how options are priced: the model behind the premium

In short

A friend of mine once asked me why a simple option on a stock cost him $4.20 when the stock itself sat at $50. I did not have a good answer that day, and I told him so. If you have ever looked at an option quote and wondered where the price came from, you are in good company. The price, called the premium, comes from a model that weighs a handful of things: where the stock sits now, the strike price, the time left, how jumpy the stock is, and the cost of money. Learn those five, and the number on the screen stops looking like magic. Look up the interest rate and the strike on your own quote before you trust any figure. Time works against the buyer every single day. That is the heart of it.

The whole of it

What it is

I once watched a neighbor of mine, a careful man, sell a field's worth of corn before it was ever planted. He took a fair price in March for a harvest that would come in October. Both of them were betting on what the weather would do. An option is a close cousin of that old handshake. It is a contract that gives the buyer the right, but not the duty, to buy or sell a stock at a set price before a set date. A call option gives the right to buy. A put option gives the right to sell. The set price is called the strike. The set date is called expiration. The money paid up front for the contract is the premium.

You have probably wondered who decides that premium. No one person does. The market sets it, trade by trade. Yet there is a model that tells you roughly where a fair price should sit, and traders use it as a starting point. The most famous one was published in 1973 by Fischer Black and Myron Scholes, and Robert Merton added to it. We call it the Black Scholes model, and it earned a Nobel Prize in economics in 1997.

Two parts make up every premium. One part is intrinsic value, which is what the option would be worth if you used it right now. The other part is time value, which is the extra amount people pay for the chance that things turn out better. A call with a $50 strike on a stock trading at $53 has $3 of intrinsic value. If the call sells for $4.20, the other $1.20 is time value. Simple as that.

How it works

If you are holding a call option, you want the stock to rise. The model asks five questions to put a price on that hope. What is the stock worth today? What is the strike? How long until expiration? How wildly does the stock swing? And what could you earn on your cash in the meantime? Each answer pushes the premium up or down.

Take the stock price first. When the stock rises, a call gains value and a put loses it. The strike works the other way. A lower strike makes a call dearer, since the buyer gets to purchase at a cheaper price. Time comes next. More days left means more chances for the stock to move, so the premium is larger. As days pass, that time value drains away. Traders call this time decay. It is a quiet leak, and it never takes a day off.

Then there is volatility, a word for how much a stock tends to jump around. A calm utility stock moves in small steps. A young tech company can lurch ten percent in a day. The wilder the stock, the more the option costs, because bigger swings raise the odds of a big payoff. Last is the interest rate. Higher rates make calls a little dearer and puts a little cheaper. The reason is that holding a call ties up less cash than owning the shares outright, and cash earns interest.

One more piece trips folks up. The model uses expected volatility, and nobody knows that number ahead of time. So the market works backward. It takes the actual premium and asks what level of volatility would produce it. That figure is called implied volatility. It shows how nervous the crowd feels.

The numbers, and where to find yours

You can find every input without paying for anything. The stock price, the strike, the expiration date, and the premium all show on the option chain that your broker provides. Implied volatility is listed there too, in most cases. The Options Clearing Corporation, which stands behind listed option contracts in the United States, publishes the contract terms and explains how an option contract works. The Cboe, the exchange that created listed options in 1973, runs an education center with plain explanations and calculators.

The interest rate input is usually a short term Treasury yield. You can read current yields on the U.S. Treasury Department's website, under daily Treasury rates. The rate changes, so I will not put a figure on it here. Check the page the day you look. Also know that one standard stock option contract covers 100 shares. A premium quoted at $4.20 means a cost of $420 for the contract, before any commission your broker charges. People forget that multiplier all the time. It is an easy slip.

A worked example

Let me tell you about a woman named Dana. She is a schoolteacher, careful with her money, and she watches a stock she knows well. It trades at $50 a share. She looks at a call option with a $50 strike that expires in 30 days. The premium shown is $2.10.

Since the strike equals the stock price, the intrinsic value is $0. The whole $2.10 is time value. One contract covers 100 shares, so 100 times $2.10 is $210. That is the most Dana can lose on this contract, because she is only buying the right and not the duty.

Now suppose the stock jumps to $56 on the last day. The call has an intrinsic value of $56 minus $50, which is $6 a share. Times 100 shares, that is $600. Subtract the $210 she paid, and her gain is $390. Her break even point at expiration is the strike plus the premium, which is $50 plus $2.10, or $52.10. Below that price, she loses money even if the stock rose a little.

Now picture a different ending. Say the stock drifts to $50.80. At expiration, the call has $0.80 a share of intrinsic value, which is $80 for the contract. Dana paid $210, so she is down $130 even though the stock went up. This is the part that surprises people. Time ate the rest.

Last, suppose the stock gets more jumpy the week she buys. If implied volatility rose, the same call might cost $2.60 instead of $2.10. That is $260 for the contract, an extra $50. She would be paying more for the same right because the crowd expects bigger swings. The stock price did not change at all. Only the mood did.

Where it goes wrong

I have seen smart people stumble here, and it is no shame. The model rests on assumptions, and the real world does not always oblige. It assumes the stock moves in a smooth, random way, with no sudden gaps. Stocks do gap. A bad earnings report can drop a price overnight, and no formula saw it coming. The basic version also assumes a steady volatility, yet volatility itself moves around. And it was built for options that can only be used at expiration, called European style. Most stock options in the United States are American style, which means the buyer may use them any time up to expiration. The model needs adjusting for that, and for dividends too.

A bigger mistake is mixing up a fair price with a good deal. The model can tell you what a premium is built from. It cannot tell you where a stock is headed. Buyers who see a cheap premium sometimes forget that cheap options often expire worthless. Sellers who collect a fat premium sometimes forget that their loss can be large. Selling a call without owning the shares, for example, carries a risk with no cap on it. The seller must deliver the shares if the buyer uses the option, no matter how high the price has climbed.

Costs add up too. The broker may charge a fee on each contract. The gap between the buying price and the selling price, called the spread, takes a bite. For thinly traded options, that bite can be big. Taxes matter as well, and the rules depend on how long the position is held and on what kind of account holds it. The IRS explains the treatment of options in Publication 550, which is worth reading before the first trade. Slow down.

Questions to answer before you leave this page

Can you name the five things that make up a premium, and say which way each one pushes the price? If a call costs $2.10 with a $50 strike, do you know your break even at expiration is $52.10? Have you looked at how the option chain at your own broker shows the premium, the strike, and the implied volatility? Do you remember that one contract covers 100 shares, so a $2.10 quote means $210 out of pocket? Can you explain to a friend why a stock can rise and the option still lose money? Do you know which short term Treasury yield the model would use today, and where to read it on the Treasury Department's site? And have you asked yourself what you would do if the whole premium were lost, and whether you could bear that without losing sleep?