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I'm currently catching up on material presented in the edX-MIT course Foundations of Mondern Finance 1, in which they present a definition of arbitrage that doesn't quite make sense to me. Informally, I understand the notion of a "free lunch": that a market participant has exploited some opportunity in an imperfect market, where a difference in price allows profit to be made without any real risk. With that in mind, I'm having a difficult time using the definition that they provide to connect the dots.

An arbitrage (free lunch) is a set of trades in the financial market such that it:

  • requires non-positive initial cash flow / investment
  • yields non-negative future payoffs
  • at least one of the inequalities is strict.

In attempting to understand, I've tried to restate the above definition in mathematical terms. Let's say that we can map each time period $0 \leq t \leq T$ to the sets $CF^0, CF^1 ,\dots, CF^T$. Here $CF^t$ is a set of scalars $\{cf^t_0, cf^t_1, \dots\}$ containing all the cash flows — positive and negative — that occurred in the time period $t$. Then we can split each $CF^t$ into two sets: $CF^{t+}$ for positive cash flows at time $t$, and $CF^{t-}$ for negative cash flows at time $t$. Finally we can define the scalar net cash flow at time $t$ as a simple sum: $ncf^t=\sum_{CF^t}{cf}$

They give a simple example of arbitrage in the course slides:

Citi’s 12-month lending rate is 1% and Chase is selling 12-month certificate of deposit (CD) at an interest rate of 1.125%.

Arbitrage trades:

  1. Borrow \$100 from Citi at interest rate of 1% per year,
  2. Buy \$100 worth of 12-month CD from Chase at 1.125% per year.

In this example, $CF^0 = \{100, -100\}$ for borrowing money and buying the CD, and $CF^1 = \{-101, 101.125\}$ for repaying the loan and cashing out the CD. That gives $ncf^0 = 0, ncf^1 = 0.125$, which clearly satisfies the conditions of an arbitrage using the definition provided. Net cash flow at $t=0$ is zero (non-positive), and the sum of all net cash flows for $t>0$ is $0.125$ (non-negative, strictly positive).

Less clear is the following example that they provide:

IBM shares are trading on New York Stock Exchange (NYSE) at \$195 and London Stock Exchange (LSE) at £120, and the pound/dollar exchange rate is at \$1.50/£. Arbitrage trades:

  1. Sell 1 share of IBM at NYSE for \$195,
  2. Convert $190 into pounds at \$1.50/£, obtaining 130,
  3. Buy 1 share of IBM at LSE at £120

The way they describe it in the course slides, everything is happening in the same time period, so the line drawn between the "initial" and "future" cash flows is hard to see. To be clear, it's not that the "free lunch" isn't obvious here either — what I'm having difficulty with is how to apply the definition as stated to prove arbitrage conditions were met.

Beyond that, a more contrived example makes it even more difficult to understand. What if, for instance, I take \$100 and burn it with a lighter, and then wake up the next day with no money in my pocket? Technically, according to the definition they provided, that's also arbitrage. $ncf^0 = -100, ncf^1 = 0$ so that would be a non-positive initial cash flow, followed by a non-negative future payoff, with the first inequality being strict. That's absolute nonsense though, so something is obviously missing here.

I guess my TLDR is:

  1. Is there something missing from the definition provided that would complete a mathematical formulation of arbitrage?
  2. If not, how can I use the above conditions to prove the presence of a "free lunch" in a rigorous manner?

Regarding point #1, I've also looked at some of the other questions on this SE dealing with arbitrage, and seen differently formulated definitions with more advanced notation. Yet being completely new to finance, I struggle to connect them to the definition I explained here, which is probably where I need some help.

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    $\begingroup$ The definition of arbitrage suffices: The initial trade (at $t_0$) nets you a profit (free of risk), and there is zero pnl at any future date with certainty. First inequality is strict $>$, second inequality is not strict $\geq$. HTH? $\endgroup$ Nov 10, 2021 at 7:19
  • $\begingroup$ Informally there are two kinds of arbitrage: (1) You can make some (>0) money now and not owe anything in any state in the future (>=0 cash flow in all states), or (2) You pay nothing now (=0) and have a chance to make positive money in some future state and lose nothing in other future states (>=0 with al least one strict inequality). $\endgroup$
    – nbbo2
    Nov 13, 2021 at 13:12

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