Definition of Arbitrage

Definition. An arbitrage is a portfolio $H$ ∈ $R^n$ such that

• $H⋅P_0≤0≤H⋅P_1$ almost surely, and

• $P(H⋅P_0=0=H⋅P_1)<1$.

where $P_0$ and $P_1$ ∈ $R^n$ represent the prices at time $t=0,1$ respectively.

Now, my question is why do we need the first condition. Suppose there is only one asset $A$ which at time $t=0$ costs $3$. Then, at $t=1$ we have $P(A=2)=\frac{1}{2}$ and $P(A=1)=\frac{1}{2}$. The portfolio $H=−1$ should be an arbitrage because it yields certain profit with no risk attached but it isn't because $H⋅P_1<0$.

Answer

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  Conceptually, an arbitrage gives you something for nothing.

  
  


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  This is a different idea than making or losing money almost surely. A risk free bond allows you to make money almost surely, but it isn't an arbitrage.

  
  

What's going wrong, the source of confusion in your example?

You've implicitly assumed the existence of a cash security, with interest rate 0 (but haven't made it explicit)

In your example, you've assumed that cash can be moved between $t=0$ and $t=1$ with an interest rate of 0. You need a security to do this.

If there exists a risk free security returning 0, then making money almost surely violates the law of one price (because you effectively have 2 different risk free rates). With unrestricted buying & selling, you can then construct an arbitrage by going long the high rate and short the low rate.

See my comment, below @noob2's quality answer.

The Law of One Price (linearity) and No Arbitrage

Two different concepts that often get conflated (even in texts) are:

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  The law of one price, that is, that the pricing function is linear. If $X$ and $Y$ are random variables representing payoffs, $\alpha$, and $\beta$ are scalars, and $f$ is the pricing function, linearity of the pricing function $f$ implies: $$ f(\alpha X + \beta Y) = \alpha f(X) + \beta f(Y) $$ The idea here is that the price of a portfolio should be linear in the price of its components.

  
  


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  The absence of arbitrage. Loosely speaking, no arbitrage requires that any security with strictly positive payoffs should have a positive price, that you can't get something for nothing.

  
  

The law of one price allows you to write the pricing function as an inner product with state prices. The absence of arbitrage implies those state prices are positive.

How the two different concepts get conflated is that violations of linearity can allow you to construct an arbitrage. People often say no arbitrage when they really mean linearity and no arbitrage.

References

Cochrane, John, Asset Pricing, 2005