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GitHub Repository: mobook/mo-book
 Path: blob/main/notebooks/04/05-cryptocurrency-arbitrage.ipynb
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Kernel: Python 3 (ipykernel)

4.5 Cryptocurrency arbitrage search

Installations and Imports

Preamble: Install Pyomo and a solver

The following cell sets and verifies a global SOLVER for the notebook. If run on Google Colab, the cell installs Pyomo and the HiGHS solver, while, if run elsewhere, it assumes Pyomo and HiGHS have been previously installed. It then sets to use HiGHS as solver via the appsi module and a test is performed to verify that it is available. The solver interface is stored in a global object SOLVER for later use.

import sys
 
if 'google.colab' in sys.modules:
    %pip install pyomo >/dev/null 2>/dev/null
    %pip install highspy >/dev/null 2>/dev/null
 
solver = 'appsi_highs'
 
import pyomo.environ as pyo
SOLVER = pyo.SolverFactory(solver)

assert SOLVER.available(), f"Solver {solver} is not available."

Problem description

Cryptocurrency exchanges are web services that enable the purchase, sale, and exchange of cryptocurrencies. These exchanges provide liquidity for owners and establish the relative value of these currencies. Joining an exchange enables a user to maintain multiple currencies in a digital wallet, buy and sell currencies, and use cryptocurrencies for financial transactions.

In this notebook, we explore the efficiency of cryptocurrency exchanges by testing for arbitrage opportunities. An arbitrage exists if a customer can realize a net profit through a sequence of risk-free trades. The efficient market hypothesis assumes arbitrage opportunities are quickly identified and exploited by investors. As a result of their trading, prices reach a new equilibrium so that any arbitrage opportunities would be small and fleeting in an efficient market. The question here is whether it is possible, with real-time data and rapid execution, for a trader to profit from these fleeting arbitrage opportunities.

Additional libraries needed: NetworkX and CCXT

This notebook uses two additional libraries: NetworkX and CCXT.

NetworkX is a Python library for the creation, manipulation, and study of the structure, dynamics, and functions of complex networks, which is installed by deafult on Google Colab.

CCXT is an open-source library for connecting to cryptocurrency exchanges. The CCXT library is not installed by default in Google Colab, so it must be installed with the following command.

%pip install ccxt

%pip install ccxt

We import all the libraries we need for this notebook in the following cell.

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import networkx as nx
import ccxt
import glob

Available cryptocurrency exchanges

Here we use the ccxt library and list current exchanges supported by ccxt.

print("Available exchanges:\n")
for i, exchange in enumerate(ccxt.exchanges):
    print(f"{i+1:3d}) {exchange.ljust(20)}", end="" if (i + 1) % 4 else "\n")
Available exchanges: 1) ace 2) alpaca 3) ascendex 4) bequant 5) bigone 6) binance 7) binancecoinm 8) binanceus 9) binanceusdm 10) bit2c 11) bitbank 12) bitbay 13) bitbns 14) bitcoincom 15) bitfinex 16) bitfinex2 17) bitflyer 18) bitforex 19) bitget 20) bithumb 21) bitmart 22) bitmex 23) bitopro 24) bitpanda 25) bitrue 26) bitso 27) bitstamp 28) bitstamp1 29) bittrex 30) bitvavo 31) bkex 32) bl3p 33) blockchaincom 34) btcalpha 35) btcbox 36) btcmarkets 37) btctradeua 38) btcturk 39) bybit 40) cex 41) coinbase 42) coinbaseprime 43) coinbasepro 44) coincheck 45) coinex 46) coinfalcon 47) coinmate 48) coinone 49) coinsph 50) coinspot 51) cryptocom 52) currencycom 53) delta 54) deribit 55) digifinex 56) exmo 57) fmfwio 58) gate 59) gateio 60) gemini 61) hitbtc 62) hitbtc3 63) hollaex 64) huobi 65) huobijp 66) huobipro 67) idex 68) independentreserve 69) indodax 70) kraken 71) krakenfutures 72) kucoin 73) kucoinfutures 74) kuna 75) latoken 76) lbank 77) lbank2 78) luno 79) lykke 80) mercado 81) mexc 82) mexc3 83) ndax 84) novadax 85) oceanex 86) okcoin 87) okex 88) okex5 89) okx 90) paymium 91) phemex 92) poloniex 93) poloniexfutures 94) probit 95) tidex 96) timex 97) tokocrypto 98) upbit 99) wavesexchange 100) wazirx 101) whitebit 102) woo 103) yobit 104) zaif 105) zonda

Representing an exchange as a directed graph

First, we need some terminology. Trading between two specific currencies is called a market, with each exchange hosting multiple markets. ccxt labels each market with a symbol common across exchanges. The market symbol is an upper-case string with abbreviations for a pair of traded currencies separated by a slash (//). The first abbreviation is the base currency, the second is the quote currency. Prices for the base currency are denominated in units of the quote currency. As an example, ETH/BTCETH/BTC refers to a market for the base currency Ethereum (ETH) quoted in units of the Bitcoin (BTC). The same market symbol can refer to an offer to sell the base currency (a 'bid') or to an offer to sell the base currency (an 'ask'). For example, xx ETH/BTC means you can buy xx units of BTC with one unit of ETH.

An exchange can be represented by a directed graph constructed from the market symbols available on that exchange. In such a graph currencies correspond to nodes on the directed graph. Market symbols correspond to edges in the directed graph, with the source indicating the quote currency and the destination indicating the base currency. The following code develops such a sample graph.

