Options Path

An **option** is a contract that gives the buyer the **right but not the obligation** to buy or sell an underlying asset at a fixed **strike price** $K$ on or before the **expiry date** $T$. The buyer pays a **premium** to the **writer** (seller), who takes on the matching obligation. **Calls** grant the right to buy, **puts** grant the right to sell. Globally, listed equity option volume hit a record 12.16 billion contracts in 2024, with the ASX listing options on around 70 leading Australian companies plus index options on the S&P/ASX 200.

A **call option** gives the holder the right to **buy** the underlying asset at the **strike price** $K$ during the life of the contract. The payoff at expiry is $\max(S_T - K, 0)$. Buyers are **bullish** on the underlying. The **writer** (seller) accepts the obligation to deliver the asset at $K$ if the buyer exercises. A long call's downside is capped at the premium $C$, while its upside is theoretically unlimited.

A **put option** gives the holder the right to **sell** the underlying asset at the **strike price** $K$ during the life of the contract. The payoff at expiry is $\max(K - S_T, 0)$. Buyers are **bearish** on the underlying or are using the put for **portfolio insurance**. The writer accepts the obligation to buy the asset at $K$ if the put holder exercises. A long put's downside is capped at the premium $P$, and the maximum payoff is $K$, reached when the stock falls to zero.

The **option premium** is the price the buyer pays to the writer in exchange for the rights conveyed by the contract. It splits cleanly into **intrinsic value** plus **time value**. Premium is paid upfront, is **non-refundable**, and equals the buyer's maximum loss. The five drivers of premium are the spot price $S$, the strike $K$, the time to expiry $T$, the volatility $\sigma$, and the risk-free rate $r$. Black-Scholes-Merton (1973) ties them together formally.

**American options** can be exercised at any business day up to and including expiry. **European options** can be exercised only at expiry. Because the American holder has every European holder's choices plus more, $C_{\text{Am}} \geq C_{\text{Eu}}$ and $P_{\text{Am}} \geq P_{\text{Eu}}$. The naming refers to **exercise style**, not where the option is listed. On the ASX, equity options are American-style while **S&P/ASX 200 (XJO) index options** are European-style and cash-settled.

**Moneyness** describes the relationship between the underlying price $S$ and the strike $K$. An option is **in-the-money (ITM)** when immediate exercise produces a positive payoff, **at-the-money (ATM)** when $S \approx K$, and **out-of-the-money (OTM)** when immediate exercise produces a zero payoff. **Deep ITM (DITM)** or **deep OTM (DOTM)** means $S$ is far from $K$. Moneyness drives both the **intrinsic value** of a premium and an option's sensitivity (delta) to the underlying.

**Intrinsic value** is the payoff an option would deliver if exercised immediately. For a call, $\text{IV} = \max(S - K, 0)$. For a put, $\text{IV} = \max(K - S, 0)$. It is always **non-negative**, equals zero for OTM and ATM options, and gives the **lower bound** on the premium of an American option. Anything the option trades above intrinsic value is **time value**.

**Time value** is the part of an option premium above its intrinsic value. It compensates the buyer for the chance that the underlying moves favourably before expiry. Time value grows with **time to expiry** $T$ and with **volatility** $\sigma$, and decays as $T$ shrinks. The rate of decay is **theta**, $\Theta = \partial V / \partial t$. At expiry, time value equals zero and the option is worth only its intrinsic value.

The **break-even price** of an option strategy is the underlying value $S_T$ at expiry where total profit equals zero. For a long call, $S_T = K + C$. For a long put, $S_T = K - P$. Above (call) or below (put) break-even, the position turns profitable. Inside that band the buyer loses, capped at the premium paid. Break-even is the **simplest decision metric** for sizing whether a directional view is large enough to be worth paying for.

A **long call** is a position created by buying a call option. The buyer pays premium $C$ for the right to buy the underlying at strike $K$ on or before expiry. **Maximum loss equals $C$** and is incurred when $S_T \leq K$. **Profit potential is unlimited** because the upside payoff $S_T - K$ grows without bound. The expiry profit curve is a hockey stick that sits flat at $-C$ up to $K$ and slopes up at $45^{\circ}$ thereafter.

