Black & Scholes · 1973 · Journal of Political Economy
Black-Scholes
The formula that priced the future
You buy a call option on Apple. Strike price $200, expires in 30 days. You pay $4.50 for it.
Why $4.50? Why not $3? Why not $6? Where does that number come from?
Before 1973, this question had no good answer. Options traded like exotic art — prices were based on intuition, experience, and negotiation. Two market makers could quote prices 50% apart for the same option. There was no formula, no consensus, no way to know if a price was "fair."
Then Fischer Black and Myron Scholes published a formula. Within years, it was programmed into every trading desk calculator. The options market exploded from $1 billion to trillions. This paper didn't just price options — it created modern derivatives markets.
The formula looks intimidating. The derivation uses stochastic calculus. But the core insight is beautiful and, once you see it, feels almost obvious. Let's build up to it from scratch.
What is an option, really?
Before we can price something, we need to be precise about what it is.
A call option gives you the right (but not obligation) to buy a stock at a specific price (the strike) on a specific date (the expiration).
A put option gives you the right to sell at the strike price.
At expiration, the math is simple. If you hold a call with strike $100:
Stock finishes at $150
You buy at $100, sell at $150
Profit: $50
Stock finishes at $80
You don't exercise (why buy at $100 what you can buy at $80?)
Payoff: $0
The payoff at expiration is just: max(Stock - Strike, 0)
This creates the famous "hockey stick" payoff diagram. Try adjusting the stock price below to see how the payoff changes:
Option Payoff at Expiration
Right to buy at strike $100. Drag the stock price to see profit/loss.
Payoff
$0.00
Net Profit
-5.00
Max Loss
-$5.00
What to notice
- • The "hockey stick" shape: unlimited upside, limited downside
- • You only profit after passing the breakeven (strike + premium)
- • Your max loss is capped at the premium you paid — that's the value of optionality
At expiration, pricing is trivial. The hard question is: what's the option worth today, when we don't know where the stock will end up?
The naive approach (and why it fails)
Your first instinct might be: estimate the probability distribution of where the stock could end up, calculate the expected payoff, and that's the option's value.
Example calculation:
- • 40% chance stock ends above $100, average payoff in those scenarios: $20
- • 60% chance stock ends below $100, payoff: $0
- • Expected value: 0.4 × $20 + 0.6 × $0 = $8
(Note: the $20 is the average payoff only when the stock finishes above $100. When it finishes below, you get nothing. The $8 expected value accounts for both outcomes.)
So the option is worth $8?
This approach has a fatal flaw: it depends on who's doing the calculation.
A cautious investor might value that gamble at only $5 — they're risk-averse and discount uncertain payoffs. An aggressive trader might pay $10 — they love upside bets. A pessimist might use lower probability estimates and get $4. An optimist might get $12.
The problem
Expected value calculations require assumptions about probabilities and risk preferences, which vary from person to person. There's no objective "fair price" — just different opinions.
This isn't just theoretical. Before Black-Scholes, option markets were thin and prices were inconsistent precisely because there was no agreed-upon method. Market makers set prices based on gut feel.
The key insight
We need a pricing method that doesn't depend on anyone's beliefs about probabilities or risk preferences. Black-Scholes found one — and it's not by calculating expected values at all.
How stocks move
Before we can understand option pricing, we need to understand how stock prices behave. The standard model — the one Black-Scholes uses — is a random walk with drift.
The basic idea:
- • Each moment, the stock moves up or down by a small random amount
- • On average, it drifts upward (stocks have positive expected returns over time)
- • The size of random moves is characterized by volatility
Mathematically, this is called Geometric Brownian Motion. The "geometric" part means percentage moves are random (a 1% move is equally likely whether the stock is at $10 or $1000), which prevents prices from going negative.
Play with the simulation below. Run multiple paths and watch how the possible outcomes fan out over time:
Random Walk Explorer
Watch how stock prices evolve randomly over time. Each path is equally likely.
What to notice
- • The paths fan out over time — uncertainty grows
- • Volatility determines the spread — higher vol = wider cone
- • Drift shifts the average upward, but individual paths can go anywhere
- • This random behavior is what makes option pricing non-trivial
What drift does
Drift shifts the distribution upward over time. Higher drift = paths trend higher on average. But for options, drift actually doesn't matter (more on this later).
What volatility does
Volatility determines the spread of outcomes. Higher volatility = wider cone of possibilities. This is crucial for options.