# create an exchange object
exchange = ccxt.binanceus()


def get_exchange_dg(exchange, minimum_in_degree=1):
    markets = exchange.load_markets()
    symbols = markets.keys()

    # create an edge for all market symbols
    dg = nx.DiGraph()
    for base, quote in [symbol.split("/") for symbol in symbols]:
        dg.add_edge(quote, base, color="k", width=1)

    # remove base currencies with in_degree less than minimum_in_degree
    remove_nodes = [
        node
        for node in dg.nodes
        if dg.out_degree(node) == 0 and dg.in_degree(node) < minimum_in_degree
    ]
    for node in reversed(list(nx.topological_sort(dg))):
        if node in remove_nodes:
            dg.remove_node(node)
        else:
            break

    # color quote currencies in gold
    for node in dg.nodes():
        dg.nodes[node]["color"] = "gold" if dg.out_degree(node) > 0 else "lightblue"

    return dg


def draw_dg(dg, rad=0.0):
    n_nodes = len(dg.nodes)
    size = int(2.5 * np.sqrt(n_nodes))
    fig = plt.figure(figsize=(size, size))
    pos = nx.circular_layout(dg)
    nx.draw_networkx(
        dg,
        pos,
        with_labels=True,
        node_color=[dg.nodes[node]["color"] for node in dg.nodes()],
        edge_color=[dg.edges[u, v]["color"] for u, v in dg.edges],
        width=[dg.edges[u, v]["width"] for u, v in dg.edges],
        node_size=1000,
        font_size=12,
        arrowsize=15,
        connectionstyle=f"arc3, rad={rad}",
    )
    nx.draw_networkx_edge_labels(
        dg, pos, edge_labels={(src, dst): f"{src}/{dst}" for src, dst in dg.edges()}
    )
    plt.axis("off")

    return plt.gca()


minimum_in_degree = 10
exchange_dg = get_exchange_dg(exchange, minimum_in_degree)
ax = draw_dg(exchange_dg, 0.01)
print(
    exchange.name
    + f" - Network obtained filtering for minimum indegree (Base Currencies) = {minimum_in_degree}"
)
print(f"Number of nodes = {len(exchange_dg.nodes()):3d}")
print(f"Number of edges = {len(exchange_dg.edges()):3d}")
Binance US - Network obtained filtering for minimum indegree (Base Currencies) = 10 Number of nodes = 155 Number of edges = 449
Image in a Jupyter notebook

Exchange order book

The order book for a currency exchange is the real-time inventory of trading orders.

A bid is an offer to buy up to a specified amount of the base currency at the price not exceeding the 'bid price' in the quote currency. An ask is an offer to sell up to a specified amount of the base currency at a price no less than a value specified given in the quote currency.

The exchange attempts to match the bid to ask order at a price less than or equal to the bid price. If a transaction occurs, the buyer will receive an amount of base currency less than or equal to the bid volume and the ask volume, at a price less than or equal to the bid price and no less than the specified value.

The order book for currency exchange is the real-time inventory of orders. The exchange order book maintains a list of all active orders for symbols traded on the exchange. Incoming bids above the lowest ask or incoming asks below the highest bid will be immediately matched and transactions executed following the rules of the exchange.

The following cell reads and displays a previously saved order book. Cells at the end of this notebook demonstrate how to retrieve an order book from an exchange and save it as a Pandas DataFrame.

filename = "Binance_US_orderbook_saved.csv"
if "google.colab" in sys.modules:
    order_book = pd.read_csv(
        "https://raw.githubusercontent.com/mobook/MO-book/main/notebooks/04/data/"
        + filename,
        index_col=0,
    )
else:
    order_book = pd.read_csv("data/" + filename, index_col=0)

display(order_book)

Modelling the arbitrage search problem as a graph

Our goal will be to find arbitrage opportunities, i.e., the possibility to start from a given currency and, through a sequence of executed trades, arrive back at the same currency with a higher balance than at the beginning. We will model this problem as a network one.

A bid appearing in the order book for the market symbol b/qb/q is an order from a prospective counter party to purchase an amount of the base currency bb at a bid price given in a quote currency qq. For a currency trader, a bid in the order book is an opportunity to convert the base currency bb into the quote currency qq.

The order book can be represented as a directed graph where the nodes correspond to individual currencies. A directed edge b→qb\rightarrow q from node bb to node qq describes an opportunity for us to convert currency bb into units of currency qq. Let VbV_b and VqV_q denote the amounts of each currency held by us, and let xb→qx_{b\rightarrow q} denote the amount of currency bb exchanged for currency jj. Following the transaction xb→qx_{b\rightarrow q} we have the following changes to the currency holdings

ΔVb=−xb→qΔVq=ab→qxb→q,\begin{align*} \Delta V_b & = - x_{b\rightarrow q} \\ \Delta V_q & = a_{b\rightarrow q} x_{b\rightarrow q}, \end{align*}

where ab→qa_{b\rightarrow q} is a conversion coefficient equal to the price of bb expressed in terms of currency qq. The capacity cb→qc_{b\rightarrow q} of a trading along edge b→qb\rightarrow q is specified by a relationship

xb→q≤cb→q.x_{b\rightarrow q} \leq c_{b\rightarrow q}.

Because the arcs in our graph correspond to two types of orders - bid and ask - we need to build a consistent way of expressing them in our ab→qa_{b\rightarrow q}, cb→qc_{b\rightarrow q} notation. Imagine now that we are the party that accepts the buy and ask bids existing in the graph.

For bid orders, we have the opportunity to convert the base currency bb into the quote currency qq, for which we will use the following notation:

ab→q=bid pricecb→q=bid volume\begin{align*} a_{b\rightarrow q} & = \text{bid price} \\ c_{b\rightarrow q} & = \text{bid volume} \end{align*}

An ask order for symbol b/qb/q is an order to sell the base currency at a price not less than the 'ask' price given in terms of the quote currency. The ask volume is the amount of base currency to be sold. For us, a sell order is an opportunity to convert the quoted currency into the base currency such that

aq→b=1ask pricecq→b=ask volume×ask volume\begin{align*} a_{q\rightarrow b} & = \frac{1}{\text{ask price}} \\ c_{q\rightarrow b} & = \text{ask volume} \times \text{ask volume} \end{align*}

The following cell creates a directed graph using data from an exchange order book. To distinguish between different order types, we will highlight the big orders with green color, and ask orders with red color.

def order_book_to_dg(order_book):
    # create a dictionary of edges index by (src, dst)
    dg_order_book = nx.DiGraph()

