<p>I am trying to formulate an understanding of arbitrage to ensure I comprehend it. There are many reference materials online, of course.</p>
<p>I envision a set of commodities <code>C</code> and a set of owners <code>O</code>. Each commodity is associated with its quantity type <code>Q_t</code>. (Water can be quantified by positive real numbers, apples by natural numbers, Bitcoin by positive rational numbers with a minimum increment of 1 Satoshi). For example, owner <code>o_1</code> is associated to a set <code>{(c_1, q_1), (c_2, q_2), ...}</code>, where <code>c_1, c_2</code> are commodities they own, and <code>q_1, q_2</code> are quantities of those commodities (which are values in the set of quantities which that commodity can take).</p>
<p>An exchange <code>E</code> is a vendor with a set of commodity buy-sell pairs: a set of tuples <code>(b, q, r)</code> where <code>b</code> is a base currency, <code>q</code> is a quote currency, and <code>r</code> is an exchange rate. If <code>b</code> is Bitcoin, <code>q</code> is USD, and <code>r</code> is 5000, that means that the vendor will buy or sell 1 Bitcoin for 5000 dollars.</p>
<p><em>(Perhaps exchanges have asymmetrical exchange values?)</em></p>
<p>A trade occurs when a quantity of one commodity and a quantity of another commodity switch owners. Let there be a <code>trade</code> function which accepts two owner-commodity-quantity triples from the set of all owners, and returns an updated set of all owners: <code>trade: (O x C x Q) x (O x C x Q) -&gt; O_all</code>, such that <code>trade((o_1, c_1, q_1), (o_2, c_2, q_2))</code> equals the set of all owners <code>O_all</code>, where for owner <code>o_1</code>, the commodity <code>c_1</code> now has value <code>q_c_1 - q_1</code> (the original quantity minus the traded quantity), and the commodity <code>c_2</code> now has value <code>q_c_2 + q_2</code>, and analogously for <code>o_2</code>.</p>
<p>Consider that the exchange rates <code>r1, r2</code> for the same currency-pair <code>(b, q)</code> on exchanges <code>E1, E2</code> are not equal; <code>r1 &lt; r2</code> for <code>(b, q, r1)</code> in <code>E1</code>, <code>(b, q, r2)</code> in <code>E2</code>. Maybe one way we can define how &quot;buying low and selling high&quot; is profitable is by defining the value of each currency in reference to some &quot;stable third currency&quot;. Maybe we can define the &quot;average market value exchange rate&quot; for all currency pairs. Selling a commodity on an exchange where the rate is above market rate, and buying a commodity on an exchange where the rate is below market rate, is therefore profitable.</p>
<p>I haven't come close to making this formal definition complete, but I was wondering if someone could comment on it, critique it, or enhance it.</p>

CC BY-SA 4.0
2024-08-31T03:35:57.873Z
Hi Julius: I'll just comment. I don't know a nice mathematical definition but one thing you need to be clearer on is what kind of arbitrage ? Are you referring just to commodities or any asset class. The general idea of arbitrage is of course that you can outlay some amount of money ( enter a trade ) at time zero and no matter what happens over time, eventually close the trade so that you receive more than you laid out. Essentially no risk, yet reward. Take a look in an asset pricing text such as cochrane or duffy. Note that it won't be defined with respect to commodities in those texts.
mark leeds
CC BY-SA 4.0
2024-08-31T03:40:50.073Z
Note that the texts I mentioned are pretty heavy in theory so I recommend that you read them from the beginning rather than just jumping in and trying to find a definition of arbitrage.
mark leeds
CC BY-SA 4.0
2024-08-31T03:44:15.660Z
duffie's is called "dynamic asset pricing theory" and cochrane's is "asset pricing". There may be pdfs out there so you don't have to purchase them. I haven't looked recently.
mark leeds
<mathematical-economics><arbitrage>
A simple mathematical definition of arbitrage
Julius Hamilton
# A simple mathematical definition of arbitrage

(Asked by: username_0 on 2024-08-29)

I am trying to formulate an understanding of arbitrage to ensure I comprehend it. There are many reference materials online, of course.

I envision a set of commodities `C` and a set of owners `O`. Each commodity is associated with its quantity type `Q_t`. (Water can be quantified by positive real numbers, apples by natural numbers, Bitcoin by positive rational numbers with a minimum increment of 1 Satoshi). For example, owner `o_1` is associated to a set `{(c_1, q_1), (c_2, q_2), ...}`, where `c_1, c_2` are commodities they own, and `q_1, q_2` are quantities of those commodities (which are values in the set of quantities which that commodity can take).

An exchange `E` is a vendor with a set of commodity buy-sell pairs: a set of tuples `(b, q, r)` where `b` is a base currency, `q` is a quote currency, and `r` is an exchange rate. If `b` is Bitcoin, `q` is USD, and `r` is 5000, that means that the vendor will buy or sell 1 Bitcoin for 5000 dollars.

*(Perhaps exchanges have asymmetrical exchange values?)*

A trade occurs when a quantity of one commodity and a quantity of another commodity switch owners. Let there be a `trade` function which accepts two owner-commodity-quantity triples from the set of all owners, and returns an updated set of all owners: `trade: (O x C x Q) x (O x C x Q) -> O_all`, such that `trade((o_1, c_1, q_1), (o_2, c_2, q_2))` equals the set of all owners `O_all`, where for owner `o_1`, the commodity `c_1` now has value `q_c_1 - q_1` (the original quantity minus the traded quantity), and the commodity `c_2` now has value `q_c_2 + q_2`, and analogously for `o_2`.

Consider that the exchange rates `r1, r2` for the same currency-pair `(b, q)` on exchanges `E1, E2` are not equal; `r1 < r2` for `(b, q, r1)` in `E1`, `(b, q, r2)` in `E2`. Maybe one way we can define how "buying low and selling high" is profitable is by defining the value of each currency in reference to some "stable third currency". Maybe we can define the "average market value exchange rate" for all currency pairs. Selling a commodity on an exchange where the rate is above market rate, and buying a commodity on an exchange where the rate is below market rate, is therefore profitable.

I haven't come close to making this formal definition complete, but I was wondering if someone could comment on it, critique it, or enhance it.

**Tags:** mathematical-economics, arbitrage
> **Comments:**
> **username_1**: Hi Julius: I'll just comment. I don't know a nice mathematical definition but one thing you need to be clearer on is what kind of arbitrage ? Are you referring just to commodities or any asset class. The general idea of arbitrage is of course that you can outlay some amount of money ( enter a trade ) at time zero and no matter what happens over time, eventually close the trade so that you receive more than you laid out. Essentially no risk, yet reward. Take a look in an asset pricing text such as cochrane or duffy. Note that it won't be defined with respect to commodities in those texts.  
> **username_1**: Note that the texts I mentioned are pretty heavy in theory so I recommend that you read them from the beginning rather than just jumping in and trying to find a definition of arbitrage.  
> **username_1**: duffie's is called "dynamic asset pricing theory" and cochrane's is "asset pricing". There may be pdfs out there so you don't have to purchase them. I haven't looked recently.  

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