CRIX - a CRyptocurrency IndeX

Simon Trimborn Wolfgang Karl Härdle

Ladislaus von Bortkiewicz Chair of Statistics

C.A.S.E.  Center for Applied Statistics

and Economics HumboldtUniversität zu Berlin http://lvb.wiwi.hu-berlin.de http://www.case.hu-berlin.de Motivation 1-1 Currencies - Cigarettes, USD, Cryptos

Everything can be a currency Everyone can oer a currency

Figure 1: Cigarette trading in postwar germany (48) Figure 2: Fridrich A. Hayek (48) CRIX - a CRyptocurrency IndeX Motivation 1-2 Digital Economy

Amazon Paypal Google Wallet Cryptocurrencies Ripple

CRIX - a CRyptocurrency IndeX Motivation 1-3

Cryptocurrencies

Decentralized, virtual, low transaction costs

NYSE, Andreesen Horowitz, DFJ: Coinbase funding (75 M$) Nasdaq: company-wide utilization of blockchain technology Citigroup: own coin development

CRIX - a CRyptocurrency IndeX Motivation 1-4

Cryptocurrencies - Facts

As of 20151011, CoinMarketCap.com I 636 cryptos I 2,022 exchange pairs I Market Cap 4.1 billion USD

Barely derivatives

Commodity Futures Trading Commission (USA) I Cryptos are commodities

CRIX - a CRyptocurrency IndeX Motivation 1-5

Challenges

Investment perspective I What is the benchmark? I How does the market evolve? I Market index necessary to compare cryptos

Classication of market I Market risk measure I Risk transfer

CRIX - a CRyptocurrency IndeX Motivation 1-6 What is the benchmark? 3000 2000 Rescaled index value Rescaled index 1000 1995−10−09 1999−08−06 2003−06−06 2007−04−06 2011−02−04 2014−12−05 Figure 3: Wilshire 5000 Total MarketDate Index, S&P500, S&P100, rescaled to a starting value of 1000 CRIX rules CRIX - a CRyptocurrency IndeX Motivation 1-7 CRIX - the benchmark

Figure 4: Screenshot: crix.hu-berlin.de (20151001)

CRIX - a CRyptocurrency IndeX Outline

1. Motivation X 2. CRIX 3. Market Comparison 4. Avoidance of Risk 5. Forecasting 6. Conclusion 7. Appendix

CRIX - a CRyptocurrency IndeX Market Index 2-1

Data

233 cryptos Time period: 20140101 - 20150901 Prices, capitalization, volumes CoinGecko, Datastream S&P500, DAX, RTSI,... US Constant Maturity yield rate: 1 M, 3 M, 6 M, 1 Y

Bitcoin

CRIX - a CRyptocurrency IndeX Market Index 2-2 CRIX - Rules I Laspeyres' idea: Pk 1 MVjt · AWjt CRIX(k) = j= t Divisor : market capitalization of crypto MVjt j k: number of constituents CWjt AWjt = Wjt : capped weight CWjt MVjt Wjt = Pk j=1 MVjt Liquidity rule: I Eligible if higher rank than 0.25 percentile I Measure regarding daily volume in USD and coins

Add. Rules Liquidity Rule Unused Bitcoins CRIX - a CRyptocurrency IndeX Market Index 2-3

CRIX - Rules II

Spine I Index members I Crucial for benchmark t

min(k) CRIX (k)t → total markett

PJ MV ·AW total market j=1 jt jt I t = Divisor I Here: J = 233

CRIX - a CRyptocurrency IndeX Market Index 2-4 CRIX - Rules III

233 P MV ·AW 1. Construct total market index: total market j=1 jt jt t = Divisor 2. Set i = 1

3. Construct CRIX (ki ), i = 1, 2, 3,..., k1 < k2 < k3 < ···

4. Compute it = total markett − CRIX (ki )t

5. Kernel density estimation for density f (it ) with leave-one-out cross validation 6. Derive 2 log Qn log BICi = − t=1 f (it ) + ki (n) 7. If i = 1: Jump to 3., else 8.

