Black-Scholes, and what it actually assumes
Black-Scholes turns five inputs into one option price: the stock price, the strike, the time to expiry, the volatility and the risk-free rate. Four of them you can read off a screen. The fifth, volatility, is the only one nobody knows, which is why the model gets run backwards more often than forwards.
You do not need to derive it. You need to know which knob moves the price and by how much, and where the model quietly lies to you.
The position this cluster runs on
September 5. UBER at $68.40. You sell the October 17 $72.50 call, 42 days out, at 32 percent implied volatility, with the risk-free rate at 4.2 percent.
The model says that contract is worth $1.53, so $153 of credit. Every page in this series takes that position apart.
Move one input at a time
| Change | New price | Effect per contract |
|---|---|---|
| Stock $68.40 to $69.40 | $1.88 | +$36 |
| Strike $72.50 to $70 | $2.40 | +$87 |
| 42 days to 35 days | $1.28 | -$25 |
| IV 32% to 37% | $1.95 | +$43 |
| Rate 4.2% to 5.2% | $1.55 | +$2 |
That table is the whole cluster in miniature. Each of those rows has a Greek attached to it: delta, then time decay, then vega, then rho. And look at the last one. A full percentage point on the risk-free rate, which is a year of Fed policy, moves this contract by two dollars. A one-dollar move in the stock moves it by thirty-six. That is why nobody talks about rho.
Why the model gets used backwards
Nobody computes a theoretical price and compares it against the market. The market price is the fact and the model is the interpreter.
Four inputs are observable. The fifth is not, and there is no such thing as "the" volatility of a stock over the next 42 days. So the working move is to take the price the contract trades at, hold the other four fixed, and solve for the volatility that reproduces it. That output is implied volatility, and it is the number the entire options market actually converses in.
There is no closed-form solve for it. It gets found numerically, by a root-finder, which is a detail that matters only when the answer comes back missing and you need to know it is your inputs and not the math.
The six assumptions, and which ones are false
Black and Scholes published this in 1973 with a set of conditions attached. Here they are, with an honest score.
- Volatility is constant and known. False, and it is the big one. Volatility moves, and it moves most violently exactly when your position is worst. Every other flaw is downstream of this.
- The stock moves continuously, with no jumps. False. Stocks gap. An earnings print is a discontinuity, and a model built on continuous paths prices it by pretending it is a lot of small moves, which is not what it is.
- Returns are lognormal. False in the tails, which is the part that costs money. Real distributions have fatter left tails than the model admits, and the volatility skew is the market patching that in.
- European exercise, so no early assignment. False for American equity options. You can be assigned any day, which matters most around ex-dividend dates.
- No dividends. The original formula assumed none. Merton extended it the same year. The worked example on these pages assumes no dividend before expiry, which is a choice made to keep the arithmetic clean, not a claim about any company.
- No transaction costs, and you can trade continuously. False. The bid-ask spread is real and on a wide chain it is larger than the edge.
So why is it still on every screen
Because being wrong in a known, stable, universally shared way is enormously useful.
The model is a translator. It turns a dollar price into a volatility, which lets you compare a $1.53 contract on UBER against a $5.28 contract on something else. It produces the Greeks, which are how anybody describes a position's risk to anybody else. And every participant is using the same wrong model, so the wrongness cancels out of the conversation and what is left is a common language.
Nobody trading options believes volatility is constant. They use a constant-volatility model and then encode everything it cannot represent in the shape of the volatility surface, which is a hack, and it works.
OptionsKing scores every candidate strike on a deterministic 0 to 100 scale, blends that score with the return on the capital the trade ties up, and shows you the highest-ranked handful. There is no minimum score. How it works covers what the ranking does and does not tell you.
Questions people actually ask
What are the inputs to Black-Scholes?
Stock price, strike price, time to expiration, volatility and the risk-free interest rate. Four are observable. Volatility is not, which is why the model is usually run backwards to solve for the volatility implied by a traded price.
Is Black-Scholes accurate?
Not literally. It assumes constant volatility, continuous price paths, lognormal returns, European exercise and no transaction costs, and most of those are false on an American equity option. It stays useful because everyone shares the same errors and the model is a common language rather than a truth claim.
Why does the risk-free rate barely matter?
On short-dated contracts it barely does. Moving the rate a full percentage point changed the 42-day $72.50 call by $2 per contract, while a $1 move in the stock changed it by $36. Rate sensitivity grows with time, so it matters on LEAPS and not on monthlies.
Does Black-Scholes work for American options?
It prices them approximately. The formula assumes exercise only at expiry, and American options can be exercised any day, which matters most for in-the-money calls around an ex-dividend date. Platforms patch this with binomial or approximation models.
Sources
Rules and thresholds above were checked against these documents on August 4, 2026. Exchange and broker rules change. Confirm anything you are about to act on with your own broker.
Keep reading
Do the math on your own trade
Every price in this article is an illustrative worked example, not a quote. Read The Greeks for the rest of the series, and the disclaimer before you act on any of it. Selling options carries real risk of loss, and the loss can be far larger than the premium you collected.