You have already seen how a simple binomial tree can price an option by mimicking the random walk of a stock. But a plain tree often wobbles toward the true price, producing an irritating saw‑tooth pattern as you add more steps. In this chapter we straighten out that wobble, learn how to get a smooth convergence with far fewer steps, and pull out the all‑important risk numbers—delta, gamma, and theta—directly from the branches, without any extra simulation.