8. If BICi−1 < BICi : stop, else jump to 3. and i = i + 1

KDE US indices

CRIX - a CRyptocurrency IndeX Market Index 2-5

Index members

Compare last 3 M Amount used for next 3 M k1 = 5 Step width: 5 constituents

Period 1 2 3 4 5 6 # constituents 25 10 10 15 15 20 # cryptos 34 38 119 172 199 213

Table 1: # CRIX constituents and all cryptos Index Members

CRIX - a CRyptocurrency IndeX Market Index 2-6 CRIX performance 1000 800 600 CRIX value 400 200

2014−02−01 2014−08−01 2015−02−01 2015−08−01 Date Figure 5: CRIX CRIXindex CRIX - a CRyptocurrency IndeX Market Comparison 3-1

CRIX vs. DAX 1.5 1.0 0.5 Annualized conditional volatility in % conditional volatility Annualized 0.0 2014−02−01 2014−06−01 2014−10−01 2015−02−01 2015−06−01 Date Figure 6: Annualized conditional volatility in % of CRIX and DAX

Methodology GARCH parameters Rolling window GARCH parameters

CRIX - a CRyptocurrency IndeX Market Comparison 3-2 CRIX and the world I 1400 1000 800 Portfolio value Portfolio 600 400 200

2014−02−01 2014−08−01 2015−02−01 2015−08−01 Date Figure 7: CRIX, S&P500, DAX, STI, RTSI, ATHEX CRIXoutmarket

CRIX - a CRyptocurrency IndeX Market Comparison 3-3 Expected Shortfall (ES)

Xt = µt + σt Zt i.i.d. with Xt ∼ (µt , σt ), Zt ∼ (0, 1)

E [Xt |Xt < xt,α] with xt,α the α-quantile

e.g. log returns of CRIX Xt Apply GARCH(1,1), obtain Zt

Tails follow Generalized Pareto Distribution (GPD) CRIX - a CRyptocurrency IndeX Market Comparison 3-4 CRIX and the world II

EVT CRIX -0.1579 SP500 -0.0528 DAX -0.0700 RTSI -0.1343 ATHEX -0.1288 CSI -0.1464 EUR to USD -0.0223 RUB to USD -0.0637

Table 2: Extreme Value Theory based ES at α = 0.01 and with threshold ut = 0.1 for CRIX SP500 DAX RTSI ATHEX CSI EUR to USD RUB to USD.

CRIX - a CRyptocurrency IndeX Avoidance of Risk 4-1

Options

Options to hedge risk

Price tag for risk avoidance

CRIX options more expensive than other markets?

Computed with Heston Nandi model

Just dependent on asset data

Heston Nandi Options

CRIX - a CRyptocurrency IndeX Avoidance of Risk 4-2 Options

KCRIX KS&P500 τ 100 150 200 250 100 150 200 250 30 0.01 0.79 9.24 35.52 0.00 0.00 0.01 22.80 90 0.56 6.10 22.66 51.10 0.00 0.00 0.30 23.48 180 3.00 14.44 36.07 66.26 0.00 0.00 1.18 24.94 365 9.91 28.64 55.39 87.99 0.00 0.03 3.97 28.93

Table 3: Put prices (European options) for CRIX and adjusted S&P500 at 20150901 both with a value of 227.22 with dierent Strikes (K) and Time to Maturities (τ) with US r CRIXhnoptions

CRIX options much more expensive Expensive to hedge risk CRIX - a CRyptocurrency IndeX Conclusion 5-1

Conclusion

CRIX represents market perfect

Crypto market much riskier than other markets

Slightly riskier than known risky market (Russia)

CRIX - a CRyptocurrency IndeX CRIX - a CRyptocurrency IndeX

Simon Trimborn Wolfgang Karl Härdle

Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E.  Center for Applied Statistics

and Economics

HumboldtUniversität zu Berlin http://lvb.wiwi.hu-berlin.de http://www.case.hu-berlin.de

6-1

Bibliography

Simon Trimborn and Wolfgang Karl Härdle CRIX or evaluating Blockchain based currencies

CRIX - a CRyptocurrency IndeX Appendix 7-1

Bitcoin

Counteract ination I Fixed max amount Anonymity Needs of users I Decision on structure

Movie: Bitcoin - Made simple

CRIX - a CRyptocurrency IndeX Appendix 7-2

Anonymity - Black market

Wallets are anonym Transactions are anonym Blockchain core feature Causes problems Figure 8: US government warning Source: www.wikipedia.org

