What an option is, and where the money comes from

OpeningRight call, total loss

On October 21, XYZ is at $100. You and a friend both think it will rise within a month. Your friend spends $10,000 on 100 shares; you spend $498 on 3 call contracts with a 105 strike expiring November 20, quoted at 1.66 each.

A month later the stock is at $104, up 4%. Your friend makes $400. At expiry the stock is not above 105, so your 3 contracts are worthless and the whole $498 is gone. You both called the direction right; one of you made money and the other lost everything.

Now a different ending: the stock rises to $115. Your friend makes $1,500, a 15% return. Each of your options is worth $10 at expiry, $3,000 for all 3; after the cost you make $2,502, a return of more than 500%.

The same view on direction, and the option turned the outcome into two extremes. The reason is that an option bets on two things a stock doesn't: how far the price rises and how fast it gets there. This chapter starts with the contract itself and ends by coming back to this question: where does the money in options actually come from?

1.1The six terms of a contract

Six terms fully define an option contract. Miss one and you don't know what you bought.

Reading a quote in full

A broker's platform usually shows the contract from the opening as “XYZ Nov 20 '26 105 Call.” The clearinghouse uses a standard 21-character code:

• First 6 characters: the underlying's symbol, padded with spaces if it is shorter than 6
• Next 6: the expiration date as YYMMDD, here November 20, 2026
• 1 character: C for a call, P for a put
• Last 8: the strike times 1000; 00105000 is 105.000

The 1.66 quote is per share. One contract costs 1.66 × 100 = $166, and 3 contracts cost $498. The contract covers 100 shares, so at a $100 stock price it controls a notional $10,000 of stock. Controlling $10,000 for $166 is the leverage we will get to later.

Who has the right, and who has the obligation

The buyer pays and gets a right, which can be exercised or abandoned. The seller is paid and takes on an obligation: when called on, the seller of a call must sell the stock at the strike, and the seller of a put must buy the stock at the strike. The premium is the seller's compensation for taking on that obligation.

Opening, closing and four endings

Your broker's order ticket asks whether you are opening or closing. Buy to Open creates a long position and Sell to Close sells it; Sell to Open creates a short position and Buy to Close buys it back.

An option position can end in only four ways: closed with an offsetting trade before expiry, exercised by the buyer, assigned to the seller, or expiring worthless. Most option positions are closed before expiry; the option itself is a price you can trade at any time. So when you buy a call, you don't need the $10,000 that exercise would take; just sell the call before expiry. But if you hold to expiry and the option is in the money, it will be exercised automatically (Chapter 2), and your account must have enough cash or margin.

1.2The four basic positions

Pair buy or sell with call or put and you get the four basic positions. However complex a strategy is, it is built from these four positions plus the stock itself.

Now some concrete numbers: stock at $100, strike 100, 30 days to expiry, implied volatility 30%. Black-Scholes prices the call at 3.59 and the put at 3.26.

Breakeven is easy to work out: for a call it is the strike plus the premium; for a put, the strike minus the premium. A long put makes its maximum profit if the stock falls to zero: (100 − 3.26) × 100 = $9,674. That is also the short put's maximum loss.

The character of the four positions fits in one sentence: buyers trade a limited loss for a larger potential gain, and sellers accept a larger potential loss for a limited gain. A short call's loss is unlimited in theory, because the stock price has no ceiling. A short put's loss looks “limited,” but $9,674 is close to the full risk of owning 100 shares.

1.3Drawing a payoff diagram at expiry

For a position made up only of options and stock, the payoff diagram at expiry is always made of straight segments. However many legs there are, you can draw it in three steps.

1. Mark the kinks: mark each strike on the horizontal axis. The line can bend only at a strike.
2. Set the slopes: each long call adds a slope of +1 to the right of its strike and 0 to the left; each long put adds −1 to the left of its strike and 0 to the right; short positions flip the sign; every 100 shares of stock add +1 everywhere. Add up the slopes in each interval.
3. Fix the height: work out the actual P&L at any one point. At a stock price equal to one of the strikes, for example, the intrinsic value of every other option is easy to compute. Then join the segments using the slopes.

