Options: Calls and Puts

Start with an everyday picture. Imagine a shop gives you a coupon: for the next month, you may buy a $100 gadget for $100, no matter what the shelf price does. You paid a few dollars for that coupon. If the gadget's price jumps to $130, your coupon is gold, you buy at $100 and you're $30 ahead. If the price drops to $80, you simply don't use the coupon and buy at $80 like everyone else. The most you ever lose is the few dollars the coupon cost. That coupon is a call option, and that lopsided outcome, big upside, tiny fixed downside, is the whole reason options exist.

Formally, an option is a contract giving you the right, but not the obligation, to buy or sell something at a price fixed in advance. "Right, but not the obligation" is the sentence to remember: you use the option only when it helps you and throw it away when it doesn't. That one-sidedness is what bends the payoff.

Every option is described by four things:

• the underlying — what the contract is on (a stock, an index, a barrel of oil);
• the strike $K$ — the fixed price you're allowed to trade at (the $100 on the coupon);
• the expiry — the date the right runs out (the coupon's "use by");
• the premium — what you pay up front to own the option (the price of the coupon).

There are two kinds. A call is the right to buy at the strike, you want it when you think the price will rise. A put is the right to sell at the strike, you want it when you think the price will fall. A put is basically an insurance policy: you pay a premium, and if the thing you own crashes in value, the put pays you back.

How to read these diagrams

Every chart below has the same two axes. The horizontal axis is the price of the underlying at expiry (call it $S$), running from low on the left to high on the right. The vertical axis is the payoff, how much the option is worth on that day. The dashed vertical line marks the strike $K$. That's it, once you can read one of these, you can read them all.

The call: a right to buy

Buy a call and you're betting the price ends up above the strike. On the expiry day, if the underlying is worth $S$, the call pays

$$\text{call payoff} = \max(S - K,\; 0),$$

which is just a fancy way of writing "whichever is bigger, $S - K$ or zero." If $S$ finished above $K$, you exercise: buy at the cheap strike $K$ something now worth $S$, pocketing $S - K$. If $S$ finished below $K$, you'd be crazy to buy at $K$ when the market sells it cheaper, so you let the call expire and it pays nothing. That gives the flat-then-rising shape:

What actually happens, step by step. Say a stock trades at $100 and you buy a call with strike $100 expiring in a month, paying a $5 premium. Three ways it can go:

• The stock climbs to $115. Your call is worth $115 - 100 = 15$. You paid $5, so you tripled your money.
• The stock drifts to $103. The call is worth $3, less than the $5 you paid, so you've lost $2, even though you were "right" about direction. Being right isn't enough; you have to be right by more than the premium.
• The stock falls to $90. The call expires worthless. You lose the $5 and not a penny more.

The put: a right to sell

A put is the exact mirror. Buy one and you're betting the price ends up below the strike. Its payoff is

$$\text{put payoff} = \max(K - S,\; 0).$$

If the price collapses to $S$ below $K$, you exercise: sell at the high strike $K$ something now worth only $S$, gaining $K - S$. If the price stays above $K$, the put expires worthless. So the payoff is tall on the left (a big crash) and flat on the right:

Payoff is not profit

Here's the trap that catches every beginner. The two charts above show payoff, what the option is worth at expiry. But you paid the premium to own it, so your real profit is the payoff minus that premium. On a chart, profit is just the payoff line slid straight down by the premium.

The point where the profit line crosses zero is the breakeven, the price you actually need to reach before you make a cent. For a call it's the strike plus the premium ($S = K + \text{premium}$); for a put it's the strike minus the premium. Between the strike and the breakeven, the option is worth exercising but still hasn't paid back what it cost you.

Don't just take the chart's word for it, drag the sliders below. Move the strike and the premium and watch the green payoff line, the amber profit line, and the breakeven point shift. Try setting the premium to zero (payoff and profit merge) or flipping to a put:

The other side of the trade: the seller

Every option has two sides. Someone sold you that coupon, and that seller (the "writer") gets the exact opposite payoff. They collect the premium up front, and in return they take on the obligation you shed. Flip the buyer's profit chart upside-down and you get the seller's.