Why volatility is everything
Here's a key insight about options: they have asymmetric payoffs.
A call option pays off when the stock goes up past the strike. If the stock crashes, you lose only the premium you paid — not more. This asymmetry means that more uncertainty (higher volatility) is actually good for option buyers.
Think about it this way:
- • High volatility: stock could end at $150 (you make $50) or $50 (you make $0)
- • Low volatility: stock ends around $100 (you make ~$0)
In the high-vol scenario, you get the upside of big moves without the symmetric downside. Options let you bet on the magnitude of moves, not just direction.
This is why options become more valuable when volatility increases. Before earnings announcements, when uncertainty spikes, option prices rise. After the announcement, when the unknown becomes known, volatility "crushes" and prices drop — even if the stock moves in the "right" direction.
The role of volatility in Black-Scholes
Of the five inputs to Black-Scholes (stock price, strike, time, interest rate, volatility), four are directly observable. Only volatility must be estimated. This is why traders obsess over volatility — it's the one unknown, the secret ingredient that determines prices.
The breakthrough: replication
So we've established the problem: expected value calculations require assumptions about probabilities and risk that vary from person to person. And we've established that volatility matters enormously for option prices.
Now we arrive at the insight that changed everything. Black and Scholes asked a different question: forget about predicting where the stock will go. Is there a way to price options without making any forecasts at all?
What if you could manufacture an option yourself, using just stocks and cash?
Here's the idea: suppose you could create a portfolio of stocks and bonds that has exactly the same payoff as the option in every possible scenario. Not on average — exactly the same, path by path.
If two things have identical payoffs in every state of the world, they must have the same price today. If they didn't, you could arbitrage: buy the cheap one, sell the expensive one, and pocket the difference with zero risk. Markets don't allow that for long.
The replication argument
If you can replicate an option using stocks and cash, then the option's price must equal the cost of the replicating portfolio. No probabilities needed. No risk preferences. Just arbitrage logic.
But can you actually replicate an option? That seems impossible — options have that hockey-stick payoff, while stocks just move up and down. How could a portfolio of stocks ever match that kinked shape?
Here's the key observation: you don't need to match the final payoff all at once. You only need to match it locally, moment by moment, as time passes.
Think about it this way. Right now, the option has some value. If the stock ticks up by $1, the option value changes by some amount — let's call it Δ (delta). If you held exactly Δ shares of stock, your portfolio would move by the same amount as the option did.
Of course, this only works for that one instant. After the stock moves, the option's sensitivity to further moves changes — delta itself changes. But here's the trick: you can just update your position. Sell some shares if delta dropped, buy some if it rose.
If you keep doing this — continuously adjusting to match the current delta — your portfolio tracks the option step by step through every twist and turn of the stock price. At the end, you arrive at exactly the same payoff.
This strategy is called delta hedging, and the process of continuously adjusting is called dynamic replication.
Delta hedging: manufacturing options
Delta (Δ) tells you how much the option price moves when the stock price moves by $1.
For example:
- • Delta = 0.5 means: if stock goes up $1, the call goes up $0.50
- • An at-the-money call has delta around 0.5
- • A deep in-the-money call has delta close to 1 (moves almost 1:1 with stock)
- • A far out-of-the-money call has delta close to 0 (barely reacts)
Here's the key: if you hold Δ shares of stock, your position moves by approximately the same amount as one call option (for small moves). So:
Holding Δ shares ≈ Holding 1 call option (locally)
But delta changes as the stock moves! This is because of the option's curved payoff (the hockey stick).
Why does delta change?
Think about it intuitively: when the stock is way below the strike, the call is almost worthless and barely reacts to stock moves (delta ≈ 0). When the stock is way above the strike, the call moves almost dollar-for-dollar with the stock (delta ≈ 1).
As the stock crosses through the strike price, delta transitions from 0 toward 1. The rate at which delta changes is called gamma (Γ).
Gamma is highest when the stock is near the strike — that's where the hockey stick bends, and small stock moves cause the biggest changes in delta.
The solution: continuously rebalance. After each stock move, update your stock position to match the new delta. If delta rises from 0.5 to 0.55, buy more shares. If it falls to 0.45, sell some.
If you do this perfectly — rebalancing infinitely often — your portfolio of stocks and cash will exactly track the option's value at every moment, no matter which path the stock takes.