The slope has another important meaning: it is how many shares the position is “equivalent to” in that interval at expiry. A slope of +1 means each contract is like holding 100 shares; a slope of 0 means the stock price doesn't matter. Delta in Chapter 4 is this slope before expiry.

Example 1: a call spread

Buy the 100 call (pay 3.59) and sell the 110 call (collect 0.66), for a net debit of 2.93. The three slopes are 0, +1, 0. Below 100 both calls expire worthless and you lose 2.93; above 110 the profit is fixed at 10 − 2.93 = 7.07; breakeven is 102.93. Maximum loss $293, maximum profit $707. Compared with buying the call alone, you give up gains above 110 in exchange for a lower cost and a closer breakeven.

Example 2: synthetic stock

Buy the 100 call (pay 3.59) and sell the 100 put (collect 3.26), for a net debit of 0.33. Left segment: the put is in the money, slope +1. Right segment: the call is in the money, slope +1. The whole thing is a straight line with slope +1, and the P&L at a stock price of 100 is −0.33. It is exactly the same as “buying 100 shares at 100.33.”

The extra 0.33 is no coincidence. Buying the stock outright means paying $100 now; with synthetic stock, the $100 is paid only at expiry (through exercise). At a 4% rate, 30 days of interest is 100 × (1 − e^{−0.04×30/365}) = 0.33. This is put–call parity, which Chapter 3 derives formally.

Example 3: covered call and short put

Own 100 shares and sell one 105 call (collect 1.66): this is a covered call. Slopes: +1 below 105 (only the stock), 0 above 105 (the stock's +1 and the short call's −1 cancel). Above 105 the profit is capped at 105 − 100 + 1.66 = 6.66; breakeven is 98.34.

Now draw “sell one 105 put” (collect 6.32): above 105 the profit is capped at 6.32, breakeven is 98.68, and below 105 the slope is +1. The two diagrams have exactly the same shape, and the 0.34 difference is interest again: the covered call needs $100 up front to buy the stock, while the short put's cash can stay in the account earning interest, 105 × (1 − e^{−0.04×30/365}) = 0.34. Many people see the covered call as “conservative” and the naked put as “aggressive,” but on the payoff diagram at expiry they are the same bet.

1.4Intrinsic value and time value

Premium splits into two parts: intrinsic value, which you would get by exercising now, and time value, which you pay for the chance that “things could still get better before expiry.”

An option with intrinsic value above zero is in the money (ITM); one whose strike equals or is closest to the current price is at the money (ATM); one with zero intrinsic value is out of the money (OTM). A call is in the money when the strike is below the stock price; a put is the opposite.

The table shows three patterns.

• Time value is largest at the money and falls off on both sides. An at-the-money option's price is all time value. A deep in-the-money option is almost pure intrinsic value, and a deep out-of-the-money option is worth almost nothing.
• At the same strike, a call has more time value than a put by the “interest on the strike.” At the 95 strike: call 1.65, put 1.33, a difference of 0.32; and 95 × (1 − e^{−0.04×30/365}) = 0.31, with the gap due to rounding. The call buyer pays the strike later and the put buyer receives the strike later; between the two, that is the interest.
• A deep in-the-money put can have negative time value. The 115 put is worth only 14.84, less than its intrinsic value of 15. Black-Scholes prices European-style options, which can't be exercised early, and receiving $115 a month sooner would earn an extra month of interest. Holders of American-style options exercise early to avoid this loss; Chapter 2 covers the mechanism in detail.

How time value disappears over time

The price of an at-the-money option (stock and strike both 100, IV 30%): 13.75 with 365 days left, 6.41 at 90 days, 3.59 at 30 days, 1.70 at 7 days, 0.63 at 1 day. Cut the time to a quarter and the price roughly halves. An at-the-money option's time value is roughly proportional to the square root of the time left, so the closer to expiry, the faster it drains each day.

1.5Three ways options differ from stock

Options and stock both bet on price, but options have three extra properties: they expire, their P&L is curved, and they come with built-in leverage.

