What is an Option?¶
Return to Thales for a moment. He paid a small deposit in winter for the right to use the olive presses after the harvest, but only if he wished. When the harvest was abundant, he exercised and profited. Had it failed, he would simply have walked away — losing the deposit and nothing more. Two features make this arrangement an option: the holder chooses whether to act, and the writer must comply once chosen against. Strip away the olives and the centuries, and the same contract is traded electronically today on every major exchange.
Formally, an option is a financial derivative giving the holder the right, but not the obligation, to buy or sell an underlying asset at a predetermined price. In everyday terms, an option is insurance with a price: the buyer pays a premium today for protection (or opportunity) tomorrow. This asymmetry — the holder chooses, the writer must comply — is the defining feature of options and the source of their mathematical richness.
Call and Put Options¶
There are two fundamental types:
Call option: The right to buy the underlying asset at the strike price \(K\) on or before the maturity date \(T\).
Put option: The right to sell the underlying asset at the strike price \(K\) on or before the maturity date \(T\).
| Call | Put | |
|---|---|---|
| Holder's right | Buy at \(K\) | Sell at \(K\) |
| Holder benefits when | \(S_T > K\) | \(S_T < K\) |
| Maximum loss (holder) | Premium paid | Premium paid |
The exact payoff formulas \((S_T - K)^+\) and \((K - S_T)^+\) are developed in § Option Payoffs.
Contract Terms¶
Every option contract specifies:
- Underlying asset: The stock, index, commodity, or other instrument the option is written on
- Strike price \(K\): The predetermined price at which the holder may buy (call) or sell (put)
- Maturity \(T\): The date at which the option expires
- Exercise style: European (exercise only at \(T\)) or American (exercise at any time \(t \leq T\))
Throughout this text, we focus primarily on European options, which can only be exercised at maturity. This restriction is not just a simplification — it makes the pricing problem analytically tractable and leads directly to the Black–Scholes formula.
Buyer vs Writer¶
Every option trade has two sides:
Buyer (holder): Pays the premium upfront. Has the right to exercise. Maximum loss is the premium.
Writer (seller): Receives the premium. Is obligated to fulfill the contract if the holder exercises. Faces potentially unlimited loss (for a naked call writer).
This asymmetry explains why writers must post margin — a deposit guaranteeing their ability to meet the obligation. The detailed risk profile and collateral mechanics are developed in § Margin and Short Positions.
Moneyness¶
An option's relationship to the current stock price is classified as:
| Classification | Call (\(S\) vs \(K\)) | Put (\(S\) vs \(K\)) | Intrinsic value |
|---|---|---|---|
| In-the-money (ITM) | \(S > K\) | \(S < K\) | Positive |
| At-the-money (ATM) | \(S \approx K\) | \(S \approx K\) | Near zero |
| Out-of-the-money (OTM) | \(S < K\) | \(S > K\) | Zero |
Moneyness determines how sensitive an option is to changes in the underlying and plays a central role in the behavior of the Greeks.
Example: Consider a European call on stock XYZ with \(K = 50\) and \(T = 3\) months. The current stock price is \(S = 53\). This call is ITM with intrinsic value \(53 - 50 = \$3\). If the premium is $5, the remaining $2 represents time value — the market's assessment that the stock could move further before expiration. This raises a natural question: how should the premium be determined? That is the subject of subsequent sections.
Moneyness in Practice: SPX Options
With the S&P 500 index near 6,600, consider call options expiring in one month:
- ITM (\(K = 6{,}500\)): Premium \(\approx 260\) points. The option has substantial intrinsic value (\(6{,}600 - 6{,}500 = 100\) points) and behaves almost like the index itself — deep ITM options track the underlying nearly dollar-for-dollar.
- ATM (\(K = 6{,}600\)): Premium \(\approx 180\) points. Intrinsic value is negligible, so nearly all of the premium is time value — the market's assessment of how far the index could move before expiration.
- OTM (\(K = 6{,}700\)): Premium \(\approx 84\) points. With zero intrinsic value, the entire premium is time value. The option is cheaper because exercise requires the index to rally more than 100 points.