Watch it in action
This is the core insight of Black-Scholes. The simulation below shows replication working in real-time. Watch the portfolio (green line) track the option value (gold line) as the stock moves randomly:
The Hedging Machine
Watch the replicating portfolio track the option value as the stock moves randomly.
Stock Price
$100.00
Option Value
$3.73
Portfolio Value
$3.73
Tracking Error
$0.00
Delta (Δ)
0.545
Shares Held
0.545
Cash Position
$-50.73
What to notice
- • The green (portfolio) line tracks the gold (option) line closely
- • Delta changes as stock moves — we constantly rebalance to stay hedged
- • At expiration, both end at the same value (the option payoff)
- • This proves: options can be replicated with just stock and cash
What's happening
At each step, we hold Δ shares of stock. As stock price changes, delta changes, so we buy/sell shares to stay hedged. The cash position adjusts to fund these trades.
The "aha" moment
At expiration, the portfolio ends up with exactly the option's payoff — whether the stock finished in-the-money or not. We manufactured the option out of stock and cash.
The profound implication
Because we can replicate an option perfectly using only stock and cash, we don't need to know anyone's:
- • Probability estimates — we're not betting on where the stock goes
- • Risk preferences — we're not asking what gambles anyone likes
- • Expected returns — the stock's drift doesn't even appear in the formula!
The option's price is simply: the cost of the hedge.
Different investors might have wildly different views on where Apple stock is going. Bulls think it'll moon; bears think it'll crash. But they'll all agree on the option's current fair value — because they all agree on what it costs to replicate it today.
This is important: Black-Scholes tells you what the option is worth now, not what it will be worth tomorrow. The price can change as the stock moves, volatility shifts, or time passes. The formula doesn't predict the future — it prices the present, given current conditions.
Risk-neutral pricing
This is often called "risk-neutral pricing" — not because anyone is actually risk-neutral, but because the replication argument makes risk preferences irrelevant. The price is pinned by arbitrage, not by opinions.
Think of it this way: the option price isn't "what people think it's worth" — it's "what it costs to manufacture." A factory doesn't price a widget based on how much customers like it; it prices based on the cost of materials and labor. Black-Scholes does the same for options: the price is the cost of the replicating portfolio, period.
This is why the formula was revolutionary. It gave an objective price in a world that seemed inherently subjective. The price doesn't come from predicting the future — it comes from showing how to replicate it.
The formula
Now we can appreciate what Black-Scholes gives us. Instead of:
- • Running thousands of simulations
- • Tracking a hedging portfolio through time
- • Hoping we rebalanced frequently enough
We get a closed-form formula:
C = S₀ · N(d₁) − K · e−rT · N(d₂)
where:
d₁ = [ln(S₀/K) + (r + σ²/2)T] / (σ√T)
d₂ = d₁ − σ√T
N(x) = cumulative standard normal distribution
The inputs
- S₀ — Current stock price
- K — Strike price
- T — Time to expiration (years)
- r — Risk-free interest rate
- σ — Volatility (the only unknown!)
What the terms mean
- S₀ · N(d₁) — Expected stock value if exercised (risk-adjusted)
- K · e−rT — Present value of strike
- N(d₂) — Probability of exercise (risk-adjusted)
Try the calculator below. Adjust the inputs and watch how the price responds:
Black-Scholes Calculator
Adjust the inputs to see how option prices and Greeks change.
Call Price
The Greeks
$0.54 per $1 move
Delta changes by 0.0554 per $1
Loses $0.054 per day
$0.114 per 1% vol change
$0.042 per 1% rate change
Call Price
$3.06
Put Price
$2.65
Put-Call Parity
VerifiedC - P = $0.4101 = S - K·e-rT = $0.4101
This identity links call and put prices. If it doesn't hold, there's an arbitrage opportunity. Traders use it to quickly derive one price from the other.
What to notice
- • Volatility effect: Drag the slider — higher vol makes both calls and puts more valuable
- • Time decay: Theta is negative — options lose value every day
- • Delta intuition: ATM options have delta ~0.5, deep ITM ~1.0, deep OTM ~0
- • Gamma peaks ATM: Delta changes fastest when near the strike
The Greeks: how option prices move
The "Greeks" measure how the option price responds to changes in each input. They're not just metrics — they're instructions for hedging.
Delta — Sensitivity to stock price
How much the option price moves per $1 move in stock. Also: how many shares to hold to hedge.
