Risk-neutral Probability, the weight that prices options without forecasts
The artificial weight that makes the discounted stock grow at the risk-free rate. Not a forecast, and the reason an option price ignores where the stock is headed.
A 2.5 minute animated lesson on the risk-neutral probability, the weight that lets us price an option without ever forecasting the stock. Built for FIN301 and ECON3003 students who have just built a one-step binomial tree.
Real investors demand a premium for risk. Risk-neutral pricing sidesteps that by reweighting the up and down moves until the stock is expected to grow at the risk-free rate, which makes the discounted stock a martingale. The formula p = (e^(r·dt) − d) / (u − d) falls straight out. A worked example with spot A$50, up 1.3, down 0.7, and a 10% rate gives p = 0.6753, checked against an expected A$55.26.
The lesson closes on the point students most often miss. p is not where the stock will actually go, and the method does not assume anyone is risk-neutral. The replication argument, not the probability story, is what makes it work. Citations to Hull (2022), Cox Ross Rubinstein (1979), and Harrison and Kreps (1979) sit on the Atlas concept page.