Learning works best on larger screens

---

id: what-are-options title: What Are Options? summary: A professional, math-aware introduction to option contracts, their payoff structure, and why they are powerful. outlook: neutral tags:

• basics
• beginner
• foundations

---

Learning goals

By the end of this lesson you should be able to:

• Define an option contract clearly and precisely.
• Distinguish rights (buyer) from obligations (seller).
• Identify the contract parts: underlying, strike, expiration, premium, contract size.
• Calculate payoff, profit, and break-even for a simple call or put.
• Explain intrinsic value, time value, and moneyness with math.

The one-sentence definition (accurate and complete)

An option is a contract that gives the buyer the right (not the obligation) to buy or sell an underlying asset at a fixed price on or before a fixed time, while the seller must honor that right if exercised.

Hidden inside that sentence are four ideas that govern everything else you will learn:

1. Right vs obligation (asymmetry of risk)
2. Fixed price (the strike, $K$)
3. Fixed time (expiration, $T$)
4. Underlying asset (usually a stock or ETF)

Notation and setup (so we can do clean math)

We will use standard options notation throughout the course:

• $S_0$: current underlying price
• $S_T$: underlying price at expiration
• $K$: strike price
• $C_0$: call premium (per share)
• $P_0$: put premium (per share)
• $T$: time to expiration (in years)
• $r$: risk-free rate (annualized)
• $\sigma$: implied volatility

Contract size matters: 1 U.S. equity option typically controls 100 shares. If the premium is 2.50 per share, the cash outlay is 250 per contract.

Calls vs puts (the two primitives)

• Call option: right to buy the underlying at the strike.
• Put option: right to sell the underlying at the strike.

Everything else in options is a combination of these two rights.

Long vs short (the buyer/seller asymmetry)

• Long option (buyer): pays premium; has the right; loss is capped at the premium.
• Short option (seller): receives premium; has the obligation; loss can be large or theoretically unlimited.

This is the fundamental asymmetry that drives option pricing, risk management, and strategy design.

Payoff vs profit (do not confuse these)

Payoff is what the option is worth at expiration. Profit accounts for the premium you paid or received.

Payoff functions at expiration

• Call payoff: $\max(S_T - K, 0)$
• Put payoff: $\max(K - S_T, 0)$

Profit for the option buyer

• Long call profit: $\max(S_T - K, 0) - C_0$
• Long put profit: $\max(K - S_T, 0) - P_0$

The premium shifts the payoff curve down for buyers and up for sellers.

Intrinsic value and time value (what you are really paying for)

An option’s price today is the sum of two components:

• Intrinsic value: value if exercised now.
  • Call: $\max(S_0 - K, 0)$
  • Put: $\max(K - S_0, 0)$
• Time value: the extra value from possibility before expiration.

So:

$$\text{Option Price} = \text{Intrinsic Value} + \text{Time Value}$$

Key intuition: time value is what you pay for uncertainty and time remaining. It is not “free.” It decays as expiration approaches.

Time value is also called extrinsic value in options jargon.

Moneyness: ITM, ATM, OTM (a precise meaning)

Moneyness compares the current price to the strike:

• In-the-money (ITM)
  • Call: $S_0 > K$
  • Put: $S_0 < K$
• At-the-money (ATM): $S_0 \approx K$
• Out-of-the-money (OTM)
  • Call: $S_0 < K$
  • Put: $S_0 > K$

Moneyness drives intrinsic value, delta, and how quickly the option price responds to price changes.

Break-even (the simplest quantitative decision point)

For a long call:

$$\text{Break-even} = K + C_0$$

For a long put:

$$\text{Break-even} = K - P_0$$

The underlying must move beyond this point by expiration for the buyer to profit.

Exercise vs assignment (what actually happens)

• Exercise: the buyer chooses to convert the option into a stock trade.
• Assignment: the seller is forced to take the other side of that trade.

Most options are not exercised; they are traded before expiration. But the right to exercise is always there for the buyer, which is why the option has value.

American vs European style (rules, not geography)

• American style: can be exercised any time before expiration (most equity options).
• European style: can be exercised only at expiration (most index options).

These rules affect pricing and optimal exercise behavior, especially around dividends.

Why options exist (three core uses)

1. Hedging (insurance)
   • A put can protect a stock position against a large drop.
2. Speculation with defined risk
   • A call/put offers directional exposure with a known maximum loss.
3. Income generation
   • Selling options can generate premium if you are comfortable with the obligation.

Options are not “good” or “bad.” They are a tool. The quality of the outcome depends on how well the tool is used.

Common beginner confusions (avoid these early mistakes)

• Confusing payoff with profit: always subtract the premium.
• Ignoring contract size: premiums are quoted per share; your cash outlay is per contract.
• Assuming high probability equals low risk: rare events dominate short option risk.
• Forgetting time: a correct directional view can still lose if it happens too slowly.

Knowledge checks (one deck, multiple questions)

Optional math: payoff functions and parityRead more

Piecewise payoff (call):

$$\text{Payoff}_{\text{call}}(S_T) = \begin{cases} 0 & S_T \le K \\ S_T - K & S_T > K \end{cases}$$

Piecewise payoff (put):

$$\text{Payoff}_{\text{put}}(S_T) = \begin{cases} K - S_T & S_T < K \\ 0 & S_T \ge K \end{cases}$$

Put-call parity (no dividends):

$$C_0 - P_0 = S_0 - K e^{-rT}$$

Parity is a no-arbitrage relationship: if it is violated, there is a theoretical trade that locks in profit. You will revisit this in the pricing lesson.