CRIX - a CRyptocurrency IndeX Appendix 7-3 The Blockchain - Spine of Bitcoin

Transaction list

Transaction processors called miners

Miners collect & publish transactions

Order is invariable

Figure 9: Spine CRIX - a CRyptocurrency IndeX Appendix 7-4 The Blockchain

Sometimes parallel chains I Due to e.g. internet lag Green block: rst block (Genesisblock)

Black blocks: main chain Purple blocks: parallel chains Figure 10: Blockchain Source: www.wikipedia.de CRIX - a CRyptocurrency IndeX Appendix 7-5

The Blockchain - Lag

Assume: 2 blocks mined simultaneously I Miner 1: Australia I Miner 2: Germany Eect of lag: I Some receive australian block I Some receive german block Parallel chain For next block: I Check which chain contains the most dicult to nd blocks I Becomes main chain

CRIX - a CRyptocurrency IndeX Appendix 7-6

Process of Transactions

Users organize process Some users (miners) create transaction list I Next block of blockchain Blocks have strict order, ensured by signature Miners search for signature Signature encrypted by cryptography

Transaction example

CRIX - a CRyptocurrency IndeX Appendix 7-7

Who accepts Bitcoin?

Overstock - Retailer Dell University of Cumbria Expedia - Travel booking agency Republican Party of Louisiana

back: Index Construction

CRIX - a CRyptocurrency IndeX Appendix 7-8

Bitcoin - The System I

Take 4 people I Alice, Bob I Gary, Grace 2 special users (miners) I Gary I Grace Alice buys a rare book from Bob and pays with Bitcoin Gary and Grace process this transaction

back

CRIX - a CRyptocurrency IndeX Appendix 7-9 Bitcoin - The System II

back

Gary

Alice Bob

Grace

block CRIX - a CRyptocurrency IndeX Appendix 7-10

Bitcoin - The System III

Gary OR Grace receive Bitcoins for service BOTH add transaction to list BOTH compute hash value (trial and error) Click for online hash generator List: one block of the blockchain Hash value: gives position of block in blockchain Contains part of hash value of last block

back

CRIX - a CRyptocurrency IndeX Appendix 7-11

CRIX - Add. Rules

High volatility: weights recalculated 1 M

Maximum weight of CRIX member: 50 %

Crypto made insensitive if trading stops

Back

CRIX - a CRyptocurrency IndeX Appendix 7-12

Liquidity Rule I

Rely often on turnover Volume Turnover = Floating Coins

Floating Coins for cryptos unclear Rule motivated by STOXX Japan 600 & AEX Family Measure relative to asset universe Small trading volume in USD but high traded coins taken into account

back

CRIX - a CRyptocurrency IndeX Appendix 7-13

Liquidity Rule II

Liquidity rule (one of these): 1.0 .25 percentile of ADTV (Average Daily Trading Volume):

ADTVi ≥ ADTV0.25

2.0 .25 percentile of ADRTC (Average Daily Relative Traded Coins):

ADRTCi ≥ ADRTC0.25

Checked every month

back

CRIX - a CRyptocurrency IndeX Appendix 7-14 Usage of Bitcoins

Percentage of last time a coin of Bitcoin was used 1.0 0.8 0.6 0.4 Percentage 0.2 0.0 2009−01−03 2011−01−03 2013−01−03 2014−11−16 Date Figure 11: one day, one week, one month, 1-3 month source: John Radcli back CRIX - a CRyptocurrency IndeX Appendix 7-15

Methodology

GARCH(1,1) 2 2 2 σt = α0 + αεt−1 + βσt−1

Full variance for time horizon T T 2 X 2 σT = σt t=1

back

CRIX - a CRyptocurrency IndeX Appendix 7-16

GARCH parameters

alpha0 alpha beta CRIX 0.00007572 0.08457491 0.88392853 DAX 0.00000257 0.09926597 0.89304179

Table 4: Parameters of the GARCH(1,1) models without mean for CRIX and DAX

back

CRIX - a CRyptocurrency IndeX Appendix 7-17 Rolling window GARCH parameters −0.002 −0.005 mean of log residuals 0 50 100 150 Index 0.8 0.4 Value of parameter Value 0.0 0 50 100 150 Index Figure 12: Upper plot: Mean of CRIX log residuals in 250 data windows, Lower plot: α, β GARCH(1,1) parameters for 250 data points rolling win- dows back CRIX - a CRyptocurrency IndeX Appendix 7-18 Index members

