how options are priced: the black-scholes model in plain words

In short

I once watched a neighbor argue over the price of a used truck for an hour, and neither man could say what the truck was really worth. Option prices can feel like that, until you see what goes into them. An option is a contract that gives its owner the right to buy or sell a stock at a set price before a set date. The Black Scholes model is a formula that turns five facts into a fair price for that contract. Those facts are the stock price, the strike price, the time left, the interest rate, and how bumpy the stock tends to be. Of the five, the bumpiness is the only one nobody can look up. That one unknown is the reason option quotes move around so much from day to day.

The whole of it

What it is

A friend of mine once asked why a coupon for a car wash cost money when it only saved him a little. I told him that the coupon was a choice, not a promise. He did not have to use it. That is the heart of an option. The buyer of a call option may buy a stock at the strike price, and the buyer of a put option may sell at that price. The buyer pays for that choice up front. That payment is called the premium.

Three men gave us the model. Fischer Black and Myron Scholes published their work in 1973 in the Journal of Political Economy, in a paper called The Pricing of Options and Corporate Liabilities. Robert Merton wrote a companion paper the same year. Scholes and Merton later received the Nobel Prize in economics for this work. Black had passed away by then.

The model does one humble job. It asks what a fair price would be for the right to buy or sell a stock later, given what we know today. It does not predict where the stock will go. It prices the choice.

How it works

If you have ever held a ticket to a game you might skip, you already know the two parts of an option's price. One part is what the ticket is worth right now. The other part is the chance that it becomes worth more before the game. The first part is called intrinsic value. The second is called time value.

Take a call with a strike of $50 on a stock trading at $55. The call lets the owner buy at $50 what sells for $55. So $55 minus $50 gives $5 of intrinsic value. If the call sells for $7, the other $2 is time value. That $2 pays for the chance of further gain.

Black and Scholes asked how to price that chance. Their answer rests on a simple picture. Over short stretches, a stock price moves up and down at random, like a cork on a pond. The cork has a drift, and it has a wobble. The size of the wobble is the volatility. Volatility is a measure of how widely the price swings, stated as a yearly percent. A calm stock has a small wobble. A jumpy one has a big wobble.

Here is the part that surprises people. A bigger wobble makes both calls and puts worth more. The owner of an option keeps the good swings and is shielded from the bad ones, since the loss is capped at the premium. More swing means more chance of a big win, and the cap on the loss stays put. Time works the same way. More days left means more room for the cork to wander.

The formula then does two things. It figures the chance that the option ends up worth something, and it weighs the payoff by that chance. It also discounts the result back to today using the interest rate, because money received later is worth a bit less than money in hand. The math inside uses logarithms and a bell curve. You do not have to work it by hand to understand what it is saying.

The numbers, and where to find yours

Four of the five inputs are easy to find. The stock price is on any quote page. The strike price and the expiration date are printed on the option itself. The interest rate is the yield on a short term United States Treasury bill, which the Treasury Department publishes on its website under daily interest rate data. The model also assumes a yield on the stock, since a stock that pays dividends loses value on the day it pays them. That figure comes from the company's own dividend history.

The fifth input is volatility, and here is the catch. Nobody knows the future wobble. So traders work backward. They take the option's actual market price, feed in the other four inputs, and solve for the volatility that makes the formula match. That number is called implied volatility. Your brokerage screen likely shows it beside each option. It tells you what the market is guessing about future swings, nothing more.

The Options Clearing Corporation publishes an official guide to option contracts, and the document is titled Characteristics and Risks of Standardized Options. Your broker must give it to every new options customer. It spells out contract sizes and rules. One standard contract usually covers 100 shares, so a quoted premium of $2 means a cost of $200 for the contract.

A worked example

Let me tell you about a retired teacher named Ruth. She owns 100 shares of a company trading at $40, and she is curious about a call with a strike of $40 that expires in six months. She is not buying anything. She just wants to see how the price comes about.

Her broker screen shows a Treasury bill yield of 4 percent, no dividend, and an implied volatility of 25 percent. Plugging those in, the model returns a call price near $3.20 a share. Ruth checks the arithmetic on cost. 100 shares times $3.20 is $320 for the one contract. The stock sits right at the strike, so all $320 is time value.

Now she changes one input at a time, the way a good student would. She raises the volatility from 25 percent to 35 percent. The model now returns about $4.35 a share. 100 shares times $4.35 is $435. The extra $115 is the market's price for a wilder ride. She puts volatility back to 25 percent and cuts the time to three months. The price falls to about $2.30 a share, which is $230 for the contract. Less time, lower price.

Ruth learned something from this little exercise. The stock price never changed, and yet the option price moved a lot. The price of a choice depends on how much room the stock has to surprise us.

Where it goes wrong

I have a fondness for any tool that admits its limits, and this one has plenty. The model assumes the stock moves smoothly, with no sudden jumps. Real stocks gap overnight when bad news lands. The model assumes volatility stays fixed until expiration. Real volatility climbs and falls, often in a single week. The model also assumes you can trade at any moment with no fees. Real trading has commissions and spreads.

The original formula was built for European options, which can only be used on the expiration date. Most stock options in the United States are American style, meaning the owner may use them any time before that date. The model needs adjusting for those, especially for puts and for stocks that pay dividends.

Then there is the volatility guess itself. If the market's number is wrong, the price is wrong, and the formula will never warn anyone. It gives a clean answer to a messy question. That tidy look can fool a person into feeling more certain than the facts allow.

Options also carry a cost that surprises new people. Time value melts away as the expiration date nears. A buyer can be right about the stock and still lose money if the move comes too slowly. Seller risk can be far larger than buyer risk, and the Options Clearing Corporation guide explains how.

Questions to answer before you leave this page

Can you say in your own words what an option buys, and why the buyer pays a premium for it? If a stock trades at $55 and a call has a $50 strike, can you work out the intrinsic value of $5? Do you know which of the five inputs cannot be looked up, and why traders solve for it backward? When your screen shows an implied volatility, do you see it as the market's guess rather than a fact? If the time left on an option is cut in half, which way would you expect the price to move? Have you read the Characteristics and Risks of Standardized Options document that your broker must provide? And when a tidy formula hands you a tidy answer, will you remember to ask what it quietly assumed?