Vincent Cheng 
No matter in traditional markets or cryptocurrency exchanges, this isn’t new things. The concept is simple. However, the actual difficulties are in the implementation.
This article just clarifies the mechanism and list some formulas to demonstrate the whole process instead of listing the process and code of implementation.
1. Brief of the mechanism
1.1 Gist
The chances of arbitrage were caused by an unreasonable quotation.
That’s what we need to seek in the markets.
1.2 Portal
If you want to skip the wordy deduction process, click here to teleport to Final Model
1.3 Similar Sample
To clarify the mechanism distinctly, I use a similar sample as an analogy, and I know there are somethings looks weird in the example, but just please focus the critical points to process. So don’t ask me where you can buy such big watermelon using 1 USD, I don’t know either.
We assume there is a “Weird Markets” like this:
There are 3 stalls which could exchange things by two-ways:
- Tomato Stall: you can buy 4 tomatoes with 1 USD. Also, you can sell 4 tomatoes with 1 USD.
- Melon Stall: you can buy 1 melon with 1 USD, and same to sell.
- No Cash Stall: there is no cash trading in this stall. But you can exchange 4 tomatoes and 1 melon with each other.
One day, the owner of Tomato Stall changes the price: now you can buy or sell 4 tomatoes with 1 USD. That caused the unreasonable quotation. Assuming you found it and begin to perform the arbitrage with 1 USD following the process:
- Buy a melon with 1 USD in Melon Stall
- Exchanges 4 tomatoes with a melon in No Cash Stall
- Sell 4 tomatoes with 2 USD in Tomato Stall
We can call this arbitrage process as positive arbitrage, as the direction of arrows in the above figure went in clock-wise.
Might you already think out there is inverse arbitrage as well, as the arrows goes in anti clock-wise. Here is it:
In this time, the owner who changes the price of one Melon to 2 USD caused the unreasonable quotation. Then we can:
- Buy 4 tomatoes with 1 USD in Tomato Stall
- Exchange a melon with 4 tomatoes in No Cash Stall
- Sell a melon with 2 USD in Melon Stall
Note:
- The name of “positive arbitrage” and “inverse arbitrage”, which just differ the process on the title, doesn’t matter with the actual procedure. You also can call the anti-clock-wise as positive arbitrage, if you want.
1.4 Sample of cryptocurrency
As we changes those 3 stall in the above sample to real trading pair, such as “BTC / USD”, “ETH / USD” and “BTC / ETH”, the actual model was shown as:
In the above sample, the trading pairs are similar to stalls , and cryptocurrencies are similar to the goods in stalls. Regarding the mapping, it’s up to you. You can regard the USD as the money what you final gain after arbitrage, either BTC or ETH. That depends on your arbitrage process and your initial capital.
Back to the gist, the chances were caused by an unreasonable quotation. Hence, we need to focus on those bid price and ask price of different trading pairs, to seek a chance, maybe within a nanosecond.
2. Seeking unreasonable quotation
2.1 Variables impact to unreasonable quotation
The prices and amounts of each trading pair obviously are the key variables which definitely impact to unreasonable quotation.
In common, there are many gears for each trading pairs in most exchanges, some are 10 gears, while some are up to 50 gears. However, we needn’t too much gear but the first, that’s because the price of first gear is nearest with dealmaker price. So for the clear demonstration, we use the first price and amount of each trading pairs to calculate.
Note:
For perfect consideration, we can’t only use the price and amount of first gear to process. Especially in the case when the price of first gear is suitable, and the amount is too small to deal, but the price of second gear is appropriate as well. I will discuss this case in the latter part.
2.2 Positive Arbitrage
2.2.1 Flow Chart
2.2.2 Background
- you have to keep a certain quantity of Mid Currency as initial capital to start the arbitrage.
- Though the initial capital is Mid Currency, the primary variable which can define the quantitative currency to initiate is the amount of Base Currency, instead of how much Mid Currency you have. The reason is that it would be easy to calculate, which will be mentioned in later part.
