If you read a scientific article on finance, you often find a phrase
like "we assume an arbitrage-free market". On the other hand, you may
have heard of people or trading firms that claim they are
practicing arbitrage. Often, the people in firms that claim they do
arbitrage happily read articles that assume there is no arbitrage. In this
node, I will attempt to clarify this seeming contradiction.
What is an arbitrage-free market?
An arbitrage-free market is a market in which there is no arbitrage.
Arbitrage is defined as the possibility to do a set of trades that
leaves you with no position, but money in the pocket. I'll illustrate this
with a simple example. Assume, for instance, that it is possible to buy
yens at 100 per dollar. Assume furthermore that it is possible to buy
euros for 150 yen a piece. If we can now buy more than 1.50 dollar per
euro, say 1.51 dollar, we can make a profit. Say we start out with 1.50
dollar. We then buy 150 yen for this. This buys us precisely 1 euro. For
this 1 euro, we buy 1.51 dollar, having gained 1 cent. This form of
arbitrage is by the way known as triangle arbitrage. If we can do this as
often as we want, we -or anyone!- could make any profit we want.
Mathematically, what we have just done is no different from stating that
1.50 = 1.51. This breaks math at a fundamental level; the "normal"
definition of a number breaks down, because they essentially all have the
same value. As such, we can't work with numbers anymore. So, one could argue
that the main reason we need to assume an arbitrage-free market is that we
otherwise couldn't do any math, or anything meaningful at all.
Arbitrage-free in practice
Of course, everyone wants to pick up this free penny. As such, a lot of
people are searching the market for these opportunities. What
they could do, for instance, is offering you to buy the dollars at for
instance .67 eurocent. If you do, they - or rather, their
computers - immediately buy 100 yen and convert these to 100
dollars, and they pocket something like .3 eurocent. In practice, the profit
per trade can be even lower, as one incurs costs in trading; one has
to pay the exchange, for instance. Furthermore, especially for a liquid
set of currencies like this, the profit would be much smaller.
In fact, one can count on it that as soon as the profit is higher than the
costs, some computer will trade and the opportunity will be gone.
On a whole, such razor-thin margin, high frequency arbitrage will make a
handsome but not spectacular profit. It is noted that this introduces
liquidity in the market.
In many cases, however, the so-called "arbitrage" that is practiced is not
true arbitrage. One could, for instance, note that if for instance the Dow
Jones Index goes up, European indices will usually close higher the
next day. By consistently betting on this, money could be made. This
is known as statistical arbitrage.
A very interesting form of arbitrage is arbitrage that isn't arbitrage
after all. For instance, consider someone buying a structured product from
a bank. As a simple example, this structured product pays 5% of interest
every year the Eurostoxx 50 index is above 2000 (it's currently close to
3000). You borrow the cash for 4%, buy a 2000 put for less than that 1%,
and presto, arbitrage!. Well, not really. See, the structured product
claims the bank will pay you back your investment. If this bank fails,
you will be left with a debt to the person you borrowed the cash from.
You are left with credit risk. Of course, if you can convince your boss
you made a profit, the only hedge you need is the O'Hare hedge.
On the one hand, academic
literature often talks about arbitrage-free markets
. On the other hand, trading firms claim they
practice arbitrage. The reason for this discrepancy
is on the one hand
that the academic
paper has to make this claim in order to be able to use
in the first place, so even if markets were not
arbitrage-free, they'd still have to claim this. On the other hand, markets
are mostly arbitrage-free: most so-called arbitrages are not total, true
arbitrages, or are on such razor-thin margins
that the profit per
trade is almost negligible. As such, they do not meaningfully distort the