Martingales for Dummies, Part 2

What to remember

• A martingale is a fair game: $E[\text{next} \mid \text{now}] = \text{now}$.
• Optional stopping: $E[X_T] = X_0$ under boundedness conditions.
• No betting strategy beats a fair game with finite money.
• Under the risk-neutral measure the discounted stock, every self-financing portfolio and every derivative are martingales; that rules out arbitrage and gives the pricing formula (the strike-12 call: 1.28).