AAltcoin-Bitcoin Arbitrage
Zura Kakushadze §† and Willie Yu (cid:93) § Quantigic (cid:114)
Solutions LLC1127 High Ridge Road † Free University of Tbilisi, Business School & School of Physics240, David Agmashenebeli Alley, Tbilisi, 0159, Georgia (cid:93)
Centre for Computational Biology, Duke-NUS Medical School8 College Road, Singapore 169857 (January 27, 2019)
Abstract
We give an algorithm and source code for a cryptoasset statistical arbitragealpha based on a mean-reversion effect driven by the leading momentum factorin cryptoasset returns discussed in https://ssrn . com/abstract=3245641 .Using empirical data, we identify the cross-section of cryptoassets for whichthis altcoin-Bitcoin arbitrage alpha is significant and discuss it in the contextof liquidity considerations as well as its implications for cryptoasset trading. Zura Kakushadze, Ph.D., is the President of Quantigic (cid:114)
Solutions LLC, and a Full Professorat Free University of Tbilisi. Email: [email protected] Willie Yu, Ph.D., is a Research Fellow at Duke-NUS Medical School. Email: [email protected] DISCLAIMER: This address is used by the corresponding author for no purpose other thanto indicate his professional affiliation as is customary in publications. In particular, the contentsof this paper are not intended as an investment, legal, tax or any other such advice, and in noway represent views of Quantigic (cid:114)
Solutions LLC, the website . quantigic . com or any of theirother affiliates. a r X i v : . [ q -f i n . P M ] A p r Introduction
There is a sizable proliferation of cryptoassets, whose number according to https://coinmarketcap . com was 2,116 as of January 18, 2019. Just as with stocks (andother asset classes), there appear to be underlying common factors for cryptoassetreturns, at least on shorter-horizons [Kakushadze, 2018]. Thus, the leading commonfactor in the daily close-to-close cryptoasset returns is the prior day’s momentum(“mom”), and on average the subsequent open-to-close return is negatively corre-lated with mom. So, there is a mean-reversion effect in daily cryptoasset returns.Can we utilize this mean-reversion effect to construct a trading signal (alpha)?The mean-reversion here is cross-sectional, so such an alpha would (ideally) involvea sizable cross-section of cryptoassets. In the case of stocks one can construct adollar-neutral mean-reversion strategy by going long a sizable number of stockswith the investment level I L and simultaneously shorting a sizable number of stockswith the investment level I S , with I S = I L (so we have dollar-neutrality), which is astandard long-short statistical arbitrage strategy. Alternatively, one can go long asizable number of stocks with the investment level I L and simultaneously short anindex futures with the investment level I S (again, I S = I L ), e.g., S&P 500 futures,in which case we have a dollar-neutral so-called S&P 500 outperformance strategy.We can attempt to do something similar with cryptoassets. However, shortinga sizable number of cryptoassets is not practicable. We can short Bitcoin futures instead. The long position then can be in a sizable number of cryptoassets otherthan Bitcoin, i.e., altcoins. We then have a Bitcoin outperformance strategy, whichwe refer to as altcoin-Bitcoin arbitrage. So, the idea is simple. We maintain a shortBitcoin position with the investment level I S (so we do not trade Bitcoin). The longposition consists of a cross-section of altcoins, which changes daily based on theirmom values, so we either establish or liquidate long altcoin positions, but nevershort them. We discuss some simple trading rules for altcoin positions in Section 2.The next question is whether the alpha is “tradable”. The main considerationsare transaction costs and liquidity. In this note we focus on liquidity considera- For the purposes of this note, cryptoassets include digital cryptography-based assets such ascryptocurrencies (e.g., Bitcoin), as well various other digital “coins” and “tokens” (minable as wellas non-minable), that have data on https://coinmarketcap . com . [Kakushadze, 2018] extends short-horizon equity factors [Kakushadze, 2015] to cryptoassets. For a discussion on Bitcoin futures, see, e.g., [Hale et al , 2018]. For some cryptoasset investment and trading related literature, see, e.g., [Alessandretti et al ,2018], [Amjad and Shah, 2017], [Baek and Elbeck, 2014], [Bariviera et al , 2017], [Bouoiyour et al ,2016], [Bouri et al , 2017], [Brandvold et al , 2015], [Bri`ere, Oosterlinck and Szafarz, 2015], [Cheahand Fry, 2015], [Cheung, Roca and Su, 2015], [Ciaian, Rajcaniova and Kancs, 2015], [Colianni,Rosales and Signorotti, 2015], [Donier and Bouchaud, 2015], [Dyhrberg, 2015], [Eisl, Gasser andWeinmayer, 2015], [ElBahrawy et al , 2017], [Gajardo, Kristjanpoller and Minutolo, 2018], [Garciaand Schweitzer, 2015], [Georgoula et al , 2015], [Harvey, 2016], [Jiang and Liang, 2017], [Kim etal , 2016], [Kristoufek, 2015], [Lee, Guo and Wang, 2018], [Li et al , 2018], [Liew and Hewlett,2017], [Liew, Li and Budav´ari, 2018], [Nakano, Takahashi and Takahashi, 2018], [Ortisi, 2016], [Shahand Zhang, 2014], [Van Alstyne, 2014], [Vo and Yost-Bremm, 2018], [Wang and Vergne, 2017]. Tables and figures summarize our backtesting results.
