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In Tomas Bjork's Arbitrage Theory in Continuous Time (or here), $\exists$ 2 inconsistent definitions of arbitrage, which is correct?

The first definition is for the single period Binomial model enter image description here

The second definition is for the multi period Binomial model enter image description here

The second suggests that there is a possibility of the portfolio value ending up zero while the first does not...isn't arbitrage a free lunch? That is you will SURELY gain? And if you don't gain, how can you still call that arbitrage?

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I think you are seeking a pragmatic definition of arbitrage instead of a theoretical definition. For practical definitions, there are two kinds of arbitrage: statistical arbitrage and deterministic arbitrage. Suppose you have a lottery with 10 identical tickets only. Each ticket sells for \$10. The single, grand prize of the lottery is \$1000. Hence, the expectation value of each ticket is +$90.

Statistical arbitrage: If you can purchase any 1 of these tickets, this would be called a statistical arbitrage opportunity since the odds are in your favor. Provided a secondary market is allowed, you can probably expect to sell this ticket to another party for close to $90.

Deterministic arbitrage: If you can purchase all 10 of these tickets, this would be called a deterministic (or pure) arbitrage opportunity since odds no longer matter and you have a lock on the profit with respect to market risk.

Take note that arbitrage scenarios are conventionally understood to refer to such opportunities with respect to market risk only:

In this first case, even though the odds are in your favor, it doesn't necessarily mean that one should take the opportunity - the risk-reward should be acceptable to your appetite, opportunity costs and risk aversion. The great extent of variability in these factors cause the actual participants who may purchase your ticket from the secondary market to offer varying prices. The market risk is non-negligible.

In the second case, you can probably sell these tickets immediately to another party for close to $900. There is still some degree of credit counterparty risk, carry costs (e.g. you have to spend some effort keeping those tickets safe and cannot simply keep those tickets lying on the ground in public), but these are negligible, so the actual participants who may purchase your tickets from the secondary market in this case would probably offer a very narrow range of prices. Hence, the market risks are effectively negligible.

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Thanks! Our profs never differentiated. I just noticed the portfolio vs possibility difference. –  BCLC Jul 6 at 23:00
    
To clarify, is the first deterministic and the latter statistical? –  BCLC Jul 7 at 16:58
    
First case, 1 ticket: statistical. Second case, all tickets: deterministic. –  madilyn Jul 7 at 20:17
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@BCLC: Sorry, I thought you were asking about my response. Yes, for your initial question, I would see the first as deterministic and the latter as statistical. However, the lines are very blur here though and I'd avoid making that distinction unless required of you - the academic school of thought behind both definitions in your question generally doesn't distinguish between statistical and deterministic arbitrage, so what they really meant in those definitions was that the 1st is a stricter condition than the 2nd. –  madilyn Jul 7 at 20:37

They are definitions, so it makes no sense in asking which one is correct. However, the second one is the one that makes most sense, and it is the one you will see in most literature.

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(You could interpret them as mathematical translations of what we commonly know as arbitrage and then ask "Which formulations are correct"?) Why? Isn't arbitrage a free lunch? That is you will SURELY gain? And if you don't gain, how can you still call that arbitrage? –  BCLC Jul 4 at 11:58
    
Yes there are multiple notions of arbitrage, for example whether the strategy is admissible (bounded for all t), in general no-arbitrage means you cannot get riskfree profit by net zero investment. –  emcor Jul 4 at 12:19
    
@emcor So doesn't it follow that arbitrage means you can get riskfree profit by net zero investment? My point is: Is it profit if the amount you will or might receive is zero? –  BCLC Jul 7 at 16:58
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@BCLC Yes you are correct, the first definition is a more strict version of the second, in general it suffices to just have nonzero probability for profit and zero profit else. –  emcor Jul 7 at 17:02

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