A lot of people and funds claim that they perform arbitrage in order to make a profit. Sometimes it is a little confusing because the term "arbitrage" is used by different individuals to mean quite different things (especially with terms such as stat arb). I think arbitrage can carry a few distinct meanings:
1. The real, good-old sure profit arbitrage
This refers to the option-pricing kind of arbitrage. The PnL is deterministic, since you are basically earning a sure profit by long-shorting the same security across different markets to exploit the price discrepancy. Examples would be a) trading the same contract listed on two exchanges when the prices diverge; and 2) synthesizing a contract (e.g. future) using something else (e.g. call and put).
2. Statistical arbitrage, looking at historical data
This refers to fitting a model to the historical performance. A classic example is pairs trading, in which the price of two assets are expected to return to the "normal" level when a deviation is observed. In a sense, this kind of arbitrage relies on the real world probability distribution as suggested by historical data.
3. Statistical arbitrage, looking at market data
This refers to fitting a model to the market prices. For example, one can try to see if the deep OTM options are over-priced or under-priced, by studying the Greeks. In a sense, this kind of arbitrage relies on the risk-neutral probability distribution as suggested by market data.
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