Cryptoassets trade continuously, 24/7. Thus, “open” on any given day means theprice right after midnight (UTC time), while “close” on any given day means theprice right before midnight (UTC time), so the open on a given day is almost thesame as the close of the previous day. All prices (open, close, high, low), volume andmarket cap are measured in dollars. The index i = 1 , . . . , N labels N different cryp-toassets cross-sectionally, while the index s = 0 , , , . . . labels the dates, with s = 0corresponding to the most recent date in the time series. So: P Cis (or, equivalently, P Ci,s ) is the close price for the cryptoasset labeled by i on the day labeled by s ; P Ois is the open price; P His is the high price; P Lis is the low price; V is is the daily dollarvolume; C is is the market cap. All our data was freely downloaded (see below).We define daily open-to-close log-returns (or “continuously compounded” re-turns), which we use to define the mom factor as in [Kakushadze, 2018] (see below): R is = ln (cid:0) P Cis /P Ois (cid:1) (1)For small values it is approximately the same as the standard (“single-period”)return defined as (cid:101) R is = P Cis /P Ois − (cid:101) R is . There are various ways to define the momentum factor (mom). For our purposeshere, we will define it as in [Kakushadze, 2018]: β mom is = R i,s +1 (3)This definition is 100% out-of-sample: we use β mom is for trading on the date labeledby s , and β mom is is computed using quantities from the prior date labeled by s + 1. The source code given in Appendix A hereof is not written to be “fancy” or optimized forspeed or in any other way. Its sole purpose is to illustrate the algorithms described in the maintext in a simple-to-understand fashion. Some important legalese is relegated to Appendix B. Assuming no special circumstances such as trading halts. High, low and volume are measured between the open and the close of a given day. .3 Trading Signal The trading signal (alpha) α is for altcoins is defined as follows: α is = θ ( − β mom is ) (4)where the Heaviside function θ ( x ) = 1 if x >
0, and θ ( x ) = 0 if x ≤
0. So, weestablish a new long altcoin position, or maintain an existing long altcoin position,if mom is negative. If mom is nonnegative, then we do not establish a new altcoinposition, and liquidate an existing long altcoin position. All altcoin positions are longor null. Meanwhile, we continuously maintain a constant short Bitcoin position. Let us assume that the constant short Bitcoin position has the investment level I S . Let the total investment level of our long altcoin position be I L . To have adollar-neutral portfolio, we must set I L = I S . Let H is be the individual altcoindollar holdings. Let us use the labels i = 2 , . . . , N for altcoins, while i = 1 willcorrespond to Bitcoin. We can define the altcoin weights as w is = H is /I L . Then wehave ( w is ≥ N (cid:88) i =2 w is = 1 (5)The simplest choice for the weights is to have equal weights for all altcoins withnonzero signals: w is = 1 n s α is (6) n s = N (cid:88) i =2 α is (7)Other weighting schemes are possible, e.g., by suppressing the weights by volatility: w is = γ s α is σ is (8) w is = (cid:101) γ s α is | β mom is | σ is (9) . . . (10)Here σ is is historical volatility (e.g., the standard deviation of the returns R is (cid:48) computed over d previous days s (cid:48) = s + 1 , . . . , s + d , or the hlv factor definedin [Kakushadze, 2018], etc.), while the normalization coefficients γ s , (cid:101) γ s are fixed viaEq. (5). In our backtests (see below) we focus on equally weighted portfolios (6). Technically, this should be short Bitcoin futures, but we assume a short Bitcoin position. Backtests