- in the arbitrage process, we trade as taker instead of maker, so means there is a certain fee during the trade following the convention of most of the exchanges. In most of the exchanges, the fee for the maker is the same for all trading pair. However some exchanges gave low or even free for a certain trading pair, so it’s better for us to remarked separately as: \(Fee_{base\_mid} , Fee_{base\_quote} , Fee_{quote\_mid}\)
2.2.3 Process
- To buy Base Currency. The amount is \(Base\_amt\), the Cost is selling \(Mid\_amt\_init\) Mid Currency
- Selling all Base Currency which you bought in the previous step, then you gain some Quote Currency, the amount is \(Quote\_amt\)
- Selling all Quote Currency which you gained in the previous step, then you retrieve the Mid Currency, the amount is \(Mid\_amt\_final\)
2.2.4 Threshold
As long as the \(Mid\_amt\_final\) is greater than \(Mid\_amt\_init\), that’s the unreasonable quotation we shoulld garb.
2.2.5 Formula
Primary variable: how much we should trade the Base Currency, which is remarked by:
\[Base\_amt\]The cost in step 1 after you sell the Mid Currency, which means how much Mid Currency you spend to buy Base Currency:
\[Mid\_amt\_init = \frac{Base\_amt * Base\_Mid_{sell\_price}}{1-Fee_{base\_mid}}\]The amount of Quote Currency after you sold the Base Currency in step 2:
\[Quote\_amt = Base\_amt * Base\_Quote_{buy\_price} * (1-Fee_{base\_quote})\]The amount of Mid Currency you retrieved in step 3:
\[Mid\_amt\_final = Quote\_amt * Quote\_Mid_{buy\_price} * (1-Fee_{quote\_mid})\]The profit after one circulation:
\[Profit = Mid\_amt\_final - Mid\_amt\_init\]Substitue all variables and get the final expression:
\[Profit = Base\_amt * [ Base\_Quote_{buy\_price} * Quote\_Mid_{buy\_price} * (1-Fee_{base\_quote}) * (1-Fee_{quote\_mid}) - \frac{Base\_Mid_{sell\_price}}{1-Fee_{base\_mid}} ]\]Threshold which makes profit:
\[Base\_Quote_{buy\_price} * Quote\_Mid_{buy\_price} * (1-Fee_{base\_quote}) * (1-Fee_{quote\_mid}) - \frac{Base\_Mid_{sell\_price}}{1-Fee_{base\_mid}} > 0\]2.2.6 Limit of amount in trade
From the above formula of profit, you may realize the value of \( Base\_amt \) is the limit which impact the final profit we gain. So in this part, let’s find out what’s the value of \( Base\_amt \).
2.2.6.1 the stock how much you can take
For an analogy, let’s back to the weird stall: after you detect the unreasonable quotation, you plan to use 2 USD as initial capital so that you could make 2 USD to 4 USD. However, when you’re ready to buy 2 melon using 2 USD in Melon Stall, there is just 1 melon in stock. so the limit of amount during the whole process emerges, which might be your initial capital, either the stock in a specific stall (the amount in the order of particular trading pair)
The stock in the above fruit stall equals the amount hang out in bid and ask order. I already mentioned that we just focus on the first gear of each trading pair. That means the amount of first gear is the limit because that’s the all we can take from that order. If we take more, then the excessive part will be charged at a different price.
In the positive arbitrage, we measured the initial variable is \( Base\_amt \). So later we ought to standardize on the amount of Base Currency.
Check the following figure, there 3 limits amount in 3 trading pairs:
-
\( Base\_Mid_{sell\_amt} \)
This’s the amount of Base Currency, no need convert.
-
\( Base\_Quote_{buy\_amt} \)
This’s the amount of Base Currency, no need convert.
-
\( Quote\_Mid_{buy\_amt} \)
This’s the amount of Quote Currency, so need to covert to the amount of Base Currency:
\[\frac{Quote\_Mid_{buy\_amt}}{Base\_Quote_{sell\_price}}\]
2.2.6.2 the money in wallet how much you can use
If you process the 3 transaction of those 3 trading pairs one by one to accomplish arbitrage, the limit defines the money how much you can use is only the initial capital. Because they are in sequence, the initial capital will ensure the amount of each cryptocurrency in each trading pair.
However, there is a truth not only in the traditional market, but cryptocurrency: the order hang out in the board is changing in every nanosecond, might it was dealt, either withdrew. To grab the chances, we need to take all 3 transactions as quick as possible. Hence we have to process them in parallel, instead of in sequence.
Since all the 3 transactions were processed in the same time, we need prepare the certain amount of each kind of cryptocurrency: Base, Mid and Quote Currency, which will impact the limit of trading amount in the arbitrage.