    # loop over each order in the order book dataframe
    for order in order_book.index:
        # if the order is a 'bid', i.e., an order to purchase the base currency
        if not np.isnan(order_book.at[order, "bid_volume"]):
            src = order_book.at[order, "base"]
            dst = order_book.at[order, "quote"]
            # add an edge to the graph with the relevant attributes
            dg_order_book.add_edge(
                src,
                dst,
                kind="bid",
                a=order_book.at[order, "bid_price"],
                capacity=order_book.at[order, "bid_volume"],
                weight=-np.log(order_book.at[order, "bid_price"]),
                color="g",
                width=0.5,
            )

        # if the order is an 'ask', i.e., an order to sell the base currency
        if not np.isnan(order_book.at[order, "ask_volume"]):
            src = order_book.at[order, "quote"]
            dst = order_book.at[order, "base"]
            # add an edge to the graph with the relevant attributes
            dg_order_book.add_edge(
                src,
                dst,
                kind="ask",
                a=1.0 / order_book.at[order, "ask_price"],
                capacity=order_book.at[order, "ask_volume"]
                * order_book.at[order, "ask_price"],
                weight=-np.log(1.0 / order_book.at[order, "ask_price"]),
                color="r",
                width=0.5,
            )

    # loop over each node in the graph and set the color attribute to "lightblue"
    for node in dg_order_book.nodes():
        dg_order_book.nodes[node]["color"] = "lightblue"

    return dg_order_book


order_book_dg = order_book_to_dg(order_book)

First, we simply print the content of the order book as a list of arcs.

# display contents of the directed graph
print(f"src   --> dst    kind            a                c")
print(f"------------------------------------------------------")
for src, dst in order_book_dg.edges():
    print(
        f"{src:5s} --> {dst:5s}   {order_book_dg.edges[(src, dst)]['kind']}"
        + f"{order_book_dg.edges[(src, dst)]['a']: 16f} {order_book_dg.edges[(src, dst)]['capacity']: 16f}    "
    )
src --> dst kind a c ------------------------------------------------------ ETH --> BTC bid 0.069735 0.012000 ETH --> ADA ask 4668.534080 0.190981 ETH --> USDT bid 1630.200000 0.950000 ETH --> BUSD bid 1630.210000 0.510000 ETH --> USDC bid 1630.160000 0.100000 ETH --> DAI bid 1629.890000 0.497400 ETH --> USD bid 1630.490000 2.564550 BTC --> ETH ask 14.335068 0.003488 BTC --> BNB ask 78.403701 0.038263 BTC --> ADA ask 66844.919786 0.032433 BTC --> SOL ask 1068.261938 0.014154 BTC --> MATIC ask 19391.118868 0.007746 BTC --> MANA ask 36968.576710 0.038113 BTC --> TRX ask 335570.469799 0.075549 BTC --> USDT bid 23373.920000 0.118619 BTC --> BUSD bid 23371.440000 0.021500 BTC --> USDC bid 23371.660000 0.020000 BTC --> DAI bid 23366.440000 0.049540 BTC --> USD bid 23373.010000 0.007463 BNB --> BTC bid 0.012743 0.050000 BNB --> USDT bid 297.857700 0.800000 BNB --> BUSD bid 297.763000 1.670000 BNB --> USD bid 297.900200 2.100000 ADA --> BTC bid 0.000015 2.000000 ADA --> ETH bid 0.000214 994.900000 ADA --> USDT bid 0.348630 1100.000000 ADA --> BUSD bid 0.348000 6582.900000 ADA --> USDC bid 0.348200 8615.400000 ADA --> USD bid 0.348800 1500.000000 SOL --> BTC bid 0.000935 1.420000 SOL --> USDT bid 21.857000 22.870000 SOL --> BUSD bid 21.830000 1.390000 SOL --> USDC bid 21.830000 343.520000 SOL --> USD bid 21.858100 55.730000 MATIC --> BTC bid 0.000052 26.200000 MATIC --> USDT bid 1.203000 1664.600000 MATIC --> BUSD bid 1.203410 623.100000 MATIC --> USD bid 1.204000 937.600000 MANA --> BTC bid 0.000027 831.000000 MANA --> USDT bid 0.631000 157.000000 MANA --> BUSD bid 0.630700 343.000000 MANA --> USD bid 0.631000 372.340000 TRX --> BTC bid 0.000003 4.000000 TRX --> USDT bid 0.069280 10824.900000 TRX --> BUSD bid 0.069290 10823.900000 TRX --> USD bid 0.069200 225602.200000 USDT --> BTC ask 0.000043 1058.348400 USDT --> ETH ask 0.000613 815.385000 USDT --> BNB ask 0.003357 238.313520 USDT --> ADA ask 2.867384 178.281000 USDT --> BUSD ask 1.000400 317048.859708 USDT --> SOL ask 0.045738 502.860500 USDT --> USDC ask 0.999800 299240.836200 USDT --> MATIC ask 0.829876 6513.748000 USDT --> MANA ask 1.581778 360.986200 USDT --> TRX ask 14.423770 750.053538 USDT --> USD bid 1.000000 10407.870000 BUSD --> USDT bid 0.999500 293433.930000 BUSD --> BTC ask 0.000043 502.601845 BUSD --> BNB ask 0.003356 399.256350 BUSD --> ETH ask 0.000613 415.843800 BUSD --> MATIC ask 0.830192 499.884100 BUSD --> USDC ask 1.000000 279879.620000 BUSD --> MANA ask 1.582028 1930.433400 BUSD --> ADA ask 2.865330 347.953000 BUSD --> SOL ask 0.045662 3965.871000 BUSD --> TRX ask 14.411298 1750.043556 BUSD --> USD bid 0.999500 79157.800000 USDC --> USDT bid 1.000100 307657.000000 USDC --> BUSD bid 0.999900 329027.800000 USDC --> BTC ask 0.000043 1192.226490 USDC --> ETH ask 0.000613 350.624150 USDC --> SOL ask 0.045704 4399.630400 USDC --> ADA ask 2.862049 850.125140 USDC --> USD bid 0.999900 5050.300000 DAI --> BTC ask 0.000043 1158.020820 DAI --> ETH ask 0.000613 2431.396900 DAI --> USD bid 0.999100 4534.660000 USD --> BTC ask 0.000043 1140.900820 USD --> ETH ask 0.000613 946.970718 USD --> USDT ask 0.999900 954935.163968 USD --> BNB ask 0.003356 399.852255 USD --> ADA ask 2.866151 1046.700000 USD --> BUSD ask 1.000100 795513.920652 USD --> MATIC ask 0.830220 1129.218750 USD --> USDC ask 1.000000 517682.170000 USD --> MANA ask 1.583281 361.489944 USD --> DAI ask 0.999900 7592.579182 USD --> SOL ask 0.045733 399.927311 USD --> TRX ask 14.409222 15562.637700

