No arbitrage iff there EMM $P^*$ theorem

Question

The definition of an arbitrage I was given:

"An arbitrage strategy is an admissible strategy with zero initial value and positive probability of a positive final value."

I think that an initial value of zero was not really necessary for an arbitrage, as long as all the values are non-negative and we have a strictly positive cash flow at sometime ( which could very well be the initial time). In fact I have seen examples in my exercises where this is the case!

Now I just finished learning the proof of the theorem stating that there is no arbitrage iff there is an equivalent martingale measure $P^*$ and in the proof we in fact make use of the fact that the initial value is zero.. So we in fact absolutely need this fact. So how come I have seen exercises where we are supposed to use NA arguments when the strategy does not have a zero initial value?

Answer 1

It's quite common to distinguish between Type A arbitrage and Type B arbitrage. We say that a trading strategy is a

• type A arbitrage if it has a positive initial cashflow and no risk of future loss.
• type B arbitrage if it has a nonnegative initial cashflow, no risk of future loss and a positive probability of future profit.

I think what you are referring to is a type A arbitrage. Obviously, definitions may vary.