Difference 1: a deadline

You can hold a stock indefinitely and wait as long as you like. An option ceases to exist after expiration, so you need to be right not only about direction but also before expiry. The direct cost of a deadline is time decay: a 30-day at-the-money call loses about 0.062 a day, or $6.2 per contract; with 7 days left it loses 0.124 a day; on the last day, 0.319. If the stock doesn't move, the buyer pays every day and the seller collects every day.

Difference 2: nonlinearity

With 100 shares, a $10 rise makes $1,000 and a $10 fall loses $1,000: perfectly symmetric. A 30-day at-the-money call is different: if the stock jumps $10 instantly, the option rises 7.30 from 3.59; if the stock drops $10 instantly, the option falls only 3.12. At expiry it is more extreme: a $10 rise makes 6.41, a $10 fall loses 3.59, and however much further the stock falls, the loss is still 3.59.

For the buyer, the P&L curve bends upward: gains accelerate as the stock rises, and losses slow down as it falls. The seller has the opposite. This curvature is called convexity, and Gamma in Chapter 4 measures it.

Difference 3: built-in leverage

For $359, one at-the-money call gives you the upside on 100 shares ($10,000) of stock. The standard measure of an option's leverage is its elasticity, also called the leverage multiple:

30-day at-the-money call: Delta 0.532, price 3.591, Ω = 0.532 × 100 / 3.591 = 14.8. If the stock rises 1%, the option rises about 15%. Repricing exactly: on a 1% rise the option gains 15.5%, and on a 1% fall it loses 14.2%. The two aren't equal; that's nonlinearity at work again.

For comparison, buying stock in a margin account lets you borrow at most half under US initial margin rules, for leverage of about 2×. Option leverage is often 10× or more, and a buyer never gets a margin call: the worst case is that the premium goes to zero.

1.6Stock or option

Expressing a bullish view with options adds a bet that the stock will “rise far enough and fast enough.” The farther out of the money the strike, the bigger the rise you need.

Let's generalize the comparison from the opening: with the stock at $100, you choose one of four strikes expiring in 30 days, and we look at the return at expiry on each $1 invested.

Look at the 105 row: if you expect exactly a 5% rise, the best performer is actually the in-the-money 95 call (+50%), followed by the at-the-money call (+39%), while the 105 call loses everything. At 110 the ranking is completely reversed. Which strike is best depends on how far you expect the stock to rise. Choosing a strike without an expected move is drawing lots.

One more fact that is easy to miss: every row in Table 1-4 is an “at expiry” result. In practice you can sell the option early. If the stock spikes before expiry, the option's price still contains time value, and closing early may beat holding to expiry. So this table is only a starting point for choosing a strike; how to manage the position while you hold it has to wait until we have covered the Greeks.

1.7Five sources of profit

Money in options comes from only five sources. Say which one you are earning before you place the order, and you will know which risk to watch.

Source 1: direction

The most intuitive source: the stock moves far enough in the direction you expect. Buy the at-the-money call (3.59); with the stock at 105 at expiry, each contract makes (5 − 3.59) × 100 = $141. This is the money the previous sections were about. It requires you to be right about three things at once: direction, size and timing.

Source 2: volatility

You can ignore direction and bet only that the stock will “move more than the market expects.” Buy a 30-day at-the-money straddle (the 100 call and the 100 put together) for 3.59 + 3.26 = 6.85, which corresponds to an implied volatility of 30%. There are two ways this makes money:

• Implied volatility rises: the next day the stock hasn't moved, but the market suddenly gets nervous and IV rises to 40%. The straddle reprices to 8.98, a gain of $213 per straddle. The straddle's sensitivity to IV is about 0.23 per point.
• Realized volatility exceeds implied volatility: IV stays the same, but the stock's realized volatility over the next 30 days is 40%. At 40%, the straddle's fair value is 9.13, 2.28 above the purchase price. On average each straddle makes about $228; on any particular path it may be more or less.

Source 3: the passage of time (sellers)

Now sell the straddle instead and collect 6.85. If the stock stays near 100 for 10 days, the straddle's value falls from 6.85 to 5.60, and the seller makes about $126 per straddle. On the day it is opened, the straddle's time decay is about 0.114 a day, and it speeds up as expiry approaches. This money looks like it “arrives automatically every day”; the price is that the seller's loss grows quickly on a big move, because the nonlinearity described earlier works against the seller.

