probability of profit and expected value in options trades

In short

A friend of mine once bragged that his options trade had a 90 percent chance of making money, and he sounded like a man who had found a loophole. If you have ever felt the pull of a number like that, you are in good company. Probability of profit tells you how often a trade might end above zero, and expected value tells you what the trade is worth on average once you count the size of every win and every loss. The two numbers can point in opposite directions. A trade can win nine times out of ten and still lose money over time. So when you see a probability, ask what the winners pay and what the losers cost, and check the real prices at your own broker.

The whole of it

What it is

I once watched a neighbor flip a coin for a dollar with his grandson, and the boy kept winning small and losing big without noticing. That is the whole lesson in one porch scene. Probability of profit, often shortened to POP, is the chance that a trade ends with a gain. It is an estimate, and it comes from a pricing model, not from a promise. Expected value, or EV, is the average result if the same bet were made many times. You get it by taking each possible outcome, multiplying it by its chance, and adding the pieces up.

You have probably seen POP on a broker screen next to an options trade. Most platforms compute it from implied volatility, which is the market's guess about how much a stock will move. The platform must make assumptions to get there. A different assumption gives a different number. So POP is a useful hint, but it is not a fact about the future.

How it works

If you are holding a trade idea right now, picture it as two piles. One pile holds what you win, and one holds what you lose. Each pile has a size and a chance attached. The EV is the chance of winning times the win, minus the chance of losing times the loss.

Here is the part that surprises people. Selling an option often gives a high POP, because the seller keeps the payment unless the stock moves a lot. But the seller took a small gain in exchange for a chance at a large loss. Buying an option often gives a low POP, because the stock has to move enough to cover the cost. But the buyer risks only what was paid. High POP and low POP are just two ways of slicing the same risk.

One more piece matters. In a fair market, the price already reflects the odds. That means the EV of a trade, before any fees, tends to sit near zero. Fees, called commissions, and the gap between the buying price and the selling price, called the bid ask spread, pull it a little lower. So a trade does not become a good one just because its POP looks tall.

The numbers, and where to find yours

Your broker's trade ticket shows the pieces. Look for the premium, which is the price of the option for each share. Remember that one standard contract covers 100 shares, so a premium of $1.50 means $150 for the contract. Look for the maximum gain, the maximum loss, and the breakeven price, which is the stock price where the trade earns exactly zero. Look for implied volatility and the POP the platform reports.

Fees matter here too. Your broker publishes its commission per contract on its pricing page. The Options Clearing Corporation publishes the official booklet called Characteristics and Risks of Standardized Options, and every broker must give it to new options customers. It explains how contracts work and what can go wrong. If you want one document to read first, that is the one. Where a rule sets a limit that changes by year, such as the amount of net capital a margin account needs, the number is the current figure, which the official source publishes each year, and the site fills in the verified figure.

A worked example

Let me tell you about a woman named Marie. She is a retired schoolteacher with a quiet streak of curiosity. She looks at a trade where she sells one put option on a stock trading near $50. A put is a contract that obliges the seller to buy 100 shares at a set price if the stock falls below it. Marie collects $100 for taking that obligation. Her platform says the POP is 80 percent.

Her best case is keeping the $100. Say the worst realistic case, if the stock falls hard, is a loss of $500 after counting the $100 she collected. She plugs in the numbers. The chance of winning is 0.80, and the win is $100, so that piece is 0.80 times $100, which is $80. The chance of losing is 0.20, and the loss is $500, so that piece is 0.20 times $500, which is $100. Her EV is $80 minus $100, which is negative $20.

Notice what happened. An 80 percent chance of winning still gave a loss of $20 on average. Now add a commission of $0.65 to open the trade. That is $0.65 times 1 contract, or $0.65, and the EV slips to about negative $20.65. Marie did not do anything foolish. She simply learned that a high POP can hide a lopsided payout.

Now flip it for a moment. Say she looked at buying a call instead. A call gives its owner the right to buy 100 shares at a set price. She pays $100, and the platform shows a POP of 30 percent. If she wins, her gain averages $400. Then 0.30 times $400 is $120. And 0.70 times the $100 she loses is $70. Her EV is $120 minus $70, which is $50 before fees. That looks nice, but a fair market would rarely hand it out. Any gap like that usually means one of her inputs is off, often the size of the average win. She tests her guess before she trusts it.

Where it goes wrong

I have made this mistake myself, so I say it with a friendly hand on your shoulder. The first trap is trusting POP as if it were certain. The model behind it assumes the stock moves in a smooth, tidy way. Real stocks gap and jump, especially after news. The tail, meaning the rare big move, is where the worst losses live.

The second trap is a small sample. EV is an average over many tries. If a person makes a handful of trades, luck can swamp the math. A trade with a positive EV can still lose five times in a row. A trade with a negative EV can win five times in a row and fool its owner.

The third trap is forgetting the costs. Commissions, the bid ask spread, and taxes all reduce the result. Gains from options held a short time are generally taxed as ordinary income, and the Internal Revenue Service explains this in Publication 550, Investment Income and Expenses. After tax results can look quite different from the figures on the screen.

The fourth trap is size. A loss that is rare but large can erase many small wins. Early exercise and assignment can also surprise a seller, since the buyer of an American style option may use it before expiration. Risk of that kind is not visible in a POP number at all.

Questions to answer before you leave this page

Have you written down both your maximum gain and your maximum loss, with a dollar sign on each, before you think about the odds? Do you know where your broker's POP number comes from, and what volatility it assumed? When you multiply each outcome by its chance and add them up, is the EV above zero after commissions and the bid ask spread? Would one bad outcome hurt you badly enough that the average no longer matters? Have you read the Options Clearing Corporation booklet, Characteristics and Risks of Standardized Options, so the rules do not surprise you? And if the numbers still look too good, what might you be missing?