitcoin Inverse Futures

OKEx Bitﬁnex

Margin Rate (q) 0.05 0.1 0.05 0.1

Short Hedge Optimal Strategy δS97.50% 97.22% 97.59% 97.34%

No-margin Strategy δ097.09% 96.01% 97.26% 96.31%

Na¨ıve Strategy ˆ

δ57.79% 55.63% 58.71% 56.58%

Average Model Prediction 84.90% 83.47% 85.47% 84.06%

Long Hedge Optimal Strategy δL97.95% 98.13% 97.99% 98.15%

No-margin Strategy δ096.46% 95.23% 95.91% 94.50%

Na¨ıve Strategy ˆ

δ61.86% 63.78% 62.72% 64.61%

Average Model Prediction 87.28% 88.28% 87.80% 88.77%

Notes. The results on the rows of “Average Model Prediction” are calculated using (2.21), which provides the

theoretical results of HE(δS) and HE(δL).

We observe the following facts from Table 8and Figure 4. First, Bitcoin futures provide a very

eﬃcient hedging tool for the OKEx and Bitﬁnex spot markets considered in the analysis, the HEs

21

are above 80% in δS, δLand δ0hedges and are close to 100% in some cases. Surprisingly, hedge

eﬃciencies under δS, δLand δ0outperform theoretical predictions (2.21) in Theorem 2.9 for OKEx

and Bitﬁnex. Second, as conﬁrmed by both Table 8and Figure 4, hedging eﬃciency of long hedges

is better than that of short hedges, which is also consistent with the relation (2.22) in Theorem

2.9. This assertion is due to the fact that the short position on Bitcoin (−S) is partially oﬀset by

the Bitcoin margin deposit δq K/F ; see Table 3or (2.15). Third, leverage imposes an asymmetric

eﬀect on short and long hedges. Elevating leverage worsens (resp. beneﬁts) the hedging eﬃciency

of short hedges (resp. long hedges), due to the increase on margin deposit.

So far we have focused on the hedging performance of Bitcoin inverse futures only; in the next

step, we incorporate the standard contracts from CME and CBOE into studies. We calculate the

out-of-sample hedging eﬃciency for the standard futures contracts traded on CME and CBOE and

present the results in Table 9. For standard futures contracts, we do not need to distinguish short

hedge and long hedge, since the ﬁat-currency USD, not Bitcoin, is taken as margins. We also report

the eﬃciency of the standard OLS (ordinary least squares) hedging strategy in Table 9. Notice

that hedging eﬃciency is around 75% for both CME and CBOE futures contracts. Such a ﬁnding

is contrary to that of Corbet et al. (2018a), in which the authors claim that Bitcoin standard

futures are not an eﬀective hedging instrument and hedging may even increases risk. Corbet

et al. (2018a) use high-frequency (one-minute tick data) futures data from Thomson Reuters Tick

History and Bitcoin price data from Thomson Reuters Eikon. The noises of market microstructure,

such as liquidity, data errors, and order imbalances, may contaminate hedging performance using

high-frequency data sample. Those eﬀects are alleviated for daily frequency samples (used in our

empirical analysis). Our ﬁndings show that rebalancing a hedging portfolio at a daily basis is a

more practical and eﬃcient approach to achieve the hedging purpose. By comparing the results in

Tables 8and 9, we easily see that standard futures contracts of CME and CBOE are less eﬀective

than those inverse futures contracts traded on less regulated exchanges (e.g., OKEx).

The last object of our empirical studies is the size of optimal hedging strategies, δSin the

S-hedge and δLin the L-hedge, and the no-margin strategy δ0(as a benchmark). We consider two

margin levels (q= 0.05 and q= 0.1) and two Bitcoin exchanges (OKEx and Bitﬁnex). We report

22

Table 9: Hedging Eﬃciency of Bitcoin Standard Futures

Variables CME CBOE

OLS Hedging Eﬃciency 75.28% 74.1%

Na¨ıve Hedging Eﬃciency 68.01% 70.44%

Average OLS Hedge Position 0.89 0.88

the summary statistics of these three strategies in Table 10 and plot their evolutions in Figure 5.

An obvious ﬁnding is δL< δ0< δS, which ﬁts with our expectation as in (2.19). However, both

δSand δLare close to δ0, especially δL. Since δS=δ0+δS

qand δL=δ0+δL

q(see (2.16)), δ0is

undoubtedly the dominating factor, comparing to δS

qand δL

q. All three strategies share almost the

same skewness and kurtosis. As the Bitcoin margin qincreases, S-hedge position δSincreases but

L-hedge position δLdecreases.

