Black-Scholes vs. Binomial Option Pricing Explorer
This page is designed for learning. It starts with the binomial model and then moves to
Black-Scholes, because the binomial model makes the hedge idea easier to see.
Then we show how Black-Scholes becomes the continuous-time version of the same pricing logic.
It is often easier to learn the binomial model first and then Black-Scholes second.
The binomial model shows the hedge idea with only two stock-price possibilities each step.
Black-Scholes uses the same no-arbitrage logic, but in continuous time.
Binomial model first
Good for learning the replicating portfolio idea.
Easy to see up state, down state, and hedge ratio.
Works well for American options too.
Shows clearly why option value comes from no arbitrage.
With more and more steps, it approaches Black-Scholes.
Then Black-Scholes
Gives one clean closed-form formula for European calls and puts.
Uses volatility, time, rates, and dividends in one formula.
Very useful for understanding Greeks.
Good for market comparison and implied volatility discussion.
More compact mathematically, but less intuitive at first than binomial.
This helped inspire the later idea of Brownian motion.
Simple idea: Binomial helps us see why the option has value.
Black-Scholes helps us see how the value is written in a single formula.
Origin story: from random dust-like motion to option pricing
The historical idea behind Black-Scholes is tied to random motion. In science, people
observed tiny particles moving in a jittery random way in fluid. That idea later became part of the
mathematics used for random price movement in finance.
Random motionBrownian motionNo-arbitrage pricingDynamic hedging
1827
Robert Brown observed random motion of tiny particles in pollen suspended in water.
This helped inspire the later idea of Brownian motion.
1973
Fischer Black and Myron Scholes published the option pricing formula that became known as
Black-Scholes.
1973
Robert Merton extended and generalized the theory and gave another way to derive the formula.
1979
Cox, Ross, and Rubinstein showed a simpler binomial approach that is easier to learn and that
approaches Black-Scholes as the number of steps becomes large.
Learning picture: Think of stock prices as moving with many tiny random pushes,
like a particle getting bumped over and over. Black-Scholes turns that idea into a pricing model.
Black-Scholes equations
Main Black-Scholes formulas
Call: C = S e^(−qT) N(d1) − K e^(−rT) N(d2)
Put: P = K e^(−rT) N(−d2) − S e^(−qT) N(−d1)
These are the core formulas for the Black-Scholes model. They give the theoretical value of a
European call and a European put using stock price, strike price, time, interest rate,
volatility, and dividend yield.
These two terms combine moneyness, time, volatility, interest rate, and dividend yield into
two standardized values used in the model.
Normal distribution terms
N(x) = cumulative standard normal distribution
n(x) = (1 / √(2π)) e^(−x² / 2)
N(x) gives the cumulative normal value, and n(x) gives the
normal density.
CRR binomial formulas
u = e^(σ√Δt)
d = 1/u
p = [ e^((r−q)Δt) − d ] / (u − d)
In the CRR model, u is the up factor, d is the down factor,
and p is the risk-neutral probability.
Quick reading guide
Higher σ → higher option value
Longer T → more time value
Higher K → lower call, higher put
These quick patterns help explain how the inputs affect option value.
Meaning of the symbols
SCurrent stock price
KStrike price
TTime to expiration in years
rRisk-free rate
σVolatility
qDividend yield
N(x)Cumulative standard normal distribution
n(x)Standard normal density
Why is N(x) important?
In Black-Scholes, N(d1) and N(d2) come from the normal distribution
because the model assumes log stock prices move in a way linked to continuous random motion.
In learning terms, N(x) converts standardized values into cumulative normal values
used inside the formula.
How Black-Scholes is derived
The full derivation can be heavy, but the core learning idea is simple:
build a hedge that removes risk over a very short time interval, then use no-arbitrage.
Show the simple learning derivation
1
Assume the stock price moves randomly over time:
dS = μSdt + σS dW
Here μ is the expected growth rate, σ is volatility,
and dW is the random Brownian-motion shock.
2
Let the option value be a function of stock price and time:
V = V(S,t)
3
Use calculus (Itô’s lemma) to write the small change in the option value:
The first bracket is the predictable part. The last term contains the random shock.
4
Form a hedged portfolio:
Π = V − ΔS
Choose
Δ = ∂V/∂S
so the random term cancels out.
5
Once the random part is removed, the portfolio is locally riskless, so it must earn the risk-free rate:
dΠ = rΠ dt
6
Rearranging gives the Black-Scholes partial differential equation:
∂V/∂t + 0.5σ²S² ∂²V/∂S² + (r−q)S ∂V/∂S − rV = 0
Solve this PDE using the option payoff at expiration, and you get the Black-Scholes formula.
7
Terminal payoff conditions:
Call payoff at expiration = max(S − K, 0)
Put payoff at expiration = max(K − S, 0)
Why the binomial model helps before this derivation
In the binomial model, you can see the same hedge idea without stochastic calculus.
You build a portfolio of stock and option so that the payoff is the same in both the up state
and the down state. That is why many classes learn binomial first and then say:
“Black-Scholes is the continuous-time version of this same idea.”
Calculator: Black-Scholes and CRR Binomial side by side
Use the same inputs for both models. The binomial model below uses the CRR setup.
As the number of steps increases, the binomial value often moves closer to Black-Scholes for European options.
BS Call
—
European theoretical value
BS Put
—
European theoretical value
CRR Call
—
binomial value
CRR Put
—
binomial value
Call Difference
—
CRR − BS
Put Difference
—
CRR − BS
Call Delta
—
Black-Scholes delta
Gamma
—
Black-Scholes gamma
Vega
—
Black-Scholes vega
Theta (daily approx.)
—
call theta / day
d1
—
standardized term
d2
—
standardized term
Blue = Black-Scholes call value. Red = Black-Scholes put value.
Purple dashed = CRR call value. Green dashed = CRR put value. Gold dashed vertical line = strike.
Show model detailsShow convergence table: binomial steps approaching Black-Scholes
Steps
CRR Call
CRR Put
Call Diff vs BS
Put Diff vs BS
Quick Quiz
Use this short quiz to check your understanding of the Black-Scholes model before moving on.
Quiz topics:
Black-Scholes purpose, European options, volatility, risk-free rate, time to expiration,
N(d1), N(d2), and the difference between model value and payoff at expiration.
Enter observed market premiums here to compare them to the model. This does not prove the market is wrong.
It simply shows whether your assumed volatility and inputs imply a higher or lower theoretical value.
Contract
Model value
Observed premium
Difference
Simple read
A large gap often means your volatility, dividend yield, or time assumption is not aligned with the market.
Useful websites and video
Helpful video
Introduction to the Black-Scholes Formula
Finance & Capital Markets • Khan Academy
Important: Black-Scholes is a model for European-style options.
The binomial model can also handle American exercise, which is one reason it is so useful in learning.