There are another 3 limits under this parallel architecture:
-
Balance in wallet of Mid Currency: \( Bal_{mid}\)
This’s the amount of Mid Currency, so need to convert to Base Currency:
\[\frac{Bal_{mid}}{Base\_Mid_{sell\_price}}\] -
Balance in wallet of Base Currency: \( Bal_{base}\)
This is the amount of Base Currency, no need to convert.
-
Balance in wallet of Quote Currency: \( Bal_{quote}\)
This’s the amount of Quote Currency, so need to convert to Base Currency:
\[\frac{Bal_{quote}}{Base\_Quote_{sell\_price}}\]
2.2.6.3 the value of \( Base\_amt \)
From the preceding description, there are 6 limits which will impact to the value of \( Base\_amt \). They are like 6 pieces of board in a bucket, the value of \( Base\_amt \) is the volumn of that bucket. Namely the lenghth of shortest board will decide the volumn, so:
\[Base\_amt = min(Base\_Mid_{sell\_amt} , Base\_Quote_{buy\_amt} , \frac{Quote\_Mid_{buy\_amt}}{Base\_Quote_{sell\_price}} , \frac{Bal_{mid}}{Base\_Mid_{sell\_price}} , Bal_{base} , \frac{Bal_{quote}}{Base\_Quote_{sell\_price}})\]2.2.7 the limit set by exchages
Normally, every exchange would set the minimum trading amount of each trading pair. here we simply remark it as \( MinEx_{base}\). It won’t be allowed if your trading amount is less than this threshold. So later in the program, this judgement should be added before arbitrage:
\[Base\_amt > MinEx_{base}\]2.3 Inverse Arbitrage
2.3.1 Flow Chart
2.3.2 Background
Same with positive arbitrage
2.3.3 Process
- To buy Quote Currency, the amount is \(Quote\_amt\). the Cost is selling \(Mid\_amt\_init\) Mid Currency
- Selling all Quote Currency which you bought in the previous step, then you gain some Base Currency, the amount is \(Base\_amt\)
- Selling all Base Currency which you gained in the previous step, then you retrieve the Mid Currency, the amount is \(Mid\_amt\_final\)
2.3.4 Threshold
As long as the \(Mid\_amt\_final\) is greater than \(Mid\_amt\_init\), that’s the unreasonable quotation we shoulld garb.
2.3.5 Formula
Primary variable: how much we should trade the Base Currency, which is remarked by: \(Base\_amt\)
The cost in step 1 after you sell the Mid Currency, which means how much Mid Currency you spend to buy Quote Currency:
\[Mid\_amt\_init = \frac{Quote\_amt * Quote\_Mid_{sell\_price}}{1-Fee_{quote\_mid}}\]The amount of Base Currency after you sold the Quote Currency in step 2:
\[Base\_amt = \frac{Quote\_amt * (1-Fee_{base\_quote})}{Base\_Quote_{sell\_price}}\]The amount of Mid Currency you retrieved in step 3:
\[Mid\_amt\_final = Base\_amt * Base\_Mid_{buy\_price} * (1-Fee_{base\_mid})\]The profit after one circulation:
\[Profit = Mid\_amt\_final - Mid\_amt\_init\]Substitue all variables and get the final expression:
\[Profit = Quote\_amt * [\frac{Base\_Mid_{buy\_price} * (1-Fee_{base\_mid})* (1-Fee_{base\_quote})}{Base\_Quote_{sell\_price}} - \frac{Quote\_Mid_{sell\_price}}{1-Fee_{quote\_mid}}]\]Threshold which makes profit:
\[\frac{Base\_Mid_{buy\_price} * (1-Fee_{base\_mid})* (1-Fee_{base\_quote})}{Base\_Quote_{sell\_price}} - \frac{Quote\_Mid_{sell\_price}}{1-Fee_{quote\_mid}} > 0\]2.3.6 Limit of amount in trade
From the above formula of profit, you may realize the value of \( Quote\_amt \) is the limit which impact the final profit we gain. So in this part, let’s find out what’s the value of \( Quote\_amt \).
2.3.6.1 the stock how much you can take
The analysis of inverse arbitrage is similar to positive arbitrage.
In the inverse arbitrage, we measured the initial variable is \( Quote\_amt \). So later we ought to standardize on the amount of Quote Currency.