Table 10: Summary Statistics of Hedging Positions

OKEx Bitﬁnex

Margin 0.05 0.1 0.05 0.1

Optimal S-hedge δSAverage 58.71 61.11 59.02 61.44

Skewness 0.8 0.8 0.79 0.8

Kurtosis 2.29 2.3 2.29 2.3

Optimal L-hedge δLAverage 54.35 52.37 54.6 52.6

Skewness 0.79 0.78 0.78 0.78

Kurtosis 2.28 2.27 2.27 2.27

No-margin δ0Average 56.46 56.46 56.74 56.74

Skewness 0.79 0.79 0.79 0.79

Kurtosis 2.28 2.28 2.28 2.28

4 Conclusion

In this paper, we focus on Bitcoin inverse futures, one of the latest derivatives in the cryptocurrency

markets, which are traded by investors across exchanges over the world. In particular, we proceed

the studies of Bitcoin inverse futures in three closely related directions: the unique non-linear

payoﬀ features, the risk structure (expectation and variance of returns on inverse futures), and

the optimal hedging problems. In the theoretical part, we obtain the expressions and ﬁrst-order

23

approximations of the risk structure, optimal short hedge and long hedge strategies, and hedging

eﬃciency of optimal strategies, all in closed forms. Through empirical studies, we not only verify

our theoretical results but also obtain new insights. First, the returns of Bitcoin intrinsic value

denominated in USD and BTC are both non-linear functions of the quoted price changes, and

Bitcoin inverse futures are more sensitive to the decrease of the quoted futures prices, which induces

the so-called asymmetry eﬀect. Such an asymmetry eﬀect will accelerate the margin account to be

washed out for long positions, when the futures price decreases. Both the risk of quoted futures

prices and spot prices contribute and inject high-order risk factors, such as skewness and kurtosis,

into the risk (variance) of the returns on Bitcoin inverse futures. Second, the hedging eﬃciency

of Bitcoin inverse futures are over 90% for both OKEx and Bitﬁnex spot markets. This result

provides convincing evidence that Bitcoin inverse futures are an eﬀective tool to hedge Bitcoin in

the segmented spot markets. In comparison, using Bitcoin standard futures from CME and CBOE

achieves about 75% of hedging eﬃciency, which is acceptable, although not as good as that of

inverse futures. The rather signiﬁcant diﬀerence in hedging eﬃciency between inverse and standard

futures oﬀers an explanation to why traders favor inverse contracts and why the market shares of

CME and CBOE in Bitcoin futures are miserable. Third, understanding asymmetry eﬀect is key to

trading Bitcoin inverse futures. Asymmetry eﬀect appears in various aspects, such as the trading

P&L, the (de)leverage eﬀect, and optimal hedging strategies in S-hedge and L-hedge.

On the one hand, Bitcoin inverse futures provide a great tool to short, hedge, or speculate

Bitcoins, which contribute to a better and more eﬃcient cryptocurrency market. On the other hand,

they are among the most complex and volatile ﬁnancial derivatives, any misunderstanding of their

payoﬀ features, risk structure or hedging problems could lead to catastrophic results in trading.

We hope this research could stimulate the interest of both practitioners and academic scholars

to study Bitcoin inverse futures. One future research direction is to understand the asymmetry

eﬀect on up and down price volatilities and the accused market manipulation in Bitcoin prices.

Another interesting extension is to investigate how Bitcoin inverse futures can be used in portfolio

management to improve risk-adjusted returns.

24

Acknowledgements

The authors would like to thank Liyan Yang and Kewei Hou for insightful comments. The research

of Jun Deng is supported by the National Natural Science Foundation of China (11501105) and

UIBE Research Funding (302/871703). The research of Bin Zou is supported in part by a start-up

from the University of Connecticut.

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Figure 2: Bitcoin Spot and Futures Prices on OKEx, Bitﬁnex, CME and CBOE

28

Figure 3: 60-day Rolling Daily Volatilities of Returns on Bitcoin Futures

Notes. Panels 1-4 report empirical standard deviations (S.D.) σ(RB) and σ(R), and their ”ﬁrst-order” approxima-

tions bσ(RB) and bσ(R), which are calculated using (2.12) and (2.13) from Theorem 2.5. Panels 5-6 report S.D. σ(R)

and σ(RF), where RFand Rare deﬁned by (2.3) and (2.6). All three panels on the left side are for OKEx, while

those on the right are for Bitﬁnex. The left y-axis is (de)leverage eﬀect and the right y-axis is the ratio of the spot

price and futures price S/F in panels 7-8. The green lines in panels 7-8 are 60-day rolling average ratios of spot

prices and futures prices. The LE (red line) and DE (blue line) are calculated by (3.1) and (3.2). When the market

is in backwardation (the ratio S/F > 1), the S.D. σ(R) (comparing to σ(RF)) is leveraged and increased by the

ratio S/F and vice versa and LE is strictly positive.

