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Black-Scholes from zero

1/7
Section 1

What is a call option?

A call option is a choice: you can buy later at a fixed price K, or walk away. That one detail creates the entire payoff shape.

If the asset finishes below the strike, you ignore the option. If it finishes above, you buy at the cheaper fixed price and pocket the difference.

$0$20$40$60K=100payoff = 0$15
$115
Payoff = max($115 − $100, 0) = $15 — buy at $100, sell at $115

Drag the slider. Below K the payoff is zero — you would never exercise. Above K the payoff rises dollar for dollar. That kink at K is the entire reason options exist.

Analogy

Think of paying a small reservation fee on a concert ticket. If resale prices explode, your reservation is valuable. If prices stay low, you walk away. The option premium is that reservation fee.

Section 2

The five inputs

Before writing the formula, make each symbol feel boring. If the symbols stay mysterious, the whole model stays mysterious.

Move each slider below and watch the call price react. Every input has a direction it pushes. Get a feel for it before we name the formula.

SSpot price$100
Where the asset is right now.
KStrike price$100
The price you can buy at later.
TTime to expiry1.00 yr
How long the option stays alive.
rRisk-free rate5.0%
What money earns while you wait.
σVolatility20%
How wide the future price range feels.
Call price
$11.91
Put: $7.03
d₁ = 0.3500 · d₂ = 0.1500

One-sentence summary: Black-Scholes prices a right whose value depends on where the asset is now (S), where you can buy (K), how long you have (T), how wide the future can be (σ), and how much waiting costs (r).

Section 3

Two big pieces

Most people meet the final formula first. That is backwards. First learn the story, then place symbols on top of it.

Click through the three layers below. Watch the English turn into math.

Story
call price = stock-like upside − cost of buying later
C = S · N(d₁) − K · e⁻ʳᵀ · N(d₂)
Hover over any part of the formula to see what it means.

The first piece is how much stock-like upside you are getting. The second piece is what you would owe for it, discounted to today. The difference is the option's value.

N(d₁) and N(d₂) are weights between 0 and 1. They come from the normal distribution. We will unpack them next.

Section 4

What are d₁ and d₂?

The part that scares most people. They are not mystical. They are scorecards — measuring how favorable the option setup is, in units of one-lifetime volatility.

N(d) is the area under the bell curve to the left of d. Drag the slider and watch how the shaded area — the weight — changes.

-3-2-10123d₂d₁
0.35
N(d₁)0.6701
N(d₂)0.5793
d₂ = d₁ − σ√T0.15

Breaking down d₁:

d₁ numerator
ln(S/K) + (r + σ²/2)T
ln(S/K) — are we above or below the strike, in log-scale?
(r + σ²/2)T — drift and volatility correction over the option's life.
d₁ denominator
σ√T
One option-lifetime of volatility. This is the ruler you measure everything in. The numerator tells you how favorable the setup is; the denominator expresses it in units of “wiggles.”
d₂
d₂ = d₁ − σ√T
Same scorecard, minus one full lifetime of volatility. N(d₁) weights the stock-like piece. N(d₂) weights the strike-payment piece.
Section 5

Work a full example

Numbers make it real. Start with friendly defaults, then change inputs and watch every intermediate step update.

ln(S/K) = ln(100/100) = 0.0000▸
Right at the strike — no built-in moneyness advantage.
(r + σ²/2)T = (0.05 + 0.0200) × 1 = 0.0700▸
Drift + volatility correction over the option lifetime.
σ√T = 0.2 × 1.0000 = 0.2000▸
One lifetime of volatility — the measuring stick.
d₁ = 0.0700 / 0.2000 = 0.3500▸
The setup is 0.35 wiggles favorable.
d₂ = 0.3500 − 0.2000 = 0.1500▸
Same score, minus one lifetime of volatility.
N(d₁) = 0.6701, N(d₂) = 0.5793
The two weights from the normal distribution lookup.
C = 100 × 0.6701 − 100 × e^(-0.0500) × 0.5793▸
$67.01 of upside minus $55.10 of discounted cost.
C = $11.91
The Black-Scholes call price.
Section 6

Why this price and no other

Black-Scholes is not a guess. Its backbone is replication: if you can copy an option using stock and cash, the option and the copy must cost the same.

Simplify to one period. The stock goes to $120 or $80. The call with K = 100 pays $20 or $0. Can we build a portfolio of stock and cash that matches those payoffs exactly?

TODAY$100STOCK$120Call pays $20STOCK$80Call pays $0stock goes upstock goes down
Replicating portfolio
120Δ + B = 20Match the up-state payoff
80Δ + B = 0Match the down-state payoff
Δ = 0.5, B = −40Half a share, borrow $40
Cost = 0.5 × 100 − 40 = $10Option must also cost $10 — or arbitrage exists

The copy costs $10. The option must also cost $10 — otherwise someone buys the cheap one, sells the expensive one, and earns risk-free profit. That is why the model is disciplined by arbitrage, not by vibes.

Black-Scholes is the smooth, continuous-time version of this copying argument — applied infinitely many times as the stock price continuously changes.

Section 7

Write it from memory

Tap each card to check yourself. If you can fill in all four, you have the formula cold.

Quick recall check — tap to see the answers:

Where to go next:

Implied volatility — using the model backward from price

Greeks reference — connecting price to hedge sensitivities

Put-call parity — the next pricing identity to learn cold

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