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Optimal Hedging with Margin Constraints and Default Aversion

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Optimal Hedging with Margin Constraints and Default Aversion Optimal Hedging with Margin Constraints and Default Aversion and its Application to Bitcoin Perpetual Futures Carol Alexandera, Jun Dengb, and Bin Zouc aUniversity of Sussex Business School and Peking University HSBC Business School. Jubilee Building, University of Sussex, Falmer, Brighton, BN1 9RH, UK. Email: [email protected]. bSchool of Banking and Finance, University of International Business and Economics. No.10, Huixin Dongjie, Chaoyang District, Beijing, 100029, China. Email: [email protected]. cDepartment of Mathematics, University of Connecticut. 341 Mansfield Road U1009, Storrs, Connecticut 06269-1009, USA. Email: [email protected]. This Version: January 6, 2021 Abstract We consider a futures hedging problem subject to a budget constraint that limits the ability of a hedger with default aversion to meet margin requirements. We derive a semi-closed form for an optimal hedging strategy with dual objectives – to minimize both the variance of the hedged portfolio and the probability of forced liquidations due to margin calls. An empirical analysis of bitcoin shows that the optimal strategy not only achieves superior hedge effectiveness, but also reduces the probability of forced liquidations to an acceptable level. We also compare how the hedger’s default aversion impacts the performance of optimal hedging based on minute-level arXiv:2101.01261v1 [q-fin.RM] 4 Jan 2021 data across major bitcoin spot and perpetual futures markets. Keywords: Cryptocurrency; Futures Hedging; Leverage; Perpetual Swap; Extreme Value Theo- rem JEL Classification: G32; G11 1 1 Introduction The cryptocurrency bitcoin is the most volatile of all financial assets held by institutional in- vestors. From 7 to 12 March 2020 its spot price fell by 60%, its 30-day implied volatility index almost touched 200% and by 31 March 2020 its 30-day statistical volatility had risen to about 170%.1 This extremely high price uncertainty imposes tremendous risk to various participants in the cryptocurrency markets and generates a strong hedging demand by investors with long posi- tions in bitcoin. These include, but are certainly not limited to: bitcoin miners such as AntPool;2 centralised and decentralised cryptocurrency exchanges like Coinbase and Uniswap; businesses and retailers that accept bitcoin as payment, e.g. Microsoft and Wikipedia; and individuals and insti- tutions who hold bitcoin in their asset portfolios.3 It is common to hedge long (short) spot price risk by buying (selling) futures contracts on the same asset, or one that is very highly correlated; see Lien and Tse (2002). There is by now a huge literature on hedging with futures, examining the use of commodities, currency, interest rates, stocks, and indexes futures as effective hedges of spot price risk. We refer to Ederington (1979) and Figlewski (1984a) for classical contributions, and Lien and Yang (2008), Mattos et al. (2008), Acharya et al. (2013), Cifarelli and Paladino (2015), Wang et al. (2015), Billio et al. (2018), and Xu and Lien (2020) for more recent works. Bitcoin futures were introduced in December 2017 and by November 2020 their monthly trading volume had reached $1.32 trillion notional.4 Nevertheless, at the time of writing there is little in- depth research into the role that bitcoin futures can play to hedge bitcoin spot risk. The two papers that study similar contracts to us agree that they are excellent hedging instruments to segmented bitcoin spot markets: Alexander et al. (2020) show that BitMEX perpetual contracts achieve outstanding hedge effectiveness against the spot risk on Bitstamp, Coinbase, and Kraken. Deng et al. (2020) show that OKEx inverse quarterly contracts are excellent hedging tools for the spot risk on Bitfinex and OKEx, and are superior to CME standard futures in terms of hedge effectiveness. However, neither of these studies accounts for the impact of margin constraints and default aversion on hedging performance.5 There are several reasons why the hedging problem is more challenging for bitcoin than it is for other assets: first, many different types of bitcoin futures are traded on multiple exchanges, and the best choice of product and exchange is a problem not often considered for traditional assets; 1Details of the bitcoin implied volatility index can be accessed here and the statistical volatility figure is from the Forbes report which can be accessed here. 2Antpool is currently one of the largest of several very large mining cooperatives that aim to optimize bitcoin revenue for their members. See Alsabah and Capponi (2020) and Cong et al. (2020a) for further analysis on the centralisation effects of these large pools. 