We downloaded the data from https://coinmarketcap . com for all 2,116 cryptoas-sets as of January 19, 2019 (so the most recent date in the data is January 18, 2019),and 2,115 cryptoassets had downloadable data, albeit for many various fields werepopulated with “?”, which we converted into NAs. In our backtests (see below),we only kept cryptoassets with non-NA price (open, close, high, low), volume andmarket cap data, with an additional filter that no null volume was allowed either. Cboe Bitcoin Futures (symbol XBT) started trading on December 10, 2017.So, technically speaking, backtesting the strategy before that time might not beparticularly meaningful. Still, although we primarily focus on the 1-year backtest(looking back from January 18, 2019), for comparison and completeness purposeswe also run 2-, 3-, 4- and 5-year lookback backtests. For the 1-year backtest we have417 cryptoassets with historical data, while for the 2-, 3-, 4- and 5-year backtestswe have 121, 67, 44 and 13 cryptoassets, respectively. For the 1-year backtest wefurther break up the universe of the 416 altcoins (which together with Bitcoin makeup the aforesaid 417 cryptoassets) into market cap tiers A,B,C,D,E,F. Thus, thealpha based on tier A on any given day goes long only the altcoins whose marketcap on the previous day ranks 2 to 30 among all cryptoassets. Similarly, for tiersB,C,D,E,F the corresponding market cap rank ranges are: 31-60, 61-100, 101-200,201-300, 301-417. In fact, based on our results (see below), running a backtest forthe full universe of all 416 altcoins would obscure the liquidity effect (see below).
For various universes mentioned above, Table 1 summarizes the market cap, the20-day average daily volume, and the daily “turnover” defined as the market cap R source code for data downloads is given in Appendix A of [Kakushadze, 2018]. This code stillworks with two minor tweaks. First, the line u <- c(x[22:28]) in the function crypto.data() now reads u <- c(x[22:27]) due to a formatting change on https://coinmarketcap . com . Sec-ond, in the function crypto.hist.prc() , right after the line shared.write.table(x, file,T) , one should now include the following (or similar) line: Sys.sleep(max(rnorm(1, 10,2), 5)) (which spaces downloads at random intervals). This is due to the apparent changeat https://coinmarketcap . com , which averts “rapid-fire” (i.e., continuous serial) downloads. This is to avoid stale prices. Further, 2 cryptoassets had apparently “artifact” stale pricesduring some periods, so they were also excluded from the corresponding backtests (see below). Which is the actual investment vehicle we implicitly assume for our short Bitcoin position. Especially considering the effect Bitcoin futures arguably had on Bitcoin (and other cryp-toassets) – see, e.g., [Hale et al , 2018]. For a nontechnical discussion, see, e.g., [Kelleher, 2018]. These counts include Bitcoin. Also, these counts exclude the aforesaid 2 cryptoassets withapparently “artifact” stale prices during some periods. Finally, in 2-year and longer backtests thecryptoasset “Circuit of Value Coin” (symbol COVAL) is excluded as it had an extraordinarily largepositive return in a short time period and including it would misleadingly “rosy-up” the results. we getROC = -26.56% and Sharpe = -1.34. Conversely, if we reverse the signal for, say,the universe 1A, we do not get a positive return. Therefore, the altcoin-Bitcoinarbitrage alpha appears to be a real effect owing to low liquidity of the altcoins forwhich it is present. Put differently, the alpha exists as it cannot be arbitraged away. So, the altcoin-Bitcoin arbitrage alpha we discuss above is essentially is “a low-liquidity premium” in altcoin returns. In practice, to arbitrage it, one must accountfor trading costs – both transaction costs and market impact. For