At the $100 contract multiplier, these premiums correspond to $26,000, $18,000, and $8,400 per contract — illustrating how moneyness directly determines the cost of an option position.
Real-World Option Contracts¶
Like futures, exchange-traded options have standardized contract sizes that determine their monetary value. The contract multiplier converts the quoted option premium into the actual dollar (or won) cost of one contract.
| Contract | Exchange | Underlying | Contract size | 1-point value |
|---|---|---|---|---|
| SPX Index Options | CBOE | S&P 500 index | $100 \(\times\) index | $100 |
| SPY Options | CBOE | SPY ETF | 100 shares | $100 |
| KOSPI 200 Options | KRX | KOSPI 200 index | 250,000 KRW \(\times\) index | 250,000 KRW |
| AAPL Options | CBOE | Apple stock | 100 shares | $100 |
For example, an SPX call option with a quoted premium of 50 points costs
per contract. Unlike futures, this amount is paid upfront as the option premium and represents the buyer's maximum possible loss. A 1-point change in the option price corresponds to $100 per SPX contract.
Contrast with futures
In a futures contract, no premium is paid at inception — both sides are symmetrically obligated, and the delivery price is set so the contract has zero initial value. In an option contract, the buyer pays a premium for the right to walk away, and the writer receives this premium as compensation for bearing one-sided risk. The contract multiplier plays the same role in both markets: it converts quoted price changes into monetary gains and losses.
Exercises¶
Exercise 1. A European call option has strike \(K = 100\) and maturity \(T = 1\) year. At maturity, the stock price is \(S_T = 115\). Should the holder exercise? What is the profit if the premium paid was $8?
Solution to Exercise 1
The holder should exercise because \(S_T = 115 > K = 100\). By exercising, the holder buys the stock at $100 and can immediately sell at $115, receiving a payoff of \(S_T - K = 15\).
The profit is payoff minus premium: \(15 - 8 = \$7\).
Exercise 2. A European put option has strike \(K = 50\) and maturity \(T = 6\) months. At maturity, \(S_T = 42\). What is the payoff? If \(S_T = 55\) instead, what is the payoff?
Solution to Exercise 2
When \(S_T = 42 < K = 50\): the holder exercises, selling at $50 an asset worth $42. Payoff \(= K - S_T = 50 - 42 = \$8\).
When \(S_T = 55 > K = 50\): the holder does not exercise (selling at $50 is worse than selling at the market price $55). Payoff \(= 0\).
Exercise 3. Explain why the maximum loss for the buyer of a call option is the premium paid, while the maximum loss for the writer of a naked call is theoretically unlimited. Relate this asymmetry to the option's payoff structure.
Solution to Exercise 3
The buyer pays the premium \(C_0\) upfront. At maturity, the payoff is \((S_T - K)^+ \geq 0\). The worst case is \(S_T \leq K\), where the payoff is zero and the buyer loses only the premium \(C_0\).
The writer receives \(C_0\) but is obligated to pay \((S_T - K)^+\) at maturity. Since \(S_T\) has no upper bound, neither does the writer's obligation. If \(S_T \to \infty\), the writer's loss \(S_T - K - C_0 \to \infty\). This unbounded loss potential is why naked call writing is considered one of the riskiest positions in finance.
Exercise 4. Classify the following options as ITM, ATM, or OTM: (a) Call with \(K = 100\), current \(S = 105\). (b) Put with \(K = 80\), current \(S = 80\). (c) Call with \(K = 60\), current \(S = 45\). (d) Put with \(K = 70\), current \(S = 55\).
Solution to Exercise 4
(a) Call with \(S = 105 > K = 100\): ITM (the call would have positive payoff if exercised now).
(b) Put with \(S = 80 = K = 80\): ATM (the put is at the boundary of having value).
(c) Call with \(S = 45 < K = 60\): OTM (exercising would mean buying at $60 what is worth $45).
(d) Put with \(S = 55 < K = 70\): ITM (the holder could sell at $70 what is worth $55, payoff \(= 15\)).