3For example, Satoshi Nakamoto, Roger Ver, Charlie Shrem, and the institutions listed in this Forbes report. 4See the CryptoCompare December 2020 Exchange Review here. 5Deng et al. (2020) claim that margins have little effect on the trading profit and loss and they implicitly assume that the hedger has an infinite supply of bitcoin to meet margin calls. 2 secondly, a key feature of bitcoin futures is the way that different exchanges apply margin calls to different products, so we need to model margin constraints when making our choice;6 and thirdly, most exchanges are at most lightly regulated and consequently the risk of default is far higher than it is on standard derivatives exchanges – so the hedger’s aversion to default needs to be part of the mathematical problem. The methodological contribution of this paper is to provide a semi-closed solution to a hedging problem that is entirely new to the finance literature, and which incorporates two key features of bitcoin markets, i.e. margin constraints and default aversion. The margin constraint imposes a positive probability that the hedger’s futures positions will be forced to liquidate during the hedge horizon, and such forced liquidations are seen as default events. The hedger seeks an optimal hedging strategy with dual objectives to minimize the risk of the hedged portfolio and the default probability. We apply the extreme value theorem to obtain the optimal strategy and show that both margin constraint and default aversion play a key role in determining its characteristics. We consider a hedger who has limited resources to meet the margin requirement while trading bitcoin futures, and hence is subject to an upper margin constraint. A second key novelty in our model is to characterize the hedger as a default averse agent. We define a default event for futures traders as the circumstance that they cannot raise enough bitcoin (or fiat currency) to meet the margin calls, i.e. a default event happens upon a forced liquidation. A hedger is called default averse if she views default as an undesirable event. There are at least two theoretical reasons to consider a default averse hedger. First, a default (forced liquidation) is an unexpected and adverse event to hedgers, causing negative impact on their financial goals. This is certainly the case for those who use futures as a hedging instrument. An early liquidation means a failure to hedge the spot risk for the chosen horizon (ending with a naked spot position). Second, when the above defined default happens, hedgers fail to pay liabilities in full amount to their counterpart in futures trading. This hurts their creditworthiness immediately and may further cause exchanges to require higher collateral or enforce more strict policies in their future trading. Including default aversion is also a natural choice in our model; because of the already imposed margin constraint, the default probability is strictly positive as long as hedgers trade futures. An empirical analysis considers different bitcoin spot exchanges and perpetual futures contracts. This way, we investigate three important topics in risk management: hedge effectiveness, default probability, and leverage. Since bitcoin markets are highly segmented, we compare results for the three most liquid bitcoin perpetual futures, BitMEX, Deribit, and OKEx, and three choices for bitcoin spot prices, i.e. the .BXBT index on BitMEX, and the two largest spot markets, Bitstamp and Coinbase. Our main empirical findings are threefold: (1) A tighter margin constraint or a 6Margin mechanisms are essential for ensuring the stability and integrity of futures markets, and there is abundant research on this topic for traditional futures like commodities, precious metals, stock indexes, and currencies; see Figlewski (1984b), Longin (1999), Cotter (2001), Daskalaki and Skiadopoulos (2016), Alexander et al. (2019) and many others. 3 higher degree of default aversion decreases the hedger’s positions in bitcoin futures, justifying the incorporation of these two features in the analysis; (2) The optimal strategy leads to a highly efficient and robust hedging performance across the considered spot and futures exchanges for most hedgers, with exception only for those with extremely tight constraint and high default aversion; (3) By following the optimal strategy, the hedger is able to control the default probability at a reasonable level, close to or less than 1% in all scenarios examined, which is also robust to the choice of bitcoin futures and spot markets. In the following: Section 2 provides the background for this study, describing the trading charac- teristics of different types of bitcoin futures on various exchanges; Section 3 formulates the optimal hedging problem under both margin constraint and default aversion, and derives the optimal hedg- ing strategy; Section 4 summarizes the bitcoin spot and futures data
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