Black-Scholes Model: How Options Are Really Priced

Two people can agree exactly where a share will be trading in a year and still argue about what an option on it is worth today. An option is not priced off where the share lands. It is priced off how far the share could plausibly wander before the option expires, and the Black-Scholes model is the standard way of turning that idea into a number you can actually use.

Where the formula came from

Fischer Black and Myron Scholes published The Pricing of Options and Corporate Liabilities in the Journal of Political Economy in 1973. Robert Merton published closely related work the same year and generalised the result, which is why you often see the model written as Black-Scholes-Merton. The timing was almost comic. The Chicago Board Options Exchange opened for business in that same year, so a proper market in standardised listed options and a credible way to price them arrived together. In 1997 Scholes and Merton received the Nobel Memorial Prize in Economic Sciences for the work. Black had died in 1995, and the prize is not awarded posthumously.

The trick is copying the option, not forecasting the share

Here is the part that surprises everyone the first time. The model never asks what you think the share will do. Black and Scholes showed that the payoff of an option can be reproduced by holding a certain quantity of the underlying share, borrowing or lending the rest, and adjusting that mix as the price moves. If a portfolio copies the option exactly, then the option has to cost what the copy costs. If it did not, you could buy the cheap one, sell the dear one, and collect the difference with no risk at all. Somebody would. That is why the expected return on the share never appears in the formula. An optimist and a pessimist who agree on how much the share bounces around will agree on the price of the option, even though they disagree completely about the company.

The six inputs, and which way each one pushes

Two of them are obvious. The current share price and the strike price together set how far the option already sits from paying out. A call with a strike above today's price only pays if the share climbs past it, so the further away that strike, the cheaper the call and the less likely it ever matters. Lift the share and the call is worth more while the put is worth less. No mystery there.

Time to expiry is the first input that is less obvious than it looks. More time means more room to travel, so a longer-dated option costs more than an otherwise identical short-dated one. But that extra value bleeds away, and it bleeds faster as expiry approaches instead of at a steady daily rate. An option with a month left surrenders a far larger share of its remaining time value each day than one with a year left. That drain has a name, theta, and it is the option greek that catches out almost every beginner who buys an option and then waits patiently for it to work.

Volatility does most of the work

Volatility in the model means the annualised standard deviation of the share's returns, which is a formal way of saying how widely the price tends to swing. Raise it and both calls and puts get more expensive. That feels wrong until you notice the asymmetry the buyer faces: the most they can lose is the premium they paid, while the upside stays open. Wider swings fatten the good tail without fattening the bad one, because the bad one is already capped. Volatility is also the only input you cannot simply look up. The share price, the strike, the days remaining, the interest rate and the expected dividends are all facts. Future volatility over the life of the option is a guess, and nearly every disagreement about what an option is worth turns out to be a disagreement about that single number.

Interest rates and dividends, the quiet two

Buying a call instead of the share itself means you postpone paying the strike, and money you have not handed over yet can sit earning the risk-free rate. Higher rates therefore shrink the present value of that future payment and lift call prices, while doing the reverse to puts. The effect is real, but next to volatility it is small. Dividends pull the other way. An option holder receives none of them, and a share tends to fall by roughly the dividend on the day it goes ex-dividend, so expected dividends drag calls down and push puts up. That is why the version of the model people actually use has a dividend input: the 1973 paper assumed the share paid nothing at all.

A worked example, with obviously round numbers

Take a share trading at 100 and a call with a strike of 100. These numbers are illustrative, picked because they are easy to hold in your head, not because they describe any real security. That option has no intrinsic value, since exercising it today would gain nothing. Everything it costs is payment for what might happen between now and expiry. Now compare two versions of it, one expiring in a month and one in a year. The one-year call costs substantially more, and every bit of that difference buys nothing except eleven extra months in which the price can move.

Change one input at a time and you can feel the machinery working. Push the strike out to 120 and the call gets much cheaper, because the share now has to climb by a fifth before the option is worth anything at expiry. Halve the assumed volatility and both strikes get cheaper, but the 120 strike loses proportionally far more, because a distant strike depends entirely on a large move actually happening. That is the whole reason cheap-looking far-out options are cheap. You are not being handed a bargain. You are being handed a low probability.

What the model assumes, and what markets actually do

The mathematics needs the share price to move in small continuous steps at a constant volatility, with no gaps and no jumps, in a market that never closes and costs nothing to trade in. Real prices refuse to behave like that. A company reports earnings after the close and opens the next morning at a completely different level. A regulator makes an announcement. Nobody rebalances a hedge continuously, and every adjustment costs money. The standard formula also prices European options, which can only be exercised at expiry, while most single-stock options listed in the United States are American style and can be exercised on any trading day before then. None of this makes the model useless. It makes it a simplification whose edges you need to know.

The volatility smile is the market saying no

If the model described reality, a single volatility number would price every option on the same share for the same expiry, whatever the strike. It does not. Work backwards from genuine market prices and each strike hands you a different volatility, typically higher for strikes well below the current price. Practitioners call the shape a smile, or a skew when it leans one way, and for equity index options that lean became a permanent fixture after the crash of October 1987. The reason is not mysterious. Markets fall faster and more violently than a smooth continuous model allows, and investors will pay real money for protection against precisely that. The smile is the price of the tail the model left out.

Implied volatility is the model run backwards

That backwards calculation is worth naming, because it is how options are really discussed. Instead of feeding a volatility in and reading a price out, you feed the observed market price in and solve for the volatility that would make the model reproduce it. The answer is the implied volatility, and professionals quote options in it rather than in currency, because a premium of two units tells you nothing until you also know the strike and the expiry, while a volatility figure is comparable across both.

Read it for what it is and no more. Implied volatility is the market's collective expectation of how much the share will move, in either direction, plus whatever extra sellers demand for carrying that risk. It says nothing about which way. A high reading in the days before an earnings release is not a forecast of bad news. It is a statement that the range of plausible outcomes just got wider, which is exactly why options over that date cost more.

How to use a model price without trusting it

Treat the output as a reference point rather than a verdict. A model price tells you what an option should cost if the share behaves the way you assumed, so its real job is to drag your own assumption into the open where you can look at it. If the market charges more than your number, you have probably not found a mispricing. You have found a market that expects more movement than you do, and the useful question is which of you has the better reason. Used that way, Black-Scholes is a thinking tool rather than a bargain detector.

So do this before you buy or sell anything. Price the option twice, once with the volatility you believe and once with the volatility the market is charging, and stare at the gap between them. If you cannot explain in one plain sentence why the two numbers differ, you do not yet have a reason for the trade, and putting real money behind it is where the damage usually starts.

Written by

Federico Romaldi

Co-Founder, Worthmap

Published: August 14, 2026

Federico is a co-founder of Worthmap, a wealth-intelligence platform built for serious investors. With a background in software engineering and a long-standing passion for value investing, he created Worthmap to bridge the gap between net-worth tracking and investment analysis.