Part V · Options · Chapter 5A

The option contract

Calls, puts and the four basic positions

Part V is the bridge between the derivative theory of Part IV and the dealer mechanics of Part VI onward. It covers the contract and its four primitive positions in this chapter, then what an option's price is made of and how to read an option chain (Chapter 5B). Next come the five primary Greeks (Chapter 5C), the higher-order Greeks (Chapter 5D), and volatility, from log returns to implied volatility (Chapter 5E). The part closes with put-call parity and the risks at expiry that the Greeks do not see (Chapter 5F), and with the volatility surface (Chapter 5G).

The treatment is deliberately ground-up, so a reader with an undergraduate background in finance can skim it. If you do not have one, know that every idea from Part VI to Part XIII assumes you have mastered what is here.

1 · A right, not an obligation

An option is a contract that gives its holder the right, but not the obligation, to buy or sell a set quantity of an underlying asset at a price fixed in advance. That price is the strike price, also called the exercise price. The right lasts until a set date, the expiry or expiration, and using it is called exercising the option.

To acquire the right, the holder pays a fee that is never refunded, called the premium. The seller of the option, called its writer, keeps the premium whatever happens at expiry. In return the writer accepts a conditional obligation: if the holder exercises, the writer must honor the other side of the contract.

This asymmetry, with rights on one side and obligations on the other, is the cornerstone of every option's value. The premium is the price of that asymmetric right.

Calls and puts

A call option gives its holder the right to buy the underlying at the strike. The holder profits when the underlying ends above the strike by more than the premium paid. The maximum loss is the premium, while the maximum gain is unbounded in theory: every dollar above the strike adds a dollar to the payoff.

A put option gives its holder the right to sell the underlying at the strike. The holder profits when the underlying ends below the strike by more than the premium paid. The maximum gain is bounded but large: the strike minus the premium, reached if the underlying goes to zero. The maximum loss is again the premium paid.

Either type can be bought, which makes you long the option, or sold (also called written), which makes you short it. Two types and two sides make the four primitive positions: long call, short call, long put and short put. Every more complex strategy in Part XII is built from these four.

In, at and out of the money

Moneyness describes where the strike $K$ sits relative to the underlying's price $S$, which tells you whether exercising right now would pay:

• A call is in-the-money (ITM) when $K < S$, and a put when $K > S$, so that exercising now would pay something.
• An option is at-the-money (ATM) when its strike is at, or very close to, the price $S$.
• A call is out-of-the-money (OTM) when $K > S$, and a put when $K < S$, so that exercising now would pay nothing.

With the index at 6,000, a 5,900 call is in the money and a 5,900 put is out of the money. Chapter 5B shows how moneyness shapes an option's price.

European and American exercise

Options are also classified by when they can be exercised. A European-style option can be exercised only at expiry, and an American-style option at any time up to expiry. The right to exercise early cannot have negative value, so an American option is worth at least as much as its European counterpart. The difference can be material when dividends or interest rates create a reason to exercise early.

US-listed options on single stocks and exchange-traded funds, such as SPY, AAPL and NVDA, are American-style. US-listed options on broad-based indexes, such as SPX, XSP, NDX, RUT and VIX, are European-style. They are also settled in cash: under cash settlement, exercise pays the in-the-money amount in dollars rather than delivering anything. The European style is a structural consequence of that cash settlement, because with no underlying to deliver, the early-exercise mechanism does not apply.

Most academic option pricing assumes European exercise because it is tractable. The practical adjustments for American exercise are well understood but add complexity. Pricing an American option means solving a free-boundary partial differential equation (PDE), in which the price at which early exercise pays must be found along with the option's value. The Black-Scholes formula of Chapter 5C is the cleaner European answer.

For the rest of this book, "option" means a European-style, cash-settled index option unless the text says otherwise: the laboratory specimen for SPX dealer mechanics. Chapter 5F takes up American-style exercise and physical settlement, where shares change hands.

