01Paying for a choice
You find a house you like for $500,000, but you won't know for three months whether your new job comes through. You offer the owner $5,000, non-refundable, for the right to buy at $500,000 any time in the next three months. If the job comes through, or house prices jump, you buy at the agreed price. If not, you walk away and the owner keeps the $5,000.
You have bought an option. You paid a fixed amount, the premium, for a right you may or may not use. Your loss is capped at $5,000; your gain has no fixed limit. The owner took the $5,000 in exchange for giving you that choice, and carries the risk of having to sell below what the house turns out to be worth.
Car insurance is the same deal from the other direction: you pay a premium for the right to hand the insurer a loss. That is a put, and we'll get to it.
02Calls and puts
- A call is the right to buy something at a set price, the strike, until a set date, the expiry. Calls gain when the price rises.
- A put is the right to sell at the strike until expiry. Puts gain when the price falls: the car-insurance deal, where the insurer has to take the wreck off your hands at the agreed value.
Every option has a buyer, who pays the premium and holds the right, and a seller (also called the writer), who receives the premium and must deliver if the buyer uses the right. Using it is called exercising the option. Listed US stock options are standard contracts: one contract covers 100 shares, so a premium quoted at $5.43 costs $543 a contract.
A call is in the money when the stock trades above the strike, at the money when it trades at the strike and out of the money below it; for a put it is the other way around.
One thing beginners often miss: you rarely exercise. Most options are sold back to the market before expiry, at whatever they are worth that day, and before expiry they are worth more than exercising would get you. Section 04 is about that worth.
03How an option pays at expiry
At expiry there is no more time and no more choice: an option is worth exactly what exercising it would get you, or nothing.
put = max(strike − stock price, 0) Profit is that value minus the premium paid. A call breaks even at strike + premium; a put at strike − premium.
A call costs $5.43 a share ($543 a contract) and a put $4.45 ($445). Where both come from is section 06. Profit per contract at expiry:
| Stock at expiry | Call | Put |
|---|---|---|
| $80 | −$543 | +$1,555 |
| $90 | −$543 | +$555 |
| $100 | −$543 | −$445 |
| $110 | +$457 | −$445 |
| $120 | +$1,457 | −$445 |
The call breaks even at $105.43 and the put at $95.55. Notice the shape: a capped loss on one side, a gain that grows on the other. And notice that the stock had to move more than 5% just for the call buyer to get the premium back.
04What an option is worth today
Before expiry, an option is worth more than what exercising it would get you right now, because there is still time for the price to move your way. So an option's premium splits into two parts:
A call with a $90 strike costs $12.00: $10.00 intrinsic (you could buy at $90 a stock worth $100) plus $2.00 of time value.
A call with a $110 strike costs $1.89, all of it time value: exercising today would mean paying $110 for a $100 stock.
The at-the-money $100 call costs $5.43, all time value too, and the most time value of the three. Time value peaks at the money, where the outcome is most uncertain.
Time value shrinks as expiry approaches, and faster at the end. With nothing else changing, that $100 call is worth $5.43 with 90 days left, $3.02 with 30 and $1.42 with a week. The slow leak is called time decay: it costs the buyer every day and pays the seller every day.
The box below draws all of this. The gold line is the profit or loss at expiry; the dashed line is what the position is worth today, before expiry. Slide the days down and watch the dashed line sink onto the gold one: that is time decay. Volatility, the other slider, is the next section.
Buying an option is calling a bet on a draw. You pay a known price for a chance at a bigger pot, and pot odds tell you whether the price is right: pay $1 to win $10 and you need to hit more than 1 time in 11. The option market sets that price for you: the premium is its price for the draw, how often it comes in times how big the pot is when it does. Buying is only a good bet if you think the draw is better than that price assumes.
05The five things that move the price
| If this rises… | Call | Put | Why |
|---|---|---|---|
| Stock price | Up | Down | The call is closer to paying; the put further from it. |
| Strike | Down | Up | A higher price to buy at is worth less; to sell at, worth more. |
| Time to expiry | Up | Usually up | More time, more chances for a big move. |
| Volatility | Up | Up | Bigger swings make big payoffs likelier, and the loss is capped either way. |
| Interest rates | Up | Down | A call lets you pay later, worth more when cash earns more. |
Volatility is the one you can't look up. Everything else is known today; how much the stock will swing between now and expiry is not. In the worked example, the same $100 call costs $3.47 at 15% volatility, $5.43 at 25% and $8.37 at 40%. Turn it around and the market price of an option tells you how much volatility traders expect. That figure is implied volatility. Cboe's VIX index applies the same idea to the whole market: the expected volatility of the S&P 500 over the next 30 days, read from S&P 500 option prices.
To put a volatility figure in dollars: 25% a year on a $100 stock is a typical move, one standard deviation, of about $25 over a year. Over 90 days it scales with the square root of time, 25% × √(90/365), or about $12.40 either way. Section 09 comes back to this number.
The Greeks, the three that matter here
- Delta: how much the option moves for a $1 move in the stock. The $100 call's delta is about 0.56, so it gains about 56 cents.
- Theta: how much value one day of time decay takes away.
- Vega: how much the option moves for a 1-point change in volatility.
Two more round out the set: gamma, how fast delta changes as the stock moves, and rho, the sensitivity to interest rates.