• ATM call ≈ 0.50 (moves ~50¢ per $1 stock move)
• Deep ITM call → 1.0 (moves like stock)
• Far OTM call → 0 (barely moves)
Gamma — Rate of change of delta
How fast delta changes as stock moves. High gamma = frequent rebalancing needed.
• Highest for ATM options near expiration
• High gamma is exciting but dangerous — big moves amplify quickly
• Low gamma = stable hedge, less work
Theta — Time decay
How much value the option loses each day. Options are wasting assets.
• Theta is negative for option buyers (you bleed money daily)
• Positive for option sellers (you earn money daily)
• Accelerates as expiration approaches — the last week is brutal
Vega — Sensitivity to volatility
How much the price changes if implied volatility moves by 1%.
• Higher for ATM options with more time
• Vol spike before earnings = options get expensive
• "Vol crush" after events = options get cheap fast
Traders' perspective: When you buy options, you're implicitly long gamma (volatility helps you) and short theta (time hurts you). These are the fundamental tradeoffs of options trading.
When Black-Scholes breaks
The formula makes assumptions. When reality violates them, prices diverge.
Constant volatility?
Black-Scholes assumes volatility is constant. In reality, it changes constantly. This creates the volatility smile — implied volatility varies across strikes. Out-of-the-money puts often have higher implied vol (people pay up for downside protection).
Log-normal returns?
The model assumes stock returns follow a bell curve. Real markets have fat tails — extreme moves happen far more often than the math predicts. Black Monday 1987 saw a 22-standard-deviation move, which should happen once every 1050 years.
Continuous trading?
Delta hedging assumes you can trade continuously. In practice, you trade discretely, and transaction costs eat into your hedge. This creates "hedging error" — the replication isn't perfect.
No jumps?
Stocks don't always move smoothly — they can jump on news. If a stock gaps from $100 to $80 overnight, you can't hedge through that discontinuity. This is why options on volatile stocks (or before earnings) trade at premiums.
Why traders still use it
Despite these limitations, Black-Scholes remains the industry standard. Not because it's perfectly accurate, but because it provides a common language. When traders quote "implied volatility," they're using Black-Scholes as a translator — converting prices into a volatility number that's comparable across options. The model is the benchmark, even when everyone knows it's imperfect.
In practice: implied volatility
Here's something interesting: traders don't actually use Black-Scholes the way the textbooks describe.
The formula takes five inputs and spits out a price. But in practice, we already know the price — it's right there on the exchange, determined by supply and demand. What we don't know is the volatility.
The reverse problem
Instead of: σ → Black-Scholes → Price
Traders solve: Price → Black-Scholes⁻¹ → σ
Given the market price, what volatility would Black-Scholes need to produce that price? This is called implied volatility (IV).
Implied volatility is powerful because it's a standardized way to quote option prices. Instead of saying "the $100 call costs $4.50," traders say "the $100 call is trading at 25 vol." This makes it easy to compare options across different strikes, expirations, and underlying stocks.
But here's the catch: if Black-Scholes were perfectly correct, all options on the same stock would have the same implied volatility. They don't.
The volatility smile
When you plot implied volatility against strike price, you get a curve — not a flat line. This pattern is called the volatility smile (or smirk, or skew).
The Volatility Smile
If Black-Scholes were perfect, IV would be constant across strikes. It isn't.
Why the smile exists
- • Skew: OTM puts trade at higher IV — people pay up for crash protection
- • Fat tails: Extreme moves happen more often than BS predicts
- • Supply/demand: Institutional hedging flows push up put prices
- • After 1987 crash, the skew became permanent — markets learned fear
The smile tells us that the market disagrees with Black-Scholes. Out-of-the-money options trade at higher implied volatilities than at-the-money options. In equities, puts especially are expensive — the "skew" reflects demand for downside protection.
Why does the smile exist?
- • Fat tails: Real returns have more extreme moves than the normal distribution predicts
- • Jumps: Stocks gap on news; Black-Scholes assumes continuous movement
- • Supply/demand: Institutions buy puts for hedging, driving up prices
- • Crash memory: After 1987, the skew became permanent
The smile is real information. It tells you what the market thinks about tail risks that Black-Scholes ignores. Traders who understand the smile can find edges; those who don't get run over.
Time decay up close
We mentioned that theta — time decay — accelerates as expiration approaches. But how dramatic is this effect? The visualization below shows exactly how option value erodes over time:
Time Decay (Theta)
Options are wasting assets. Watch how the decay accelerates as expiration approaches.
