Multi-Period Binomial Trees and American Option Pricing

Two extensions make this practically important. First, American options can be priced by testing for early exercise at each node: the option value is the maximum of the continuation value and the immediate payoff. This is why the binomial tree is still used in production systems for American options (and Bermudan/barrier variants) even after Black-Scholes is available for European options. Second, calibrating the up/down factors $u = e^{\sigma\sqrt{\Delta t}}$ and $d = 1/u$ to market-observed volatility $\sigma$ connects the discrete tree to the continuous-time Black-Scholes world — and as $n \to \infty$, the binomial price converges to the Black-Scholes price.

• apply backward induction to a two-step binomial tree to price a European option;
• extend backward induction to an American option by testing for early exercise at each node;
• state the CRR parameterisation $u = e^{\sigma\sqrt{\Delta t}}$, $d = e^{-\sigma\sqrt{\Delta t}}$ and $p = (a - d)/(u - d)$ where $a = e^{r\Delta t}$;
• apply the four-formula set (eqs. 13.15–13.18) to build a volatility-calibrated binomial tree;
• explain how delta changes through the tree and why this requires portfolio rebalancing;
• state the convergence result: as $n \to \infty$, the binomial European price converges to the Black-Scholes price.

1. Two-step binomial tree for a European option
2. European put: two-step example
3. American options: adding the early-exercise test
4. Delta and dynamic hedging in the tree
5. Calibrating the tree to market volatility
6. Convergence to Black-Scholes