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The Mountain Path Academy
The Mountain Path Academy
Finance Ā· Risk Management Ā· Quantitative Analytics
Derivatives Ā· FRM Core

Binomial Option Pricing

Risk-Neutral Valuation on a Price Tree

European & American Options Ā· CRR Ā· Early Exercise Ā· Convergence

Prof. V. RavichandranProfessor of Finance Ā· Corporate Finance, Banking & Academia
Learner-Friendly Study GuideMBA Ā· CFA Ā· FRM Ā· Derivatives Analytics

The binomial model breaks an optionā€™s life into discrete steps. At each step, the underlying moves up or down. The model prices the terminal payoff first, then works backward using no-arbitrage risk-neutral valuation.

1Why Use a Binomial Tree?

A tree makes the pricing logic visible at every node. It handles European exercise, American early exercise, dividends and changing inputs more naturally than a single closed-form formula.

Transparent

Inspect the stock price, payoff, continuation value and exercise decision at every node.

Flexible

Value calls or puts with European or American exercise.

Convergent

Under CRR assumptions, a fine tree approaches the Blackā€“Scholes price.

Core sequence
Build the stock tree, calculate terminal payoffs, then discount risk-neutral expected values backward to the root.

2Building the Stock-Price Tree

Let each step have length \(\Delta t=T/n\). From any node, the stock moves up by factor \(u\) or down by factor \(d\):

\[S_{i,j}=S_0u^jd^{i-j}\]

Here, \(i\) is the step number and \(j\) is the number of up moves. A recombining tree satisfies \(ud=1\), so an up-then-down path reaches the same price as a down-then-up path.

\(S_0\)ā†’\(S_0u\) or \(S_0d\)ā†’Terminal stock prices

3Risk-Neutral Probability

Choose the probability that makes the expected stock growth equal to the risk-free carry after dividend yield:

\[p=\frac{e^{(r-q)\Delta t}-d}{u-d},\qquad 1-p=\frac{u-e^{(r-q)\Delta t}}{u-d}\]
What \(p\) means
Risk-neutral \(p\) is not a forecast of the real-world probability that the stock will rise. It is a pricing weight that enforces no-arbitrage.

The one-step no-arbitrage condition requires \(d<e^{(r-q)\Delta t}<u\), ensuring \(0<p<1\).

4Terminal Payoff and Backward Induction

\[C_T=\max(S_T-K,0),\qquad P_T=\max(K-S_T,0)\]

At each earlier node, discount the risk-neutral expected value over one step:

\[V_{i,j}=e^{-r\Delta t}\left[pV_{i+1,j+1}+(1-p)V_{i+1,j}\right]\]

Repeat until the root node. The root value \(V_{0,0}\) is the binomial option price.

5One-Step Worked Example

Suppose \(S_0=100\), \(u=1.20\), \(d=0.85\), \(K=100\), \(r=5\%\), \(q=0\), and \(T=1\). The terminal call pays ā‚¹20 after an up move and ā‚¹0 after a down move.

\[p=\frac{e^{0.05}-0.85}{1.20-0.85}=0.575\]
\[C_0=e^{-0.05}\left[0.575(20)+0.425(0)\right]\approx 10.94\]

The one-step theoretical call value is approximately ā‚¹10.94.

Replication intuition
The option can also be valued by forming a delta position in the stock and borrowing or lending so that the portfolio matches both terminal payoffs.

6American Exercise

An American option may be exercised at any node. Compare immediate exercise with continuation value:

\[V_{i,j}^{Am}=\max\left(\text{intrinsic value},\ e^{-r\Delta t}[pV_u+(1-p)V_d]\right)\]
Early-exercise rule
Exercise when intrinsic value exceeds continuation value. American puts may be exercised early when deep in the money. A non-dividend-paying American call is generally not exercised early under standard assumptions.

7Coxā€“Rossā€“Rubinstein Specification

When volatility is an input, CRR chooses reciprocal up and down factors:

\[u=e^{\sigma\sqrt{\Delta t}},\qquad d=e^{-\sigma\sqrt{\Delta t}}=\frac{1}{u}\]

These factors match the variance of the continuous-time process as steps become small. Dividend yield enters through the risk-neutral probability.

SetupTree inputs
Manual factors, no yieldSpecify \(u\) and \(d\); set \(q=0\)
Manual factors with yieldSpecify \(u\), \(d\), and \(q\)
CRR volatility modeCompute \(u,d\) from \(\sigma\); include \(q\) in \(p\)

8Interactive Two-Step Pricer

Adjust spot, strike, maturity and volatility. The calculator uses a two-step CRR tree, a 6% rate and 1% dividend yield.

European binomial call and put calculator
Call value
Put value
Risk-neutral p
Up factor
Down factor
Steps2

9Convergence and Limitations

As the number of CRR steps increases, a European option price converges toward the Blackā€“Scholes value under matching assumptions. Convergence may oscillate, so compare several step counts.

Discretisation

A coarse tree can produce unstable prices and early-exercise boundaries.

Input risk

Volatility, dividends and rates may change across time and states.

Market frictions

Transaction costs, liquidity and discrete hedging remain outside the basic model.

Strength of the framework
The binomial model turns no-arbitrage, replication and early exercise into an auditable calculation. More steps improve resolution but do not repair poor inputs or missing market features.

10Live Binomial Application

The Mountain Path lab supports calls and puts, European and American exercise, manual tree factors, CRR volatility mode, dividend yield, node inspection, convergence analytics and Excel export.

Binomial Option Pricing Model

Build the tree, inspect every node, compare early exercise and benchmark a fine CRR tree against Blackā€“Scholes.

Open the binomial pricing lab ā†—
Prepared for The Mountain Path Academy Ā· Practical reference: Binomial Option Pricing Model