29

Appendix A Tables of Bitcoin Futures

Table A.1: CBOE and CME Bitcoin Futures Speciﬁcations

CBOE Bitcoin Futures CME Bitcoin Futures

Listing Date 10-Dec-17 18-Dec-17

Symbol XBT BTC

Quantity 1 BTC 5 BTC

Valuation Gemini auction price BTC Reference Rate (BRR)

Tick Size $10.00 $5.00

Tick Value $10.00 $25.00

Margin Requirement 44% 37% - 40.7%

Settlement Cash settled Cash settled

Trading Hours Sun. 5:00 pm to Mon. 3:15 pm (CT)

Tue. 3:30 pm to Fri. 3:15 pm (CT)

Sunday - Friday

5:00 pm - 4:00 pm (CT)

Notes. Accessed at 1:54pm UTC, August 22, 2019. Source: www.investopedia.com and www.cmegroup.com.

Table A.2: Top 5 Cryptocurrency Exchanges by Futures Trading Volume

Exchange Volume (24h $bn)

BitMEX 9.13

OKEx 3.71

bitFlyer 1.13

Deribit 1.08

CryptoFacilities 0.09

Notes. Accessed at 1:05 pm UTC on June 26, 2019 from https://www.sk3w.co/bitcoin_futures.

Table A.3: OKEx Bitcoin Futures Speciﬁcations

Key Speciﬁcations OKEx Bitcoin Futures

Subject Bitcoin Index

Contract Value BTC 100 USD

Leverage 10X or 20X

Quotation Unit Index Point

Minimum Price Intervals 0.01 Point

Contract Expirations Weekly, Bi-weekly, Quarterly

Delivery Time 16:00, Friday of the Expiry Week

Settlement Date Same as Delivery Date

Settlement Method Settled by Bitcoin

Notes. Accessed at 1:54pm UTC, August 22, 2019. Source: www.okex.com.

32

Appendix B Proofs

Proof of Theorem 2.5.Using deﬁnitions (2.3) and (2.5), we get

RB=1

Ft−1

Ft+∆t

=∆Ft

Ft(Ft+ ∆Ft)=1

Ft

RF

1 + RF

=1

Ft

∞

X

i=0

(−1)iR1+i

F,(B.1)

where we have used the Taylor expansion of function 1

1+xto derive the last equality. By taking

condition expectation and conditional variance at time tin (B.1), we obtain (2.7) and (2.9) in

Theorem 2.5.

From the deﬁnition of Rin (2.6), we derive

R=1

Ft−1

Ft+∆tSt+∆t=∆Ft

Ft

St+∆t

Ft+∆t

=RF

St+ ∆St

Ft+ ∆Ft

=RF

St/Ft+ ∆St/Ft

1+∆Ft/Ft

=RF

St/Ft+e

R

1 + RF

=RFSt

Ft

+e

R∞

X

i=0

(−1)iRi

F

=St

Ft

∞

X

i=0

(−1)iR1+i

F+

∞

X

i=0

(−1)ie

RR1+i

F,

which naturally implies the results in (2.8) and (2.10). The ﬁrst-order approximations then follow

easily by taking terms with index i= 0 in equations (2.7)-(2.10). The proof is now complete.

Proof of Theorem 2.8.We provide proofs to the optimal S-hedge problem and show δSis indeed

given by (2.16). The case of the optimal L-hedge problem can be solved using similar arguments.

For notational simplicity, we suppress time tsubscript for all notations in the proof.

Using the deﬁnitions of k,mand nin (2.18), we have

Var(∆VS) = Var((1 + nδ)∆S−mδ∆F)

= (1 + nδ)2Var(∆S) + m2δ2Var(∆F)−2mδ(1 + nδ)Cov(∆S, ∆F)

=δ2(n2Var(∆S) + m2Var(∆F)−2mn Cov(∆S, ∆F))

+ 2δ(nVar(∆S)−mCov(∆S, ∆F)) + Var(∆S).(B.2)

33

The ﬁrst-order condition gives

δS=mCov(∆S, ∆F)−nVar(∆S)

n2Var(∆S) + m2Var(∆F)−2mn Cov(∆S, ∆F)

=δ0+n·2δ2

0−nkδ0−k

1−2nδ0+n2k,

where δ0=ρ√kand ρare deﬁned in (2.18). The second-order condition yields

n2Var(∆S) + m2Var(∆F)−2mn Cov(∆S, ∆F)≥0.

Hence, the ﬁrst-order condition is also suﬃcient, and δSgiven by (2.16) is indeed the optimal

S-hedge strategy.

Proof of Theorem 2.9.From (B.2), we get

Var(∆VS(δ)) = a·(δ+b/2a)2−a·(b/2a)2+Var(∆S),

where a=n2Var(∆S) + m2Var(∆F)−2mnCov(∆S, ∆F) and b= 2nVar(∆S)−2mCov(∆S, ∆F).

Using the notations of aand b, we obtain

δS=−b

2aand Var(∆V(δS)) = Var(∆S)−a(δS)2.

Recall the deﬁnition of hedging eﬃciency in (2.20), the result of HE(δS) in (2.21) then follows. A

similar argument leads to HE(δL) in (2.21).

If ρ≥0, simple calculation yields

HE(δS)−HE(δL) = −4ρn√k(1 + n2k)

1−2nδ0+n2k)(1 + 2nδ0+n2k≤0.

The proof is now complete.

34