Check the following figure, there 3 limits amount in 3 trading pairs:
-
\( Quote\_Mid_{sell\_amt} \)
This’s the amount of Quote Currency, no need to convert.
-
\( Base\_Quote_{sell\_amt} \)
This’s the amount of Base Currency, so need to convert the amount of Quote Currency:
\[Base\_Quote_{sell\_amt} * Base\_Quote_{buy\_price}\] -
\( Base\_Mid_{buy\_amt} \) This’s the amount of Base Currency, so need to covert to the amount of Base Currency:
\[Base\_Mid_{buy\_amt} * Base\_Quote_{buy\_price}\]
2.3.6.2 the money in wallet how much you can use
The analysis of inverse arbitrage is similar to positive arbitrage.
There are another 3 limits under this parallel architecture:
-
Balance in wallet of Mid Currency: \( Bal_{mid}\)
This’s the amount of Mid Currency, so need to convert to Quote Currency:
\[\frac{Bal_{mid}}{Quote\_Mid_{sell\_price}}\] -
Balance in wallet of Base Currency: \( Bal_{base}\)
This is the amount of Quote Currency, so need to convert to Quote Currency:
\[Bal_{base} * Base\_Quote_{buy\_price}\] -
Balance in wallet of Quote Currency: \( Bal_{quote}\)
This’s the amount of Quote Currency, so no need to convert.
2.3.6.3 the value of $ Quote\_amt $
The analysis of inverse arbitrage is similar with that of positive arbitrage, so:
\[Quote\_amt = min(Quote\_Mid_{sell\_amt} , Base\_Quote_{sell\_amt} * Base\_Quote_{buy\_price} , Base\_Mid_{buy\_amt} * Base\_Quote_{buy\_price} , \frac{Bal_{mid}}{Quote\_Mid_{sell\_price}} , Bal_{base} * Base\_Quote_{buy\_price} , Bal_{quote})\]2.3.7 the limit set by exchanges
Normally, every exchange would set the minimum trading amount of each trading pair. here we simply remark it as \( MinEx_{quote}\). It won’t be allowed if your trading amount is less than this threshold. So later in the program, this judgement should be added before arbitrage:
\[Quote\_amt > MinEx_{quote}\]3. Final Model
3.1 Model for positive arbitrage
3.1.1 Occasion
When the following expression is ture, we can perform positive arbitrage:
\[Base\_Quote_{buy\_price} * Quote\_Mid_{buy\_price} * (1-Fee_{base\_quote}) * (1-Fee_{quote\_mid}) - \frac{Base\_Mid_{sell\_price}}{1-Fee_{base\_mid}} > 0\]3.1.2 Profit
The profit after positive arbitrage is:
\[Profit = Base\_amt * [ Base\_Quote_{buy\_price} * Quote\_Mid_{buy\_price} * (1-Fee_{base\_quote}) * (1-Fee_{quote\_mid}) - \frac{Base\_Mid_{sell\_price}}{1-Fee_{base\_mid}} ]\]3.1.3 Propelled Quantity of Base Currency
The value of \( Base\_amt \) among the above expression is:
\[Base\_amt = min(Base\_Mid_{sell\_amt} , Base\_Quote_{buy\_amt} , \frac{Quote\_Mid_{buy\_amt}}{Base\_Quote_{sell\_price}} , \frac{Bal_{mid}}{Base\_Mid_{sell\_price}} , Bal_{base} , \frac{Bal_{quote}}{Base\_Quote_{sell\_price}})\]3.1.4 Exception when should skip
However, here is the exception, which you should skip this time, when:
\[Base\_amt < MinEx_{base}\]3.2 Model for inverse abritrage
3.2.1 Occasion
When the following expression is ture, we can perform inverse arbitrage:
\[\frac{Base\_Mid_{buy\_price} * (1-Fee_{base\_mid})* (1-Fee_{base\_quote})}{Base\_Quote_{sell\_price}} - \frac{Quote\_Mid_{sell\_price}}{1-Fee_{quote\_mid}} > 0\]3.2.2 Profit
The profit after inverse arbitrage is:
\[Profit = Quote\_amt * [\frac{Base\_Mid_{buy\_price} * (1-Fee_{base\_mid})* (1-Fee_{base\_quote})}{Base\_Quote_{sell\_price}} - \frac{Quote\_Mid_{sell\_price}}{1-Fee_{quote\_mid}}]\]3.2.3 Propelled Quantity of Quote Currency
The value of \( Quote\_amt \) among the above expression is:
\[Quote\_amt = min(Quote\_Mid_{sell\_amt} , Base\_Quote_{sell\_amt} * Base\_Quote_{buy\_price} , Base\_Mid_{buy\_amt} * Base\_Quote_{buy\_price} , \frac{Bal_{mid}}{Quote\_Mid_{sell\_price}} , Bal_{base} * Base\_Quote_{buy\_price} , Bal_{quote})\]3.2.4 Exception when should skip
However, here is the exception, which you should skip this time, when:
\[Quote\_amt < MinEx_{quote}\]4. Implementation
The development is not complicated if we just focus on monitoring the unreasonable quotation, excluding the perform the process of arbitrage.