Source 4: mispricing

When the price relationship between options breaks, there is relative mispricing. Section 1.3 showed that the call minus the put at the same strike and expiration should equal 0.33. Suppose you see the 100 call at a mid price of 3.80 and the put at 3.26: a difference of 0.54, or 0.21 too much. Sell the call, buy the put and buy 100 shares, and at expiry you sell the stock at 100 wherever it trades. After 0.33 of interest on the $99.46 tied up for 30 days, you lock in about 0.21 × 100 = $21.

In reality, market makers' algorithms take almost all of these opportunities. Nearly every “mispricing” you see on screen has a sound explanation: an upcoming dividend, shares that are hard to borrow, the early-exercise value of American-style options, a stale quote, or a bid–ask spread so wide that trading three legs costs far more than $21. For individual traders, the more realistic use of this source is picking contracts that are relatively cheap: for the same view, for example, choosing the expiration or strike with relatively low implied volatility.

Source 5: the volatility risk premium

A large body of research finds that the implied volatility of index options is, on average, higher than the volatility realized afterward. The reason isn't hard to see: investors are willing to overpay for insurance, and sellers demand compensation for bearing crash risk. This persistent gap is called the volatility risk premium, and it is the main source of sellers' long-run returns.

Some numbers give a feel for its shape. Suppose SPY is at $780 and you sell a 30-day at-the-money straddle at IV 18%, collecting 32.12. If realized volatility over the next 30 days is only 15%, the straddle's fair value is 26.80, an average gain of about $533 per straddle. But in a month with realized volatility of 40%, fair value becomes 71.24, an average loss of about $3,912. One bad month takes 7.3 good months to make up. The premium is real, but it is payment for bearing tail risk; Chapter 6 discusses it in detail.

In real trading, one position often bets on several sources at once. Buying a call before earnings bets on direction, but also on “IV won't fall too much,” and the second bet is often overlooked (Case 2). Selling a covered call earns mainly from the passage of time and the risk premium, while also carrying a mildly bullish direction. Before placing an order, list each source involved and write down what you lose if each one goes wrong. You will find that the main risk in many “directional trades” is actually volatility.

1.8Win rate, payoff ratio and expectancy

If options are fairly priced, the expected P&L of all four positions is close to zero: buyers have a low win rate and a high payoff ratio, and sellers have a high win rate and a low payoff ratio.

“Fairly priced” has a precise meaning here: assume the stock price at expiry follows the Black-Scholes lognormal distribution, with volatility equal to the implied volatility in the option's price. Under that assumption you can calculate exactly each position's probability of profit and its average win and average loss.

Selling the 90 put wins nearly 90% of the time, which sounds tempting. But its average loss of 3.53 is 8.8 times its average win of 0.40. 0.898 × 0.40 − 0.102 × 3.53 ≈ 0: win rate and payoff ratio cancel out exactly. Buying the 110 call is the reverse: it loses almost nine times out of ten, and when it wins, it makes more than 7 times its cost on average.

The table makes one point: with fair pricing, there is no “sure thing” structure. Choosing to buy or to sell changes only the shape of the P&L distribution, not the expectancy. For expectancy to turn positive, there must be something you see more accurately than the market: direction (Source 1), future volatility compared with IV (Sources 2 and 5), or a price relationship that really is broken (Source 4). The passage of time creates no expectancy by itself; it is just the way sellers collect compensation for risk.

How big an edge can a view on direction give you? Suppose you are convinced the stock's expected annual return is 10%, 6 points above the 4% risk-free rate. The 30-day at-the-money call's probability of profit rises only from 33.9% to 36.0%, and its average value at expiry from 3.60 to 3.87: about $27 more per contract, or about 7.5% of the premium. Even with an expected return as high as 20%, the probability of profit is only 39.7%. Over a one-month term, being “bullish for the long run” gives you very little edge; what really changes expectancy is your judgment about near-term events, the size of the move and volatility.

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