2 · The four positions at expiry

At expiry an option is worth exactly what exercising it pays. Write $S_T$ for the underlying's price at expiry, $K$ for the strike and $P$ for the premium, each per share or per point of the index. A call pays the amount by which $S_T$ exceeds $K$, or nothing:

Here $\max(a, b)$ means the larger of the two numbers, so a payoff is never below zero. A put pays the amount by which $S_T$ falls short of $K$, or nothing:

The buyer's profit or loss (P&L) at expiry is the payoff minus the premium. The writer's P&L is the premium minus the payoff, which is the buyer's P&L with its sign flipped. Before costs, whatever the holder gains the writer loses, which gives the four positions in this table:

The breakeven is the price at expiry where the P&L is zero. For a long call above the strike, set $S_T - K - P = 0$ and solve to get $S_T = K + P$. The short call's P&L is the same expression negated, so it crosses zero at the same price. A short position breaks even exactly where the long one does, because one side's gain is the other side's loss.

The short call's maximum loss is unbounded in theory and, in practice, bounded only by how high the underlying can go. The short put's maximum loss, $K - P$, is substantial, because the put can be exercised against you with the underlying near zero.

The position picker below draws these four lines for any strike and premium you choose. Pick a position with the buttons, then move the strike from 70 to 130 and the premium from $0.50 to $20. The chart shows the P&L at expiry per share, which the figure calls the payoff, with profit shaded green and loss red. Dashed lines mark the strike and the breakeven, whose value the readouts give along with the maximum risk per contract of 100 shares.

position long call short call long put short put

strike K

$100

premium

$3.00

position = long call

breakeven = $103.00

max risk = $300 per contract

Long call. Right to buy at strike. Long premium.

Switching between the long and the short side of the same option reflects the line across zero and leaves the breakeven where it was: $K + P$ for either call and $K - P$ for either put. Raising the premium pushes the breakeven away from the strike, so that the buyer needs a bigger move and the writer has more room. At the starting values the maximum risk readout shows the asymmetry in dollars: $300 for either long position, $9,700 for the short put, and unbounded for the short call.

3 · What each position is a bet on

Each position suits a different view of the market, its natural thesis. The thesis involves both direction and implied volatility (IV), the volatility that an option's price implies (Chapter 5E). The loss of a naked short call, one written without holding the underlying to cover it, is unbounded.

The asymmetry between the two sides is the fundamental feature of options trading. The long sides have defined risk and theoretically unlimited reward, while the short sides have limited reward and large or unlimited risk.

Every "income strategy" you meet is some structured version of selling premium. It accepts the asymmetric risk in exchange for collecting positive expected value through the variance risk premium, the excess of what options charge over the volatility that follows. Chapter 1C showed why a high win rate cannot prove that expected value is positive, and Chapter 10C weighs the evidence for the premium. Every "lottery ticket" strategy is the opposite, paying premium up front for a chance at the convex upside. The two are mirror images, traded at different population frequencies for different reasons.

Why these four matter

Every options strategy you meet is a specific combination of these four primitives, with the underlying itself added in some. That covers a covered call, an iron condor, a jade lizard, a calendar spread and a broken-wing butterfly. Knowing the four cold, including their P&L asymmetry and natural thesis, is the difference between memorizing strategy names and understanding what you are trading. Part XII uses this composition principle to take apart fifty-odd named strategies, a deconstruction that is trivial once you have internalized the four primitives.

4 · Contract size: the multiplier

Option prices are quoted per share or per index point, but you trade whole contracts. The contract multiplier converts one into the other:

• US equity and ETF options: a multiplier of 100, so one contract controls 100 shares and a premium of $3.00 on the chain is $300 of cash per contract.
• SPX index options: $100 times the index level, since there are no shares to count.
• ES futures options: $50 per index point, on the E-mini S&P 500 futures contract.
• VIX options: $100 per point of the VIX index.

Converting any quoted amount into dollars takes one multiplication, with $m$ the multiplier and $n$ the number of contracts:

The multiplier is the most common source of "I thought I bet smaller than that" mistakes in retail accounts.

The XSP multiplier saves you from sizing errors that would be catastrophic on SPX. It does not save you from misunderstanding what you have sold.

The discipline is to multiply the per-contract values by the multiplier and the number of contracts every time you place an options order, and to do it before clicking submit. The two-second pause has saved more retail accounts than any indicator stack.

5 · Check yourself