Daily decay @ 90d
$0.03
Daily decay @ 30d
$0.05
Daily decay @ 7d
$0.11
Why decay accelerates
- • Time value comes from optionality — the chance to benefit from future moves
- • Less time = less chance for big moves = less optionality value
- • The square root of time in the formula means: halving time doesn't halve value
- • The last week can wipe out half the remaining premium — beware of "gamma risk"
This is why experienced traders avoid holding options through the last week before expiration unless they have strong conviction. The theta bleed becomes brutal. Option sellers love this; option buyers fear it.
The impact
The Chicago Board Options Exchange opened in April 1973. The Black-Scholes paper was published in May 1973. The timing was no coincidence — the exchange knew they needed a pricing model for the market to function.
Before Black-Scholes
- • Options market: ~$1 billion
- • Prices: negotiated, inconsistent
- • Hedging: ad hoc, intuitive
- • Most investors avoided options entirely
After Black-Scholes
- • Options market: trillions of dollars
- • Prices: algorithmic, consistent
- • Hedging: systematic, delta-neutral portfolios
- • Derivatives became a mainstream tool
The formula didn't just price options — it enabled the entire derivatives industry. Credit default swaps, exotic options, structured products — all built on the foundation of risk-neutral pricing and dynamic replication.
Historical note
The original paper was rejected by multiple journals. Black and Scholes had to publish it themselves, with help from Milton Friedman (who found it "too mathematical" but recognized its importance).
Myron Scholes and Robert Merton (who contributed key mathematical insights) won the Nobel Prize in Economics in 1997. Fischer Black had died in 1995 — the Nobel is not awarded posthumously. The prize committee acknowledged that Black would undoubtedly have shared the award.
The math (for the curious)
Click to expand the derivation
Step 1: Model the stock
We assume the stock follows Geometric Brownian Motion:
dS = μS dt + σS dW
where μ is drift (expected return), σ is volatility, and dW is a Wiener process (random noise). This means percentage changes are normally distributed.
Step 2: Consider a hedged portfolio
Construct a portfolio: long one option (value V), short Δ shares:
Π = V − ΔS
The goal is to choose Δ so that the portfolio is riskless — immune to small stock moves.
Step 3: Apply Ito's Lemma
Ito's Lemma is like the chain rule for stochastic calculus. If V is a function of S and t:
dV = (∂V/∂t)dt + (∂V/∂S)dS + ½(∂²V/∂S²)σ²S²dt
The last term is the "Ito correction" — it's what makes stochastic calculus different from regular calculus. It arises because the quadratic variation of Brownian motion is non-zero.
Step 4: Eliminate the random term
The change in our portfolio is:
dΠ = dV − ΔdS
If we set Δ = ∂V/∂S, the dS terms cancel! We're left with:
dΠ = (∂V/∂t + ½σ²S²∂²V/∂S²) dt
This is deterministic — no randomness! The portfolio changes by a known amount over each instant. A riskless portfolio.
Step 5: No arbitrage → Risk-free return
A riskless portfolio must earn the risk-free rate r. Otherwise you could arbitrage.
dΠ = rΠ dt
Substituting Π = V − (∂V/∂S)S and equating with Step 4:
∂V/∂t + ½σ²S²∂²V/∂S² + rS∂V/∂S − rV = 0
This is the Black-Scholes PDE. Notice: μ (drift) doesn't appear! The expected return on the stock is irrelevant for pricing.
Step 6: Solve with boundary conditions
For a call option, the boundary condition at expiration is:
V(S, 0) = max(S − K, 0)
Solving the PDE (using transforms or Feynman-Kac) yields the Black-Scholes formula:
C = S·N(d₁) − Ke−rT·N(d₂)
The N(d₁) and N(d₂) terms emerge naturally from solving the heat equation (which the Black-Scholes PDE can be transformed into).
The remarkable insight
We started with a random process (stock prices). We combined it cleverly with its derivative (the option) to create a riskless portfolio. The no-arbitrage requirement then pinned down the option price exactly. No beliefs about future prices needed — just the cost of replication.
Original Paper
The Pricing of Options and Corporate Liabilities
Fischer Black, Myron Scholes
Journal of Political Economy, Vol. 81, No. 3 (May-Jun., 1973), pp. 637-654
Read the original paper →