low-liquidityaltcoins the market impact can quickly become prohibitive when attempting toexecute sizable trades. In fact, for the 2-year, 3-year, 4-year and 5-year lookbacks(where the number of altcoins with historical data is smaller) the na¨ıve pickup inthe performance is due to the fact that most of the altcoins in these universes, eventhough they have been around for a while, are lower-cap, lower-liquidity cryptoassets(which is a telltale sign for persistence), with the notable exception of XRP (Ripple),which is what “Max” in the market cap corresponds to in Table 1 for these universes.Can the altcoin-Bitcoin arbitrage alpha – or the momentum indicator on whichit is based – be useful outside outright arbitraging it (which may be challenging dueto the liquidity considerations)? Perhaps. In a sideways cryptoasset market, thisindicator could be used as a guide for executing trades for lower-liquidity altcoinsin other contexts. Thus, statistically, we expect that there is a mean-reversioneffect, and if its yesterday’s momentum is positive, today an altcoin (on average)is expected to trade lower, and if its yesterday’s momentum is negative, today saidaltcoin (on average) is expected to trade higher. This can then conceivably be usedas a “shorter-horizon” (daily) execution signal for longer-horizon trades. It should bementioned, however, that the aforesaid alpha is a statistical effect, which is expectedto work better for a sizable cross-section of altcoins, so using it as a “shorter-horizon”execution signal for such sizable cross-sections of altcoins would make more sensethan for a single (or a few) altcoin(s). In this regard, let us mention [Liew, Liand Budav´ari, 2018], whose conclusion is that forecasting (using machine learningtechniques) short-horizon single-cryptoasset returns (for the top 100 cryptoassets bymarket cap) appears to be challenging. This also bodes well with our findings here. In this case we have a quasi-static (the altcoin universe can change with market cap fluctua-tions) dollar-neutral portfolio, which is obtained by setting α is ≡ i = 2 , . . . , N ) in Eq. (6). R Source Code: Trading Signal
In this appendix we give R (R Project for Statistical Computing, . r-project . org/ ) source code for computing the altcoin-Bitcoin arbitrage trading sig-nal. The sole function crypto.arb() reads the aggregated data files cr.prc.txt (close price), cr.open.txt (open price), cr.high.txt (high price), cr.low.txt (lowprice), cr.vol.txt (dollar volume), cr.cap.txt (market cap), cr.name.txt (namesof the cryptoassets in the same order as all the other files), and cr.mnbl.txt (1 ifthe name is minable, otherwise 0) generated by the function crypto.prc.files() of [Kakushadze, 2018] (also, see fn.12 hereof). Internally, crypto.arb() computesthe av (average volume), size (market cap), mom (momentum), hlv (intraday volatil-ity) factors of [Kakushadze, 2018] and the trading signal based on mom (togetherwith the trading universe based on size ). The inputs of crypto.arb() are days (the length of the selection period used in fixing the cryptoasset universe by apply-ing the aforesaid non-NA data and non-zero volume filters, which period is further“padded” – see below), back (the length of the skip period, i.e., how many daysto skip in the selection period before the lookback period), lookback (the lengthof the lookback period over which the backtest is run), d.r (the extra “padding”added to the selection period plus one day, so the moving averages can be computedout-of-sample; we take d.r = 20 ), d.v (the av moving average length; we take d.v= 20 ), d.i (the hlv moving average length; we take d.i = 20 ), ix.upper (the rankof the highest market cap (as of the previous day) altcoin to include in the tradinguniverse), and