Next, we draw the graph itself.

draw_dg(order_book_dg, 0.05)
plt.show()
Image in a Jupyter notebook

Trading and arbitrage

With this unified treatment of bid and ask orders, we are ready to formulate the mathematical problem of finding an arbitrage opportunity. An arbitrage exists if it is possible to find a closed path and a sequence of transactions in the directed graph resulting in a net increase in currency holdings. Given a path

i0→i1→i2→⋯→in−1→in\begin{align*} i_0 \rightarrow i_1 \rightarrow i_2 \rightarrow \cdots \rightarrow i_{n-1} \rightarrow i_n \end{align*}

the path is closed if in=i0i_n = i_0. The path has finite capacity if each edge in the path has a non-zero capacity. For a sufficiently small holding wi0w_{i_0} of currency i0i_0 (because of capacity constraints), a closed path with i0=ini_0 = i_n represents an arbitrage opportunity if

∏k=0n−1aik→ik+1>1.\begin{equation} \prod_{k=0}^{n-1} a_{i_k\rightarrow i_{k+1}} > 1. \end{equation}

If all we care about is simply finding an arbitrage cycle, regardless of the volume traded, we can use one of the many shortest path algorithms from the networkx library. To convert the problem of finding a path meeting the above condition into a sum-of-terms to be minimized, we can take the negative logarithm of both sides to obtain the condition:

−log⁡(∏k=0n−1aik→ik+1)=−∑k=0n−1log⁡(aik→ik+1)<0,\begin{align*} -\log(\prod_{k=0}^{n-1} a_{i_k\rightarrow i_{k+1}}) = - \sum\limits_{k = 0}^{n-1} \log (a_{i_k\rightarrow i_{k+1}}) < 0, \end{align*}

In other words, if we assign the negative logarithm as the weight of arcs in a graph, then our problem just becomes translated into the problem of searching for a cycle with a total sum of weights along it to be negative.

Find order books that have arbitrage opportunities

A simple cycle is a closed path in which no node appears twice. Simple cycles are distinct if they are not cyclic permutations (essentially, the same set of nodes but with a different start and end points) of each other. One could check for arbitrage opportunities by checking if there are any negative simple cycles in the graph.

However, searching for a negative-weight cycle through searching for an arbitrage opportunity can be a daunting task - a brute-force search over all simple cycles has complexity (n+e)(c+1)(n + e)(c + 1) which is impractical for large-scale applications. A more efficient search based on the Bellman-Ford algorithm is embedded in the NetworkX function negative_edge_cycle that returns a logical True if a negative cycle exists in a directed graph. 

order_book_dg = order_book_to_dg(order_book)
nx.negative_edge_cycle(order_book_dg, weight="weight", heuristic=True)
True

The function negative_edge_cycle is fast, but it only indicates if there is a negative cycle or not, and we don't even know what kind of a cycle is it so it would be hard to use that information to perform an arbitrage.

Luckily, the networkx library includes the function find_negative_cycle that locates a single negative edge cycle if one exists. We can use this to demonstrate the existence of an arbitrage opportunity and to highlight that opportunity on the directed graph of all possible trades. The following cell reports the cycle found and the corresponding trading return measured in basis points (1 basis point = 0.01%). The same trading cycle is marked with thicker arcs in the graph.

def sum_weights(cycle):
    return sum(
        [
            order_book_dg.edges[edge]["weight"]
            for edge in zip(cycle, cycle[1:] + cycle[:1])
        ]
    )


order_book_dg = order_book_to_dg(order_book)
arb = nx.find_negative_cycle(order_book_dg, weight="weight", source="USD")[:-1]
bp = 10000 * (np.exp(-sum_weights(arb)) - 1)

for src, dst in zip(arb, arb[1:] + arb[:1]):
    order_book_dg[src][dst]["width"] = 5

print(
    f"Trading cycle with {len(list(arb))} trades: {' --> '.join(list(arb))}\nReturn = {bp:6.3f} basis points "
)
ax = draw_dg(order_book_dg, 0.05)
plt.show()
Trading cycle with 3 trades: USDC --> USDT --> BUSD Return = 5.002 basis points
Image in a Jupyter notebook

Note this may or may not be the trading cycle with maximum return. There may be other cycles with higher or lower returns, and that allow higher or lower trading volumes.

Brute force search arbitrage with simple cycles

Not all arbitrage cycles are the same - some yield a higher relative return (per dollar invested) than the others, and some yield a higher absolute return (maximum amount of money to be made risk-free) than others. This is because the amount of money that flows through a negative cycle is upper bounded by the size of the smallest order in that cycle. Thus, if one is looking for the best possible arbitrage sequence of trades, finding just 'a cycle' might not be enough.

A crude way to search for a good arbitrage opportunity would be to enumerate all possible simple cycles in a graph and pick the one that's best according to whatever criterion we pick. A brute force search over for all simple cycles has order (Nnodes+Nedges)(Ncycles+1)(N_{nodes} + N_{edges})(N_{cycles} + 1) complexity, which is prohibitive for large order books. Nevertheless, we explore this option here to better understand the problem of finding and valuing arbitrage opportunities.

In the following cell, we compute the loss function for all simple cycles that can be constructed within a directed graph using the function simple_cycles from the networkx library to construct a dictionary of all distinct simple cycles in the order book. Each cycle is represented by an ordered list of nodes. For each cycle, the financial return is computed, and a histogram is constructed to show the distribution of potential returns. Several paths with the highest return are then overlaid on the graph of the order book.

Again, note that no account is taken of the trading capacity available on each path.

The next cell iterates over all simple cycles in a directed graph. This can take a long time for a large dense graph.