06Black-Scholes in one page
Start with the idea, not the formula. An option's fair price is the average of what it pays across every place the stock could finish, each weighted by how likely it is, then brought back to today's money. How likely each outcome is depends on how much the stock swings: the volatility you assume. For a European call on a stock that pays no dividend, that average comes out as:
d1 = [ln(S / K) + (r + σ² / 2) × T] / (σ × √T)
d2 = d1 − σ × √T S = stock price, K = strike, T = years to expiry, r = risk-free interest rate, σ = volatility, N(·) = the chance a standard normal variable falls below that number. The put follows from the call by put-call parity (next section).
Read it as two pieces. S × N(d1) is today's value of the stock you receive in the cases where the option pays off; K × e−rT × N(d2) is today's value of what you expect to pay for it. N(d2) is the model's chance that the call finishes in the money, under its own assumptions rather than a forecast of the real odds, and N(d1) is its delta.
S = $100, K = $100, T = 90/365 = 0.2466, r = 4%, σ = 25%.
d1 = [0 + (0.04 + 0.03125) × 0.2466] / (0.25 × 0.4966) = 0.1415; d2 = 0.1415 − 0.1241 = 0.0174.
N(d1) = 0.55627; N(d2) = 0.50693; K × e−rT = $99.02.
C = 100 × 0.55627 − 99.02 × 0.50693 = 55.63 − 50.20 = $5.43.
One puzzle remains: the formula grows the stock at the risk-free rate, not at whatever return investors expect from it. That was the 1973 insight of Fischer Black and Myron Scholes and, in a companion paper, Robert Merton: by holding the right number of shares against the option and adjusting as the price moves, a trader can cancel the option's risk. A riskless position can only earn the risk-free rate, so nobody's view of where the stock is heading enters the price. Only its volatility does.
The model assumes things markets don't quite do: constant volatility, smooth prices with no jumps, no trading costs. Traders know this, and they patch it the practical way: they use the formula to translate prices into implied volatility and back. When options at different strikes imply different volatilities, that pattern (the skew) is the market telling you where it thinks the model is wrong, usually that crashes are likelier than the bell curve allows.
07Put-call parity
Hold a call plus enough cash to pay the strike at expiry, and you end up with the stock if it is above the strike, or the cash if it is below. Hold a put plus the stock, and you end up with exactly the same thing. Two portfolios with the same payoff must cost the same, so calls and puts on the same stock, strike and expiry are locked together:
$5.43 − $4.45 = $0.98, and $100 − $99.02 = $0.98. If the two sides ever drift apart, a trader can buy the cheap side and sell the dear one for a riskless profit, and that trading closes the gap.
Rearrange it and parity shows something useful: owning a stock and selling a call against it (a covered call) has the same payoff as selling a put at that strike and holding cash. The premium a covered call brings in is not free income. It is pay for carrying the stock's downside, exactly as a put seller does.
08How buyers and sellers make money
Buyers pay the premium up front. The most they can lose is that premium; the upside is large on a call and runs to the strike on a put. They need the stock to move far enough, and soon enough, to beat both the premium and time decay. Many options expire worthless, and the buyer's whole premium goes with them.
Sellers take the premium up front and keep it if the option expires worthless. They win small and often, but each loss can be large: a seller of a call who doesn't own the stock (a naked call) has no limit on the loss if the stock keeps rising. A seller of a put has agreed to buy the stock at the strike however far it falls: the loss runs all the way down to a stock price of zero, less the premium. Whether sellers come out ahead over time depends on whether the premiums they collected were priced high enough for the moves that actually came.
Options are, like futures, a zero-sum contract between buyer and seller before fees. The skill is in the price: buying volatility the market has priced too cheap, or selling it too dear.
Option sellers play like a tight player who wins a lot of small pots: steady results, then the occasional hand that gives back a month of gains. Option buyers play like someone chasing draws: mostly small losses, sometimes a big pot. Neither style wins by itself. What wins is getting a better price than the odds deserve, the same expected value arithmetic as any bet.
09Four classic combinations
Each of these is in the box above; pick it to see the shape. Figures use the $100 stock, 90 days and 25% volatility.
- Covered call: own the stock and sell a call above the price. Selling the $110 call brings in $1.89 a share: income now, in exchange for giving up any gain above $110. Best case $11.89 a share; it breaks even at $98.11.
- Protective put: own the stock and buy a put below it. A $90 put costs $1.12 and caps the loss at $11.12 a share however far the stock falls: insurance, with the premium as the cost.
- Bull call spread: buy the $100 call and sell the $110 call. It costs about $3.55 instead of $5.43 and can make at most about $6.45 a share. A cheaper bet on a rise, with the upside capped.
- Long straddle: buy the $100 call and the $100 put. It costs $9.88 and pays if the stock moves far either way, below $90.12 or above $109.88. A bet on movement, not direction. The catch: the market knows about coming news too, so the straddle is already priced for a big move. At 25% volatility, $9.88 is about the average size of the 90-day move, up or down, that the market expects (about 0.8 of the $12.40 typical move from section 05). The straddle pays only if the move beats what was already priced in.
10What can go wrong
- Buyers can lose the whole premium, and often do: being right about direction but wrong about timing still loses.
- Sellers can lose far more than the premium; a naked call has no limit.
- Volatility can fall after you buy. Option prices often drop after an earnings report even when the stock moves your way, because the uncertainty that was priced in is gone. It is the straddle's catch from section 09.
- Early exercise: options on individual US stocks can be exercised on any day up to expiry (American style), so a seller can be assigned at any time, not only at expiry. Options on the S&P 500 index itself (SPX) can be exercised only at expiry (European style) and settle in cash.
- Leverage: a small premium controls 100 shares, so gains and losses are large relative to the money put in.