I’ll be repetitive because it’s good practice.

Here’s a simple game.

I ask you to reach into a bag with 4 balls.

• If you pull out a green ball, I pay you $3
• If you pull any other color, you pay me $1

Assume the game is fair (ie has an expectancy of zero).

What’s the probability of pulling a green ball?

• Answer

The risk-neutral or arbitrage-free probability is the one that makes a proposition fair to just receiving the risk-free rate on your money.

In the above example, the bet is settled immediately so the risk-free rate can be ignored.

But for other random processes that occur over time, like the unfolding of a stock price path, we want to generalize the formula to account for the risk-free rate. We are effectively discounting the payoffs to present value where all comparisons are apples-to-apples and then implying the probabilities.