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Binomial Option Pricing

Cox-Ross-Rubinstein · Discrete-time lattice · $\mathbb{Q}$-measure
Family 02 Pricing DP on tree American OK

Cox, Ross & Rubinstein (1979) discretised the continuous Black-Scholes diffusion into a recombining binomial tree with up-factor $u$, down-factor $d$, and risk-neutral probability $p$. The tree prices European and American options by backwards induction, converges to Black-Scholes as the number of steps grows, and needs no PDE solver. For practitioners, it is the simplest non-trivial pricing engine that handles early exercise — something Black-Scholes cannot do.

Educational Purpose
This page is an educational demonstration of discrete-time option pricing. It is not investment, hedging, or trading advice. The CRR tree inherits Black-Scholes’ limiting assumptions (constant $\sigma$, constant $r$, no dividends) at the continuous limit; the discrete version additionally introduces discretisation error that decays as $O(1/N)$.

The model

Discrete tree, risk-neutral probabilities, backwards induction

OR family: Derivative pricing Solver class: DP on lattice Measure: Risk-neutral $\mathbb{Q}$ Realism: ★★★ Exact (under CRR assumptions)

Notation

SymbolMeaning
$N$Number of time steps
$\Delta t$Step size, $\Delta t = T/N$
$u$Up-move factor per step
$d$Down-move factor per step; $d = 1/u$ in CRR
$p$Risk-neutral up-probability
$S_{n,j}$Underlying price at node $(n,j)$: $n$ steps, $j$ up-moves
$V_{n,j}$Option value at node $(n,j)$

CRR parameters

Tree parameters (Cox, Ross & Rubinstein, 1979) $$ u = e^{\sigma\sqrt{\Delta t}}, \qquad d = e^{-\sigma\sqrt{\Delta t}} = 1/u $$ $$ p = \frac{e^{r\Delta t} - d}{u - d} $$
$p$ is the risk-neutral up-probability that makes the discounted underlying a $\mathbb{Q}$-martingale. Requires $d < e^{r\Delta t} < u$ (no-arbitrage).

Backwards induction

At maturity ($n = N$), option value equals payoff: $V_{N,j} = g(S_{N,j})$ where $g(S) = (S-K)^+$ for a call, $(K-S)^+$ for a put. Step backward from $n=N-1$ to $n=0$:

European option (backwards induction) $$ V_{n,j} \;=\; e^{-r\Delta t} \bigl[\, p \, V_{n+1,j+1} + (1-p) \, V_{n+1,j} \,\bigr] $$
Discounted expected value under $\mathbb{Q}$. At node $(0,0)$, $V_{0,0}$ is the option price.
American option (backwards induction with early-exercise check) $$ V_{n,j} \;=\; \max\Bigl\{\, g(S_{n,j}), \;\; e^{-r\Delta t} \bigl[\, p \, V_{n+1,j+1} + (1-p) \, V_{n+1,j} \,\bigr] \,\Bigr\} $$
At every node, take the greater of exercising now (intrinsic value $g$) or holding (continuation value). This is the key advantage over Black-Scholes.

Convergence

As $N \to \infty$, the CRR tree converges to the Black-Scholes price at rate $O(1/N)$ (Leisen & Reimer 1996 give $O(1/N^2)$ for improved tree parameters). Practical values: $N = 100$ gives 3-4 decimal places of accuracy; $N = 10^4$ reaches machine precision for European options but takes quadratic memory without sparse tricks.

Interactive tree

Visualise the lattice and price convergence; compare to Black-Scholes

CRR Binomial Tree

★★★ Exact under discrete model
100
100
1.00
3.00%
25.00%
6
CRR price
—
BSM price (European)
—
Difference
—
$u$ factor
—
$p$ (RN prob.)
—

Each node is labelled with the underlying price $S_{n,j}$. Node colour intensity encodes option value at that node (gold = high, dim = low).

Convergence: CRR European price as a function of $N$, with the Black-Scholes price as a horizontal reference. American prices appear above the BS line (early-exercise premium).

Assumptions & uses

When binomial beats Black-Scholes — and when it doesn’t

When to prefer the binomial tree

  • American options. Early exercise has no Black-Scholes closed form; the tree handles it natively via the $\max$ at each node.
  • Discrete dividends. Known cash dividends at specific dates are easy to incorporate by shifting the tree after the ex-dividend node.
  • Pedagogy. No stochastic calculus required — the risk-neutral argument reduces to a one-period portfolio of stock and bond.
  • Path-independent exotic payoffs with finite states.

When to prefer Black-Scholes or Monte Carlo

  • European options. Closed-form BSM is exact, instant, and doesn’t require tree construction.
  • Path-dependent payoffs (Asian, barrier, lookback) usually break tree recombination; Monte Carlo handles them cleanly.
  • Multi-factor models. Trees become hyper-dimensional (Boyle 1988); Monte Carlo scales linearly in number of factors.
  • Very short-dated options need many small steps — memory grows $O(N^2)$ without explicit recombination.

Key references

Seminal papers and extensions

Cox, J. C., Ross, S. A. & Rubinstein, M. (1979).
Option Pricing: A Simplified Approach.
Journal of Financial Economics, 7(3), 229–263. doi:10.1016/0304-405X(79)90015-1
Rendleman, R. J. & Bartter, B. J. (1979).
Two-State Option Pricing.
The Journal of Finance, 34(5), 1093–1110. doi:10.1111/j.1540-6261.1979.tb00058.x
Boyle, P. P. (1988).
A Lattice Framework for Option Pricing with Two State Variables.
Journal of Financial and Quantitative Analysis, 23(1), 1–12. doi:10.2307/2331019
Leisen, D. P. J. & Reimer, M. (1996).
Binomial Models for Option Valuation — Examining and Improving Convergence.
Applied Mathematical Finance, 3(4), 319–346. doi:10.1080/13504869600000015
Black, F. & Scholes, M. (1973).
The Pricing of Options and Corporate Liabilities.
Journal of Political Economy, 81(3), 637–654. doi:10.1086/260062 (limiting case)
Hull, J. C. (2021).
Options, Futures, and Other Derivatives, 11th ed., chs. 13, 21.
Pearson. ISBN 978-0-13-693997-9.
Shreve, S. E. (2004).
Stochastic Calculus for Finance I: The Binomial Asset Pricing Model.
Springer. ISBN 978-0-387-40100-3.

Related sub-applications

The two continuous-time pricing engines and one stochastic method.

Finance landscape Black-Scholes Monte Carlo Real options
Reminder — educational purpose only
The CRR tree is exact only under its assumptions. Real markets exhibit stochastic volatility, jumps, and transaction costs — all violated here. Not investment advice; historical model performance never implies future results.