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Risk Structure and Optimal Hedging of Bitcoin Inverse Futures

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In the Bitcoin futures markets, the dominating contracts are inverse contracts. Unlike standard 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.
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Risk Structure and Optimal Hedging of Bitcoin Inverse Futures
Jun DengHuifeng PanShuyu ZhangBin Zou§
September 8, 2019
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:
School of Banking and Finance, University of International Business and Economics, Beijing, China. Email:
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:
§Department of Mathematics, University of Connecticut, Storrs, CT, USA. Email:
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
2See news report
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
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
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)t0and F= (Ft)t0the 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
define FB= (FB
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+∆tStand ∆Ft=Ft+∆tFtfor all t0.
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 t0in 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
one long position opened at tand closed at t+ ∆t, ∆FB
tis given by
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
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
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
360000104591-Futures-Account-Profit- Loss.
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/F01/Ft) for one
long position and (1/Ft1/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
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.
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.
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+∆tFt
R:= St+∆tSt
R:= FB
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
(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
Theorem 2.5. We obtain the time tconditional expectation and variance of RBand Rby
Et(RB) = 1
Et(R) = St
F) +
Vart(RB) = 1
F, R1+j
Vart(R) = S2
F, R1+j
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(RF) + Ethe
tVart(RF)2Covt(RF, R2
F):= d
R·RF) + Vart(e
R·RF) := d
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
Rboth contribute to the risk (variance) of inverse futures’ return, and these factors are intertwined.
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
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
Table 3: Positions of S-hedge and L-hedge (unit in BTC)
Spot Futures Margin Account
S-hedge 1 δ δ qK
L-hedge -1 δ δ qK
VLthe corresponding P&L, denominated in USD. We derive ∆VSby
VS=1 + δqK
'1 + δqK
FtStδK St
where we have used the approximation St+∆t/Ft+∆t'St/Ftto derive the second (approximation)
equality.5Similarly, we obtain ∆VLby
Ft1St+δK St
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
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
in the rest of this section.
Theorem 2.8. We obtain optimal strategies δSand δLto Problem 2.7 by
qand δL=δ0+δL
where δ0,δS
qand δL
qare defined by
δ0=ρk, δS
120+n2k, δL
with parameters given by
ρ=Cov(∆S, F)
pVar(∆S)Var(∆F), k =Var(∆S)
m2Var(∆F), m =KS
F2, n =qK
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
markets, we expect that
q>0 and δL
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
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 20+n2k)(δS)2and HE(δL) = 1
k(1 + 20+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 ρ0.(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
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
Table 4: Information of Data Used in Empirical Studies
Exchange Data Source Time Range
Inverse Futures OKEx From 2018/10/07 to 2019/07/26
Standard Futures CBOE From 2017/12/12 to 2019/06/19
Standard Futures CME From 2017/12/18 to 2019/07/03
Bitcoin Spot OKEx & Bitfinex 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 ∆SF
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
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
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).
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
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)
Deleverage Effect (DE) = max(Vart(RF)Vart(R),0)
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)
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
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(t100), 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
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
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
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.
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
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.
Figure 4: Model Predicted Hedging Efficiency of Optimal Strategies
Notes. Model predicted hedging efficiency HE(δS) and HE(δL) are calculated using (2.21) under two margin levels
q= 0.05 and q= 0.1. The left panel is for OKEx and the right for Bitfinex.
Figure 5: Optimal Short and Long Hedging Positions in Futures
Notes. The hedging strategies δS,δLand δ0are given by (2.16) and (2.17). We use the full available historical data
as the in-sample data to estimate them.
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: and
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
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:
Appendix B Proofs
Proof of Theorem 2.5.Using definitions (2.3) and (2.5), we get
Ft(Ft+ ∆Ft)=1
1 + RF
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
St+ ∆St
Ft+ ∆Ft
St/Ft+ ∆St/Ft
1 + RF
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 + )∆SF)
= (1 + )2Var(∆S) + m2δ2Var(∆F)2(1 + )Cov(∆S, F)
=δ2(n2Var(∆S) + m2Var(∆F)2mn Cov(∆S, F))
+ 2δ(nVar(∆S)mCov(∆S, F)) + Var(∆S).(B.2)
The first-order condition gives
δS=mCov(∆S, F)nVar(∆S)
n2Var(∆S) + m2Var(∆F)2mn Cov(∆S, F)
where δ0=ρkand 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)2a·(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
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 ρ0, simple calculation yields
HE(δS)HE(δL) = 4ρnk(1 + n2k)
120+n2k)(1 + 20+n2k0.
The proof is now complete.
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