You can invoke the API of exchanges provided, or even using requests package in Python to fetch the data from the website if there is no available API from exchanges. Then you can calculate the value base on the preceding formula in a loop to perform the monitor.
Regarding the implementation, I output the monitoring value into a file, then Zabbix on the backend will fetch and display on the site.
5. Development of Performing Arbitrage
The above model is constructed on the theory and is meaningful in the practical area. However, the model of theory is just a beginning, and the actual difficulties are on the stage of development. To clarify this, let’s think about several following questions:
Q: How should we fetch the quotation from exchanges?
A: API is preferable if there was any in exchanges. We just enclose API invoking in a class for each exchange, then calculate the value base on those parameter values we fetch from API. But there is no API provided, and we have to scrape the data from the website of exchanges using Python response package or other packages.
Q: The chance is emerging and elapsed with a nanosecond, so does it means we have to fetch those quotations within a very short interval?
A: Yes, the interval has to be shorter than 1 second. The shorter, the better. It’s better for no interval; namely, your program runs continuously in the background.
Q: Most sites will set the limit of API for releasing the server load. Invoking API to fetch quotation too frequently surely would cause the ban from server, so how?
A: Register more account from exchanges, there are several API assigned from exchanges under each account. Setting up an IP pool and using the polling mechanism.
Q: How if there is WebSocket available in exchanges?
A: Under our model, the WebSocket is better than API. Any changes on the quotation will push to us via WebSocket. We can push the real-time quotation via 3 channels fo WebSocket into a FILO list(first-in-last-out) for calculating the value of the expression. One more upside is there would be no limit from API, which can release some workload to get around the limit of API.
Q: How to realize the process 3 transactions in parallel?
A: Using multiple processes in Python.
Q: Is there any signal to communicate between different processes?
A: No need. Actually, there is no time for them to communicate. Timing is money, so the duty of each process is just finished their trading transaction ASAP.
Q: How if 2 transactions ran successfully, but another transaction failed? Is there any rollback?
A: There is no practical rollback in the markets. In other words, rollback is with enormous cost, like you want to sell an order after you buy it, then you have to use a lower price which would cause losses to you. If the situation what you mentioned above occurred, that would be a fatal issue in your coding. A fatal error will put in the log and stop the running. After you increase the robust abilities, then rerun it.
Q: How if 2 transactions ran successfully, but when another trading pair is about to deal, the quotation is changed. So how?
A: It is the most significant risk in the running, and just happened in the long term. The changes in the markets are in high-speed, that would cause what I mentioned above: the chances is emerging and elapsed within a nanosecond. So if the situation what you said above occurred, that means the chance elapsed already. Might someone take the order in front of our hands, either the owner of that order withdrew it in front of our eyes. But, we have no choice, and what we could do is GO AHEAD. I know, it must cause loss this time, but if we stop with the situation: 2 done and 1 not yet, it would break the balance and would cause a bigger loss potentially because you have an uncompleted order and nobody knows the quotation in the future. The more serious impact is that it would make your code lose the ability of sustainable running.
Q: How can I grab the chance with accomplishing all 3 transactions before others take some order in front of me?
A: You can refer the following point to speed up the process: Optimize your code, simplify some means, but be careful about the robust of your code. Using a fast network provider. Don’t use API proxy in a slow and remote network. You should gain the same IP proxy in the same network with your server. Put your server under the server room of exchanges. Plug a cable from your server to the server of exchanges.