A **short call** is written by selling a call option. The writer collects premium $C$ upfront and accepts the obligation to sell the underlying at strike $K$ if the holder exercises. **Maximum profit equals $C$** and is achieved when $S_T \leq K$. A **naked** short call has **unlimited loss** potential, since $S_T$ can rise without bound. The position is **bearish to neutral** and expresses a view that the stock will not rally past $K + C$.

A **long put** is created by buying a put option. The buyer pays premium $P$ for the right to sell the underlying at strike $K$ on or before expiry. **Maximum loss equals $P$** and is incurred when $S_T \geq K$. **Maximum profit equals $K - P$** and is reached only if $S_T = 0$. The expiry profit curve is the mirror of a long call, sloping down to $K - P$ to the left of the strike and sitting flat at $-P$ to the right.

A **short put** is written by selling a put option. The writer collects premium $P$ upfront and accepts the obligation to **buy** the underlying at strike $K$ if the holder exercises. **Maximum profit equals $P$**, reached when $S_T \geq K$. **Maximum loss equals $K - P$**, reached if the stock falls to zero. Unlike a naked short call, the downside is **bounded** because share prices cannot go below zero. The view is **bullish to neutral**, often used to buy stock at a target price while earning income.

A **long straddle** combines buying a call and buying a put at the **same strike** $K$ and **same expiry** $T$. The position profits from a **large price move in either direction** and loses if the underlying stays near the strike. **Maximum loss equals the total premium** $C + P$, reached when $S_T = K$. **Profit is unbounded** on the upside and capped at $K - C - P$ on the downside. The two break-evens are $K \pm (C + P)$. A straddle is a textbook **long-volatility play**.

A **short straddle** is created by writing a call and writing a put at the **same strike** $K$ and **same expiry** $T$. The writer collects both premia upfront and profits if the underlying **stays close to $K$**. **Maximum profit equals $C + P$**, achieved when $S_T = K$. **Maximum loss is unlimited** to the upside via the short call and bounded but large to the downside via the short put. The strategy is a **short-volatility** play, often called a short-vega income trade.

A **synthetic long stock** is built by **buying a call and selling a put** at the same strike and expiry. By put-call parity the combined payoff equals owning the stock minus the present value of the strike, $C - P = S_0 - K e^{-rT}$. The position behaves like a leveraged stock holding without actually buying shares. Traders use synthetic stock when the underlying is hard to borrow or short, when capital efficiency matters, or as one leg of a conversion arbitrage. The mirror trade, **short call + long put**, creates a **synthetic short stock**.

A **synthetic long call** is built by **owning the stock and buying a put** at the desired strike. The put floors the loss at $K - S_0 - P$ and the stock provides unlimited upside, exactly the payoff profile of a long call. Put-call parity confirms the equivalence, $S_0 + P = C + K e^{-rT}$, so the synthetic call differs from a direct call only by a **zero-coupon bond** of face value $K$. This construction is also known as a **protective put** and is widely used by investors who already hold a stock and want downside insurance.

**Put-call parity** is a no-arbitrage condition that pins the prices of European call and put options with identical strike and expiry to the current stock price and the present value of the strike. The relation is $C + K e^{-rT} = P + S_0$, first formalised by Stoll (1969) and extended by Merton (1973). Any deviation from this equality allows a **riskless profit**, so in liquid markets violations are fleeting. The relation applies strictly to **European options**. For American options on non-dividend-paying stocks, only an inequality holds.

When put-call parity $C + K e^{-rT} = P + S_0$ does not hold, an arbitrageur can construct a **riskless profit** equal to the violation. The strategy is to **buy the cheaper portfolio**, **sell the dearer portfolio**, and pocket the difference. Two symmetric trades are called the **conversion** (long stock, long put, short call) and the **reverse conversion** (short stock, short put, long call). Transaction costs, borrowing rates, and dividends create a **no-arbitrage band** within which small violations are not profitable to exploit.