Risk-neutral Valuation: A Gentle Introduction (1)

Abstract

Risk-neutral valuation is simple, elegant and central in option pricing theory. However, in teaching risk-neutral valuation, it is not easy to explain the concept of 'risk-neutral' probabilities. Beginners who are new to risk-neutral valuation always have lingering doubts about the validity of the probabilities. What do the probabilities really mean? Are they real or fictional? Where do they come from? What is the relationship between the risk-neutral probabilities and the actual probabilities? Does it mean that all investors are risk-neutral? When is it appropriate to use the risk-free rate as the discount rate? From a pedagogical point of view, in the beginning it is best to avoid the use of probabilities because probabilities can be a barrier to understanding. Instead, it is far preferable to introduce the idea of state prices and then show that the approach with risk-neutral probabilities is equivalent to the use of state prices. In this teaching note, we use simple one-period examples to explain the intuitive ideas behind risk-neutral valuation. It is a gentle introduction to risk-neutral valuation, with a minimum requirement of mathematics and prior knowledge. We will provide the motivation and the rationale for calculating state prices and we will show that the risk-neutral approach is simply another way of looking at the issue of state prices.