Binomial Option Pricing (Cox–Ross–Rubinstein)

Victor Miller · GitHub · LinkedIn

An implementation of the Cox-Ross-Rubinstein binomial tree for European and American options.

Table of Contents

• Key results
• How it works
• What is inside
• Getting started
• Repository structure
• License

Key results

All checks use $S_0 = 100$, $K = 100$, $T = 1$, $\sigma = 20%$, $r = 5%$, and $n = 5000$ steps.

Check	Tree	Reference
European call	10.4502	Black-Scholes: 10.4506
Call minus put	4.8771	$S_0 - Ke^{-rT}$: 4.8771
Call hedge ratio	0.6368	$N(d_1)$: 0.6368
American put	6.0902	European put: 5.5731

The tree price converges to Black-Scholes as $n$ grows; the error decays like $1/n$.

How it works

The stock moves up by $u = e^{\sigma\sqrt{\Delta t}}$ or down by $d = 1/u$. The up probability

$$p = \frac{e^{r\Delta t} - d}{u - d}$$

is chosen so the stock earns the risk-free rate on average. The option is priced backwards from its payoff, discounting at $r$ each step. American options add one check per node: the value is the maximum of keeping the option and exercising it.

What is inside

1. European options: european_option_binomial_model prices a call or a put with the CRR tree and returns the price plus the hedge ratio $\Delta = (C_u - C_d) / (S_0 (u - d))$.
2. American options: american_option_binomial_model adds the early-exercise check at each node: keep the option or exercise it, whichever is worth more.
3. Put-call parity: a call and a put at the same strike and maturity must satisfy $C - P = S_0 - Ke^{-rT}$; the tree reproduces it exactly.
4. Black-Scholes reference: black_scholes_option_pricer gives the closed-form price and hedge ratio used as benchmark.
5. Convergence: the figure above plots the tree error against the $1/n$ reference slope.

Getting started

pip install -r requirements.txt

Open binomial_model.ipynb and run all cells.

Repository structure

binomial_model/
├── binomial_model.ipynb   # model, validation, figures
├── README.md
├── requirements.txt
├── LICENSE.txt
├── .gitignore
└── figures/
    └── convergence.png

License

MIT