ix.lower (the rank of the lowest market cap (as of the previous day)altcoin to include in the trading universe). The function crypto.arb() internallycomputes and plots/outputs the daily P&L and annualized ROC and Sharpe ratio. crypto.arb <- function (days = 365, back = 0, lookback = days,d.r = 20, d.v = 20, d.i = 20, ix.lower = NA, ix.upper = 2) { read.prc <- function(file, header = F, make.numeric = T) { x <- read.delim(file, header = header)x <- as.matrix(x)if(make.numeric)mode(x) <- "numeric"return(x) } calc.mv.avg <- function(x, days, d.r) { if(d.r == 1)return(x[, 1:days]) In our backtests we always set back = 0 . Also note that mnbl and av are not used internally. <- matrix(0, nrow(x), days)for(i in 1:days)y[, i] <- rowMeans(x[, i:(i + d.r - 1)], na.rm = T)return(y) } prc <- read.prc("cr.prc.txt")cap <- read.prc("cr.cap.txt")high <- read.prc("cr.high.txt")low <- read.prc("cr.low.txt")vol <- read.prc("cr.vol.txt")open <- read.prc("cr.open.txt")mnbl <- read.prc("cr.mnbl.txt")name <- read.prc("cr.name.txt", make.numeric = F)d <- days + d.r + 1prc <- prc[, 1:d]cap <- cap[, 1:d]high <- high[, 1:d]low <- low[, 1:d]vol <- vol[, 1:d]open <- open[, 1:d]take <- rowSums(is.na(prc)) == 0 & rowSums(is.na(cap)) == 0 &rowSums(is.na(high)) == 0 & rowSums(is.na(low)) == 0 &rowSums(is.na(vol)) == 0 & rowSums(is.na(open)) == 0 &rowSums(vol == 0) == 0ret <- log(prc[take, -d] / prc[take, -1])ret.d <- prc[take, -d] / prc[take, -1] - 1prc <- prc[take, -1]cap <- cap[take, -1]high <- high[take, -1]low <- low[take, -1]vol <- vol[take, -1]open <- open[take, -1]mnbl <- mnbl[take, 1]name <- name[take, 1]if(back > 0) { ret <- ret[, (back + 1):ncol(ret)] et.d <- ret[, (back + 1):ncol(ret)]prc <- prc[, (back + 1):ncol(prc)]cap <- cap[, (back + 1):ncol(cap)]high <- high[, (back + 1):ncol(high)]low <- low[, (back + 1):ncol(low)]vol <- vol[, (back + 1):ncol(vol)]open <- open[, (back + 1):ncol(open)] } days <- lookbackav <- log(calc.mv.avg(vol, days, d.v))hlv <- (high - low)^2 / prc^2hlv <- 0.5 * log(calc.mv.avg(hlv, days, d.i))take <- rowSums(!is.finite(hlv)) == 0 { x <- -sign(mom[, i]) } pnl1 <- pnlpnl <- pnl[days:1]pnl <- cumsum(pnl)plot(1:days, pnl, type = "l",col = "green", xlab = "days", ylab = "P&L")roc <- round(mean(pnl1) * 365 / 2 * 100, 2) } B DISCLAIMERS
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100 200 300 − . − . − . − . . days P& L Figure 1: P&L (in the units where the short Bitcoin position is normalized to 1) forthe lookback 1A portfolio (see Table 2). 16
100 200 300 − . − . − . − . − . − . . days P& L Figure 2: P&L (in the units where the short Bitcoin position is normalized to 1) forthe lookback 1B portfolio (see Table 2). 17
100 200 300 − . − . . . days P& L Figure 3: P&L (in the units where the short Bitcoin position is normalized to 1) forthe lookback 1C portfolio (see Table 2). 18
100 200 300 − . − . − . . . days P& L Figure 4: P&L (in the units where the short Bitcoin position is normalized to 1) forthe lookback 1D portfolio (see Table 2). 19
100 200 300 . . . . . . days P& L Figure 5: P&L (in the units where the short Bitcoin position is normalized to 1) forthe lookback 1E portfolio (see Table 2). 20
100 200 300 days P& L Figure 6: P&L (in the units where the short Bitcoin position is normalized to 1) forthe lookback 1F portfolio (see Table 2). 21
200 400 600 days P& L Figure 7: P&L (in the units where the short Bitcoin position is normalized to 1) forthe lookback 2 portfolio (see Table 2). 22
200 400 600 800 1000 days P& L Figure 8: P&L (in the units where the short Bitcoin position is normalized to 1) forthe lookback 3 portfolio (see Table 2). 23
500 1000 1500 days P& L Figure 9: P&L (in the units where the short Bitcoin position is normalized to 1) forthe lookback 4 portfolio (see Table 2). 24