Martingales and the importance of risk-neutral probabilities

This paper explains the role of risk-neutral probabilities in the binomial model, showing how they ensure that the discounted stock price is a martingale and that arbitrage is avoided. It introduces martingales, submartingales, and supermartingales as key probabilistic tools for understanding expected price movements. By connecting these ideas, the paper provides an accessible foundation for option pricing and highlights why the risk-neutral measure is central in quantitative finance.