Period 1 2 3 4 5 6 7 8 9 10 1 btc ltc msc nxt nmc doge mec wdc ftc zet 2 btc xrp ltc ppc doge nxt msc nmc pts qrk 3 btc xrp ltc ppc doge pts xpm mec vtc ybc 4 slr blk pts hbn pot prt e zeit cbx fair 5 slr pts safe rdd prt grcx pnd cbx karm mona 6 btc ltc nxt ppc xc zet ftc mec ifc ybc 7 btc ltc nxt xrp ppc dash doge nmc msc blk 8 btc ltc xrp bts nxt ppc dash doge nmc safe 9 btc ltc xrp bts nxt doge ppc dash nmc safe 10 btc xrp ltc bts doge nxt ppc dash nmc safe 11 btc xrp ltc bts doge nxt ppc xcp safe dash 12 btc xrp ltc bts safe doge nxt str ppc xcp 13 btc mtc xrp ltc xpy bts safe str doge nxt 14 btc xrp ltc bts doge sync safe dash xpy str 15 btc xrp ltc bts dash doge nxt safe str xpy 16 btc xrp ltc dash bts str doge safe nxt banx 17 btc xrp ltc dash str doge bts nxt safe banx 18 btc xrp ltc bts doge str dash nxt banx ppc

Table 5: First 10 CRIX constituents in the respective periods back CRIX - a CRyptocurrency IndeX Appendix 7-19

Heston Nandi GARCH(1,1) Option Pricing model for European Options

Log returns follow process: Xt 1 2 (1) Xt = rf + (µ − 2)σt + σt Zt with risk free rate, 0 1 , the mean and 2 a rf Zt ∼ N( , ) µ σt GARCH(1,1) process.

GARCH(1,1) process: 2 2 2 (2) σt = ω + βσt−1 + α(Zt−1 − γσt−1)

back

CRIX - a CRyptocurrency IndeX Appendix 7-20

Risk neutrality

Transform (1) and (2) Receive risk neutral versions 1 2 ∗ Xt = rf + (µ − 2)σt + σt Zt and

2 2 ∗ ∗ 2 σt = ω + βσt−1 + α(Zt−1 − γ σt−1)

with ∗ and ∗ . Zt = Zt + µσt γ = γ + µ

back

CRIX - a CRyptocurrency IndeX Appendix 7-21 Log-linear generator function

θ exp 2
f (θ) = St At;T ,θ + Bt;T ,θσt+1 where 1 log 1 2 At;T ,θ = At+1;T ,θ + θrf + Bt+1;T ,θω − 2 ( − αBt+1,T ,θ) 2 1 1 2 1/2(θ − γ) Bt;T ,θ = θ(µ − + γ) − γ + βBt+1;T ,θ + 2 2 1 − 2αBt+1;T ,θ with terminal conditions

AT ;T ,θ = 0

BT ;T ,θ = 0

back

CRIX - a CRyptocurrency IndeX Appendix 7-22 Heston Nandi Options European Call option 1 Ct = 2St exp{−r(T − t)} Z ∞ K −iθf ∗(iθ + 1) + < dθ π 0 iθ 1 1 Z ∞ K −iθf ∗(iθ)  − K exp{−r(T − t)} + < dθ 2 π 0 iθ

where C Call price, St asset price, K strike price and < real part of a complex number. f ∗ the risk neutral version of f . Put-Call parity for Put price:

Pt = Ct + K exp{−r(T − t)} − St back CRIX - a CRyptocurrency IndeX Appendix 7-23

Kernel Density Estimation (KDE)

Compute pdf with KDE n   1 X x − Xi fˆ (x) = K h nh h i=1 with K(u) Gaussian kernel, h bandwidth

Bandwidth selection with Wand & Jones plug-in estimator

back

CRIX - a CRyptocurrency IndeX Appendix 7-24

Attributions

Cigarette trading in postwar germany, Bundesarchiv, Bild 183-R79014 / CC-BY-SA

By The original uploader was DickClarkMises at English Wikipedia [CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons

CRIX - a CRyptocurrency IndeX