# convert order book to a directed graph
order_book_dg = order_book_to_dg(order_book)


# compute the sum of weights given a list of nodes
def sum_weights(cycle):
    return sum(
        [
            order_book_dg.edges[edge]["weight"]
            for edge in zip(cycle, cycle[1:] + cycle[:1])
        ]
    )


# create a dictionary of all simple cycles and sum of weights
cycles = {
    tuple(cycle): 10000 * (np.exp(-sum_weights(cycle)) - 1)
    for cycle in nx.simple_cycles(order_book_dg)
}

print(
    f"There are {len(cycles)} distinct simple cycles in the order book, {len([cycle for cycle in cycles if cycles[cycle] > 0])} of which have positive return."
)

# create histogram
plt.hist(cycles.values(), bins=int(np.sqrt(len(cycles))))
ax = plt.gca()
ax.set_ylabel("count")
ax.set_xlabel("Basis Points")
ax.set_title("Histogram of Returns for all Simple Cycles")
ax.grid(True)
ax.axvline(0, color="r")
plt.show()
There are 203147 distinct simple cycles in the order book, 974 of which have positive return.
Image in a Jupyter notebook

Next, we sort out the negative cycles from this list and present them along with their basis-points (1% is 100 basis points) return.

arbitrage = [
    cycle for cycle in sorted(cycles, key=cycles.get, reverse=True) if cycles[cycle] > 0
]

n_cycles_to_list = 5

print(f"Top {n_cycles_to_list}\n")
print(f"Basis Points             Arbitrage Cycle")
for cycle in arbitrage[0 : min(n_cycles_to_list, len(arbitrage))]:
    t = list(cycle)
    t.append(cycle[0])
    print(f"{cycles[cycle]:6.3f}         {len(t)} trades: {' --> '.join(t)}")
Top 5 Basis Points Arbitrage Cycle 14.774 8 trades: USDT --> ADA --> BTC --> ETH --> USD --> BUSD --> USDC --> USDT 14.747 8 trades: USDT --> TRX --> BTC --> ETH --> USD --> BUSD --> USDC --> USDT 14.699 7 trades: USDT --> ADA --> BTC --> USD --> BUSD --> USDC --> USDT 14.673 7 trades: USDT --> TRX --> BTC --> USD --> BUSD --> USDC --> USDT 13.772 7 trades: USDT --> ADA --> BTC --> ETH --> USD --> USDC --> USDT

In the end, we draw an example arbitrage cycle on our graph to illustrate the route that the money must travel.

n_cycles_to_show = 1

for cycle in arbitrage[0 : min(n_cycles_to_show, len(arbitrage))]:
    print(
        f"Trading cycle with {len(t)} trades: {' --> '.join(t)}\nReturn = {cycles[cycle]:6.3f} basis points "
    )
    # get fresh graph to color nodes
    order_book_dg = order_book_to_dg(order_book)

    # color nodes red
    for node in cycle:
        order_book_dg.nodes[node]["color"] = "red"

    # makes lines wide
    for edge in zip(cycle, cycle[1:] + cycle[:1]):
        order_book_dg.edges[edge]["width"] = 4

    ax = draw_dg(order_book_dg, rad=0.05)

    t = list(cycle)
    t.append(cycle[0])
    plt.show()
Trading cycle with 8 trades: USDT --> ADA --> BTC --> ETH --> USD --> BUSD --> USDC --> USDT Return = 14.774 basis points
Image in a Jupyter notebook

Pyomo model for arbitrage with capacity constraints

The preceding analysis demonstrates some practical limitations of relying on generic implementations of network algorithms:

  • First, more than one negative cycle may exist, so more than one arbitrage opportunity may exist, i.e. an optimal strategy consists of a combination of cycles.

  • Secondly, simply searching for a negative cycle using shortest path algorithms does not account for capacity constraints, i.e., the maximum size of each of the exchanges. For that reason, one may end up with a cycle on which a good `rate' of arbitrage is available, but where the absolute gain need not be large due to the maximum amounts that can be traded.

Instead, we can formulate the problem of searching for a maximum-gain arbitrage via linear optimization. Assume that we are given a directed graph where each edge i→ji\rightarrow j is labeled with a 'multiplier' ai→ja_{i\rightarrow j} indicating how many units of currency jj will be received for one unit of currency ii, and a 'capacity' ci→jc_{i\rightarrow j} indicating how many units of currency ii can be converted to currency jj.

We break the trading process into steps indexed by t=1,2,…,Tt = 1, 2, \ldots, T, where currencies are exchanged between two adjacent nodes in a single step. We denote by xi→j(t)x_{i\rightarrow j}(t) the amount of currency traded from node ii to jj in step tt. In this way, we start with the amount wUSD(0)w_{USD}(0) at time 00 and aim to maximize the amount wUSD(T)w_{USD}(T) at time TT. Denote by OjO_j the set of nodes to which outgoing arcs from jj lead, and by IjI_j the set of nodes from which incoming arcs lead.

A single transaction converts xi→j(t)x_{i\rightarrow j}(t) units of currency ii to currency jj. Following all transactions at event tt, the trader will hold vj(t)v_j(t) units of currency jj where

vj(t)=vj(t−1)+∑i∈Ijai→jxi→j(t)−∑k∈Ojxj→k(t)v_{j}(t) = v_{j}(t-1) + \sum_{i\in I_j} a_{i\rightarrow j}x_{i\rightarrow j}(t) - \sum_{k\in O_j} x_{j\rightarrow k}(t)

