how options are priced: the model behind the number

In short

A friend of mine once asked why a stock option cost $4.20 when the stock itself sat at $50. I did not have a good answer, and I told him so. If you have ever looked at an options quote and wondered where the number came from, you are in good company. The price is not a guess, and it is not a gift from the exchange. It comes from a handful of inputs that anyone can look up. Five of them matter most: the stock price, the strike price, the time left, the expected swing in the stock, and the interest rate. Change one input and the price moves in a predictable way. Learn those five, and the quote stops being a mystery.

The whole of it

What it is

I once watched a man at a county fair try to guess the weight of a hog. He did not know the hog. But he knew what hogs usually weigh, and he made a fair guess. That is close to what an options model does. It does not know where the stock will land. It asks what range of prices is likely, and then it puts a fair price on the right to buy or sell inside that range.

An option is a contract. A call option gives the holder the right to buy 100 shares at a set price, called the strike, on or before a set date. A put option gives the holder the right to sell 100 shares at the strike. The seller of the contract must honor that right if the holder uses it. For taking on that duty, the seller collects a payment called the premium. The premium is the number on your screen.

The famous model for that number is the Black Scholes model. Fischer Black and Myron Scholes published it in 1973, and Robert Merton added to the work. It earned a Nobel Prize in economics in 1997. The idea is simple at heart. The fair price is what the seller would need to charge to cover the cost of protecting against the swings in the stock.

How it works

You have probably noticed that some options are cheap and some are dear, even on the same stock. The model explains why. It takes in five inputs and gives back one price.

The first input is the stock price today. A call gains value when the stock rises. A put gains value when the stock falls. The second is the strike price. A call with a strike far below the stock price is worth more, because the holder can already buy cheap. The gap between stock and strike is the core of the price. Traders call that gap intrinsic value.

The third input is time. More days left means more chances for the stock to move your way. So a contract with six months left costs more than one with six days left, even when everything else matches. The extra part of the price that comes from time is called time value, and it melts away as the expiration date nears.

The fourth input is volatility. That is a measure of how wildly the stock tends to swing. A calm stock rarely moves far, so its options are cheap. A jumpy stock can move a lot, so its options cost more. Here is the odd part. The model does not know the future swing. Traders read the market price of the option and work backward to find the swing it implies. That is called implied volatility. It is a guess the crowd makes together.

The fifth input is the interest rate. It plays a smaller role. A higher rate nudges call prices up and put prices down, because money tied up in shares could otherwise earn interest. Dividends the stock pays also nudge the price, since the stock usually drops by the dividend on the payout date.

Put them together, and the model produces one number. Not magic. Just arithmetic.

The numbers, and where to find yours

Let me say a word about where to look. The stock price, the strike, and the expiration date all appear on any options chain at your brokerage. The Options Clearing Corporation, which guarantees every listed contract in the United States, publishes the contract rules. The Options Industry Council, which it supports, has free lessons on pricing at optionseducation.org. Read the booklet called Characteristics and Risks of Standardized Options. Brokers must give it to new options customers.

For the interest rate, the model usually uses a short term Treasury yield. You can find daily yields on the Treasury Department's website, treasury.gov, under interest rate data. For volatility, most brokerage screens show implied volatility right beside each contract. Your own broker will list it as a percent.

Any rule set by law, such as the size of the tax on a gain, changes by year and by your situation. For the tax rate on short term gains, check IRS Publication 550, Investment Income and Expenses. For the holding period that sets short term versus long term treatment, the figure is the current figure, which the official source publishes each year. The agency sets that, not the model.

A worked example

Let me tell you about a woman named Marcy. She follows a stock priced at $50. She looks at a call with a $45 strike that expires in three months. The quote reads $6.10.

Marcy does the first step by hand. Intrinsic value is the stock price minus the strike. That is $50 minus $45, which is $5. So of the $6.10 premium, $5 is intrinsic value.

She subtracts to find the rest. $6.10 minus $5 is $1.10. That $1.10 is time value, the price of the chance that the stock climbs further before the date arrives.

Now she checks the cost of one contract. Each contract covers 100 shares. So $6.10 times 100 shares is $610. That is the most she can lose if she buys it, since the premium is all she pays.

Next she looks at a different call, with the same strike and date, on a calmer stock also priced at $50. Its quote is $5.40. The intrinsic value is still $5. So the time value is $5.40 minus $5, which is $0.40. The calm stock has far less time value. Why? The crowd expects smaller swings. Same stock price, same strike, same date, and a very different premium.

Marcy sees the lesson. The gap between $1.10 and $0.40 is the market's opinion of how jumpy each stock will be. That is the whole story of implied volatility in one small comparison.

Where it goes wrong

I would be a poor porch friend if I told you the model never fails. It does. The Black Scholes model assumes the stock moves in a smooth, bell shaped way. Real stocks do not. They gap on bad news and lurch on good news. The model also assumes volatility stays fixed, yet it changes every day. Those assumptions are why the model gives a fair starting point and not a promise.

Another trap catches careful people. A low premium feels like a bargain. It often is not. A cheap option far from the strike may carry a small chance of ever paying off. The price is low because the odds are low. The buyer of such a contract can lose the whole premium, and it happens often.

Time works against the buyer, too. Every day, a little time value drains out of the price. Traders call that decay. Picture ice melting on a warm porch. A contract can lose some of its price from one week to the next, with the stock price unchanged. The buyer holds the same right, but it is worth a little less.

Last, the quote on your screen is not the price you always get. The gap between what buyers bid and sellers ask can be wide on thinly traded contracts. Fees and taxes come on top of that. The broker charges a commission or a per contract fee, and the buyer must report any gain on a tax return.

Questions to answer before you leave this page

Can you name the five inputs that set an option's price, and say which way each one pushes it? If a call has a $45 strike and the stock sits at $50, how much of the premium is intrinsic value, and how much is time value? What is implied volatility, and where on your own brokerage screen would you find it? How many shares does one contract cover, and what does a $6.10 premium cost for a whole contract? Where would you look up the contract rules, and who guarantees that the seller honors the deal? And what does the buyer lose each day that passes, even when the stock does not move at all?