One of the harder ideas in fixed income is risk-neutral probabilities. In this video, I'd like to specifically illustrate, and define, what we mean by risk-neutral probabilities. I will do this in three steps. The first one is just a simple example of a coin toss, where my objective is to illustrate what we mean by risk-neutral probabilities. These are the probabilities that equate the expected discounted value to the market price of the instrument. Then, having defined the concept, in the second sheet I've replicated Bruce Tuckman's example in Chapter 7 where he retrieves the risk-neutral probabilities that are implied by a 1-year zero-coupon bond. We'll take those risk-neutral probabilities and go to the third sheet and use them to price or value an option on that same bond so that's a contingent claim. Then we'll see why there is some magic to these risk-neutral probabilities.