For every edge i→ji\rightarrow j, the sum of all transactions must satisfy

∑t=1Txj→k(t)≤cj→k\sum_{t=1}^T x_{j\rightarrow k}(t) \leq c_{j\rightarrow k}

The objective of the optimization model is to find a sequence of currency transactions that increase the holdings of a reference currency. The solution is constrained by assuming that the trader cannot short sell any currency. The resulting model is

max⁡vUSD(T)s.t.vUSD(0)=v0vj(t)=vj(t−1)+∑i∈Ijai→jxi→j(t)−∑k∈Ojxj→k(t)∀j∈N,t=1,2,…,Tvj(t−1)≥∑k∈Ojxj→k(t)∀j∈N,t=1,2,…,T∑t=1Txj→k(t)≤cj→k∀(j,k)∈E,t=1,2,…,Tvj(t),xi→j(t)≥0∀t\begin{align*} \max \quad & v_{USD}(T) \\ \text{s.t.} \quad & v_{USD}(0) = v_0 \\ & v_{j}(t) = v_{j}(t-1) + \sum_{i\in I_j} a_{i\rightarrow j}x_{i\rightarrow j}(t) - \sum_{k\in O_j} x_{j\rightarrow k}(t) & \forall j\in N, t=1, 2, \ldots, T \\ & v_j(t-1) \geq \sum_{k\in O_j} x_{j\rightarrow k}(t) & \forall j\in N, t = 1, 2, \ldots, T \\ & \sum_{t=1}^T x_{j\rightarrow k}(t) \leq c_{j\rightarrow k} & \forall (j, k) \in E, t = 1, 2, \ldots, T \\ & v_{j}(t), x_{i\rightarrow j}(t) \geq 0 & \forall t \end{align*}

where the constraints model:

  • the initial amount condition,

  • the balance equations linking the state of the given node in the previous and subsequent time periods,

  • the fact that we cannot trade at time step tt more of a given currency than we had in this currency from time step t−1t - 1. This constraint 'enforces' the time order of trades, i.e., we cannot trade in time period tt units which have been received in the same time period.

  • the maximum allowed trade volumes,

  • the non-negativity of the variables.

The following cell implements this optimization model using Pyomo.

def crypto_model(order_book_dg, T=10, v0=100.0):
    m = pyo.ConcreteModel("Cryptocurrency arbitrage")

    # length of the trading chain
    m.T0 = pyo.RangeSet(0, T)
    m.T1 = pyo.RangeSet(1, T)

    # currency nodes and trading edges
    m.NODES = pyo.Set(initialize=list(order_book_dg.nodes))
    m.EDGES = pyo.Set(initialize=list(order_book_dg.edges))

    # currency on hand at each node
    m.v = pyo.Var(m.NODES, m.T0, domain=pyo.NonNegativeReals)

    # amount traded on each edge at each trade
    m.x = pyo.Var(m.EDGES, m.T1, domain=pyo.NonNegativeReals)

    # total amount traded on each edge over all trades
    m.z = pyo.Var(m.EDGES, domain=pyo.NonNegativeReals)

    # "multiplier" on each trading edge
    @m.Param(m.EDGES)
    def a(m, src, dst):
        return order_book_dg.edges[(src, dst)]["a"]

    @m.Param(m.EDGES)
    def c(m, src, dst):
        return order_book_dg.edges[(src, dst)]["capacity"]

    @m.Objective(sense=pyo.maximize)
    def wealth(m):
        return m.v["USD", T]

    @m.Constraint(m.EDGES)
    def total_traded(m, src, dst):
        return m.z[src, dst] == sum([m.x[src, dst, t] for t in m.T1])

    @m.Constraint(m.EDGES)
    def edge_capacity(m, src, dst):
        return m.z[src, dst] <= m.c[src, dst]

    # initial assignment of v0 units on a selected currency
    @m.Constraint(m.NODES)
    def initial(m, node):
        if node == "USD":
            return m.v[node, 0] == v0
        return m.v[node, 0] == 0

    @m.Constraint(m.NODES, m.T1)
    def no_shorting(m, node, t):
        out_nodes = [dst for src, dst in m.EDGES if src == node]
        return m.v[node, t - 1] >= sum(m.x[node, dst, t] for dst in out_nodes)

    @m.Constraint(m.NODES, m.T1)
    def balances(m, node, t):
        in_nodes = [src for src, dst in m.EDGES if dst == node]
        out_nodes = [dst for src, dst in m.EDGES if src == node]
        return m.v[node, t] == m.v[node, t - 1] + sum(
            m.a[src, node] * m.x[src, node, t] for src in in_nodes
        ) - sum(m.x[node, dst, t] for dst in out_nodes)

    return m

Using this function, we are able to compute the optimal (absolute) return from an order book while respecting the order capacities and optimally using all the arbitrage opportunities inside it.

order_book_dg = order_book_to_dg(order_book)
v0 = 10000.0
T = 8
m = crypto_model(order_book_dg, T=T, v0=v0)
SOLVER.solve(m)
vT = m.wealth()

print(f"Starting wealth = {v0:0.2f} USD")
print(f"Weath after {T:2d} transactions = {vT:0.2f} USD")
print(f"Return = {10000 * (vT - v0)/v0:0.3f} basis points")
Starting wealth = 10000.00 USD Weath after 8 transactions = 10009.01 USD Return = 9.006 basis points

To track the evolution of the trades throughout time, the script in the following cell illustrates, for each currency (rows) the amount of money held in that currency at each of the time steps t=0,…,8t = 0, \dots, 8. It is visible from this scheme that the sequence of trades is not a simple cycle, but rather a more sophisticated sequence of trades which we would not have discovered with simple-cycle exploration alone, especially not when considering also the arc capacities.

for node in m.NODES:
    print(f"{node:5s}", end="")
    for t in m.T0:
        print(f" {m.v[node, t]():11.5f}", end="")
    print()
ETH 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 BTC 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00004 0.00000 BNB 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 ADA 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 2.00000 0.00000 0.00000 SOL 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 MATIC 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 MANA 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 TRX 0.00000 0.00000 -0.00000 -0.00000 0.00000 0.00000 4.00000 -0.00000 0.00000 USDT 0.00000 4953.27900 5046.73030 0.00000 4955.75660 5049.25470 0.00000 4958.23550 0.00000 BUSD 0.00000 -0.00000 4955.26110 5048.74980 0.00000 4957.73970 5050.30000 0.00000 0.00000 USDC 0.00000 5046.22570 0.00000 4955.26110 5048.74980 0.00000 4957.73970 5050.30000 0.00000 DAI 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 USD 10000.00000 -0.00000 0.00000 0.00000 0.00000 -0.00000 0.00000 0.00000 10009.00600

To be even more specific, the following cell lists the sequence of transcations executed.

print("\nTransaction Events")
for t in m.T1:
    print(f"t = {t}")
    for src, dst in m.EDGES:
        if m.x[src, dst, t]() > 1e-6:
            print(
                f" {src:8s} -> {dst:8s}: {m.x[src, dst, t]():14.6f} {m.a[src, dst] * m.x[src, dst, t]():14.6f}"
            )
    print()
Transaction Events t = 1 USD -> USDT : 4953.774300 4953.278972 USD -> USDC : 5046.225700 5046.225700 t = 2 USDT -> BUSD : 4953.279000 4955.261104 USDC -> USDT : 5046.225700 5046.730323 t = 3 USDT -> BUSD : 5046.730300 5048.749800 BUSD -> USDC : 4955.261100 4955.261100 t = 4 BUSD -> USDC : 5048.749800 5048.749800 USDC -> USDT : 4955.261100 4955.756626 t = 5 USDT -> BUSD : 4955.756600 4957.739696 USDC -> USDT : 5048.749800 5049.254675 t = 6 USDT -> ADA : 0.697500 2.000000 USDT -> BUSD : 5048.279900 5050.300020 USDT -> TRX : 0.277320 4.000000 BUSD -> USDC : 4957.739700 4957.739700 t = 7 ADA -> BTC : 2.000000 0.000030 TRX -> BTC : 4.000000 0.000012 BUSD -> USDC : 5050.300000 5050.300000 USDC -> USDT : 4957.739700 4958.235474 t = 8 BTC -> USD : 0.000042 0.976057 USDT -> USD : 4958.235500 4958.235500 USDC -> USD : 5050.300000 5049.794970

We next illustrate the arbitrage strategy in the graph by marking all the corresponding arcs thicker.

# for each currency we took only one ask and one bid, this is why we are unique between each pair of nodes

# report what orders to issue
for src, dst in m.EDGES:
    if m.z[src, dst]() > 0.0000002:
        order_book_dg.nodes[src]["color"] = "red"
        order_book_dg.nodes[dst]["color"] = "red"
        order_book_dg[src][dst]["width"] = 4

draw_dg(order_book_dg, 0.05)
plt.show()
Image in a Jupyter notebook

If we want to be even more precise about the execution of the trading strategy, we can formulate a printout of the list of orders that we, as the counter party to the orders stated in the order book, should issue for our strategy to take place.

# report what orders to issue
print("Trading Summary for the Order Book")
print(f"  Order Book   Type    Capacity         Traded")
for src, dst in m.EDGES:
    if m.z[src, dst]() > 0.0000002:
        kind = order_book_dg.edges[(src, dst)]["kind"]
        s = f"{src:>5s} -> {dst:<5s} {kind} {m.c[src, dst]:12.5f} {m.z[src, dst]():14.5f}"
        s += "  >>>  "
        if kind == "ask":
            base = dst
            quote = src
            symbol = base + "/" + quote
            price = 1.0 / order_book_dg.edges[(src, dst)]["a"]
            volume = m.z[src, dst]() / price
            s += f"sell {volume:15.6f} {symbol:11s} at {price:12.6f}"

        if kind == "bid":
            base = src
            quote = dst
            symbol = base + "/" + quote
            price = order_book_dg.edges[(src, dst)]["a"]
            volume = m.z[src, dst]()
            s += f"buy {volume:16.6f} {symbol:11s} at {price:12.6f}"
        print(s)
Trading Summary for the Order Book Order Book Type Capacity Traded BTC -> USD bid 0.00746 0.00004 >>> buy 0.000042 BTC/USD at 23373.010000 ADA -> BTC bid 2.00000 2.00000 >>> buy 2.000000 ADA/BTC at 0.000015 TRX -> BTC bid 4.00000 4.00000 >>> buy 4.000000 TRX/BTC at 0.000003 USDT -> ADA ask 178.28100 0.69750 >>> sell 2.000000 ADA/USDT at 0.348750 USDT -> BUSD ask 317048.85971 20004.04600 >>> sell 20012.050820 BUSD/USDT at 0.999600 USDT -> TRX ask 750.05354 0.27732 >>> sell 4.000000 TRX/USDT at 0.069330 USDT -> USD bid 10407.87000 4958.23550 >>> buy 4958.235500 USDT/USD at 1.000000 BUSD -> USDC ask 279879.62000 20012.05100 >>> sell 20012.051000 USDC/BUSD at 1.000000 USDC -> USDT bid 307657.00000 20007.97600 >>> buy 20007.976000 USDC/USDT at 1.000100 USDC -> USD bid 5050.30000 5050.30000 >>> buy 5050.300000 USDC/USD at 0.999900 USD -> USDT ask 954935.16397 4953.77430 >>> sell 4953.278972 USDT/USD at 1.000100 USD -> USDC ask 517682.17000 5046.22570 >>> sell 5046.225700 USDC/USD at 1.000000

In the end, we can illustrate the time-journey of our balances in different currencies using time-indexed bar charts.

# display currency balances
balances = pd.DataFrame()
for node in order_book_dg.nodes:
    if sum(m.v[node, t]() for t in m.T0) > 0.0000002:
        for t in m.T0:
            balances.loc[t, node] = m.v[node, t]()

balances.plot(
    kind="bar",
    subplots=True,
    figsize=(8, 10),
    xlabel="Transaction",
    ylabel="Currency Units",
)
plt.gcf().tight_layout()
plt.show()
Image in a Jupyter notebook

Exercises for the reader

The previous notebook cells made certain assumptions that we need to consider. The first assumption was that there was at most one bid and one ask order between any pair of currencies in an exchange. This assumption was based on the number of orders we downloaded from the online database, but in reality, there may be more orders. How would the presence of multiple orders per pair affect our graph formulation? How can we adjust the MILO formulation to account for this?

Another assumption was that we only traded currencies within one exchange. However, in reality, we can trade across multiple exchanges. How can we modify the graph-based problem formulation to accommodate this scenario?

Further, we have assigned no cost to the number of trades required to implement the strategy produced by optimization. How can the modeled be modified to either minimize the number of trades, or to explicitly include trading costs?

Finally, a tool like this needs to operate in real time. How can this model be incorporated into, say, a streamlit application that could be used to monitor for arbitrage opportunities in real time?

Real Time Downloads of Order Books from an Exchange

The goal of this notebook was to show how network algorithms and optimization can be utilized to detect arbitrage opportunities within an order book that has been obtained from a cryptocurrency exchange.

The subsequent cell in the notebook utilizes ccxt.exchange.fetch_order_book to obtain the highest bid and lowest ask orders from an exchange for market symbols that meet the criteria of having a minimum in-degree for their base currencies.

def get_order_book(exchange, exchange_dg):
    def get_orders(base, quote, limit=1):
        result = exchange.fetch_order_book("/".join([base, quote]), limit)
        if not result["asks"] or not result["bids"]:
            result = None
        else:
            result["base"], result["quote"] = base, quote
            result["timestamp"] = exchange.milliseconds()
            result["bid_price"], result["bid_volume"] = result["bids"][0]
            result["ask_price"], result["ask_volume"] = result["asks"][0]
        return result

    # fetch order book data and store in a dictionary
    order_book = filter(
        lambda r: r is not None,
        [get_orders(base, quote) for quote, base in exchange_dg.edges()],
    )

    # convert to pandas dataframe
    order_book = pd.DataFrame(order_book)
    order_book["timestamp"] = pd.to_datetime(order_book["timestamp"], unit="ms")

    return order_book[
        [
            "symbol",
            "timestamp",
            "base",
            "quote",
            "bid_price",
            "bid_volume",
            "ask_price",
            "ask_volume",
        ]
    ]


minimum_in_degree = 10

# get graph of market symbols with mininum_in_degree for base currencies
exchange_dg = get_exchange_dg(exchange, minimum_in_degree)

# get order book for all markets in the graph
order_book = get_order_book(exchange, exchange_dg)
order_book_dg = order_book_to_dg(order_book)

# find trades
v0 = 10000.0
m = crypto_model(order_book_dg, T=12, v0=v0)
vT = m.wealth()

print(f"Potential Return = {10000*(vT - v0)/v0:0.3f} basis points")
display(order_book)
Potential Return = 0.000 basis points

The following cell can be used to download additional order book data sets for testing.

from datetime import datetime
import time

# get graph of market symbols with mininum_in_degree for base currencies
minimum_in_degree = 5
exchange_dg = get_exchange_dg(exchange, minimum_in_degree)

# search time
search_time = 20
timeout = time.time() + search_time

# threshold in basis points
arb_threshold = 1.0

# wait for arbitrage opportunity
print(f"Search for {search_time} seconds.")
while time.time() <= timeout:
    print("Basis points = ", end="")
    order_book = get_order_book(exchange, exchange_dg)
    order_book_dg = order_book_to_dg(order_book)

    v0 = 10000.0
    m = crypto_model(order_book_dg, T=12, v0=10000)
    vT = m.wealth()
    bp = 10000 * (vT - v0) / vT
    print(f"{bp:0.3f}")

    if bp >= arb_threshold:
        print("arbitrage found!")
        fname = f"{exchange} orderbook {datetime.utcnow().strftime('%Y%m%d_%H%M%S')}.csv".replace(
            " ", "_"
        )
        order_book.to_csv(fname)
        print(f"order book saved to: {fname}")

print("Search complete.")
Search for 20 seconds. Basis points = 0.149 Search complete.

Bibliographic Notes

Crytocurrency markets are relatively new compared to other markets, and relatively few academic papers are available that specifically address arbitrage on those markets. Early studies, such as the following, reported periods of large, recurrent arbitrage opportunities that exist across exchanges, and that can persist for several days or weeks.

Makarov, I., & Schoar, A. (2020). Trading and arbitrage in cryptocurrency markets. Journal of Financial Economics, 135(2), 293-319.

Subsequent work reports these prices differentials do exist, but only at a fraction of the values previously reported, and only for fleeting periods of time.

Crépellière, T., & Zeisberger, S. (2020). Arbitrage in the Market for Cryptocurrencies. Available at SSRN 3606053. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3606053

The use of network algorithms to identify cross-exchange arbitrage has appeared in the academic literature, and in numerous web sites demonstrating optimization and network applications. Representative examples are cited below.

Peduzzi, G., James, J., & Xu, J. (2021, September). JACK THE RIPPLER: Arbitrage on the Decentralized Exchange of the XRP Ledger. In 2021 3rd Conference on Blockchain Research & Applications for Innovative Networks and Services (BRAINS) (pp. 1-2). IEEE. https://arxiv.org/pdf/2106.16158.pdf

Bruzgė, R., & Šapkauskienė, A. (2022). Network analysis on Bitcoin arbitrage opportunities. The North American Journal of Economics and Finance, 59, 101562. https://doi.org/10.1016/j.najef.2021.101562

Bruzgė, R., & Šapkauskienė, A. (2022). Dataset for Bitcoin arbitrage in different cryptocurrency exchanges. Data in Brief, 40, 107731.

The work in this notebook is related to materials found in the following web resources.

https://anilpai.medium.com/currency-arbitrage-using-bellman-ford-algorithm-8938dcea56ea

Crypto Trading and Arbitrage Identification Strategies

A more complete analysis of trading and exploiting arbitrage opportunities in decentralized finance markets is available in the following paper and thesis.

Byrne, S. An Exploration of Novel Trading and Arbitrage Methods within Decentralised Finance. https://www.scss.tcd.ie/Donal.OMahony/bfg/202021/StephenByrneDissertation.pdf

Levus, R., Berko, A., Chyrun, L., Panasyuk, V., & Hrubel, M. (2021). Intelligent System for Arbitrage Situations Searching in the Cryptocurrency Market. In CEUR Workshop Proceedings (pp. 407-440). http://ceur-ws.org/Vol-2917/paper32.pdf

In addition to the analysis of arbitrage opportunities, convex optimization may also have an important role in the developing of trading algorithms for crypocurrency exchanges.

Angeris, G., Agrawal, A., Evans, A., Chitra, T., & Boyd, S. (2021). Constant function market makers: Multi-asset trades via convex optimization. arXiv preprint arXiv:2107.12484. https://baincapitalcrypto.com/constant-function-market-makers-multi-asset-trades-via-convex-optimization/ and https://arxiv.org/pdf/2107.12484.pdf