Selling looks tempting because you get paid immediately and, most of the time, the option expires worthless and you keep the whole premium. But look at the shape: the seller's gain is capped at the premium, while the loss runs without limit if the price moves against them. That asymmetry is the single most important thing to understand about options.

What decides the premium

If two things drive the payoff, three things drive its price. An option costs more when:

1. It's closer to paying off. A call whose strike is right at today's price is worth more than one far above it, it has a better shot at finishing in the money.
2. There's more time left. More time means more chances for the price to swing your way, so a three-month option costs more than a one-week one. That extra is the option's time value, and it bleeds away as expiry nears.
3. The underlying is jumpier. A wild, volatile stock is more likely to make a big move in your favour, so its options cost more. This "jumpiness" is volatility, and it's the single biggest lever on an option's price.

Turning those intuitions into an actual number, a fair premium, is exactly what The Black-Scholes Model does, and how much of the price comes from expected jumpiness is captured by Implied Volatility. How the premium changes as the price, time, and volatility move is measured by the The Option Greeks.

Moneyness, intrinsic value, and time value

Traders have shorthand for where the price sits relative to the strike, called moneyness:

• In the money (ITM): exercising right now would pay something ($S > K$ for a call, $S < K$ for a put).
• At the money (ATM): the price is sitting right at the strike, $S \approx K$.
• Out of the money (OTM): exercising would pay nothing.

An option's price is made of two parts. Intrinsic value is what you'd get by exercising today, the in-the-money amount, and it can't be negative. Time value is everything on top of that, the money you pay for the chance the price moves further your way before expiry. A $100-strike call on a $103 stock trading for $5 has $3 of intrinsic value and $2 of time value. On expiry day, time value is zero and only intrinsic value is left.

The four basic positions

You can buy (go long) or sell / write (go short) either kind of option, four positions in all:

Position	Premium	Best case	Worst case	You want
Long call	you pay	unlimited gain	lose the premium	price up, a lot
Long put	you pay	gain up to the strike	lose the premium	price down
Short call	you receive	keep the premium	unlimited loss	price flat or down
Short put	you receive	keep the premium	loss down to the strike	price flat or up

Read the pattern down the table: the buyer always has a small, known, fixed loss (the premium) and a large possible gain. The seller always has a small, fixed gain (the premium) and a large possible loss. Buying options is like buying a lottery ticket or an insurance policy; selling them is like being the lottery or the insurance company, you win small and often, but a rare big move can hurt.

Worked example

You buy a call with strike $K = 100$ for a premium of $5.

• If the stock ends at $S = 120$: payoff $= \max(120 - 100, 0) = 20$. Subtract the $5 premium and your profit is $15, a $5 bet turned into $15.
• If the stock ends at $S = 95$: payoff $= \max(95 - 100, 0) = 0$. You let it expire and lose just the $5 premium.
• Breakeven is where profit is zero: $S = K + \text{premium} = 105$. Above $105 you make money; below it you don't, even between $100 and $105 the call is worth exercising but hasn't yet earned back its cost.

Now a put with strike $K = 100$ bought for $4. If the stock crashes to $S = 80$: payoff $= \max(100 - 80, 0) = 20$, minus the $4 premium leaves a $16 profit. Its breakeven is $S = K - \text{premium} = 96$. In both cases the buyer's loss is capped at the premium while the gain can be large, and that asymmetry is the entire appeal.

Common pitfalls

• Payoff is not profit. The hockey-stick payoff ignores the premium you paid. Slide the whole line down by the premium to get profit, that's what sets the breakeven, not the strike.
• Being right on direction isn't enough. You have to be right by more than the premium before expiry, or you still lose money. A stock that inches up can leave a call buyer in the red.
• Buying and selling are not symmetric. The seller has the mirror-image payoff: a capped gain (the premium) and a potentially unlimited loss. Selling a naked call can lose far more than it ever makes.
• Time is the enemy of the buyer. An option is a wasting asset. Even if the price never moves, its time value drains toward zero as expiry approaches, this is theta, one of the The Option Greeks.
• Cheap is cheap for a reason. A far-out-of-the-money option is cheap because it's unlikely to pay. Its low price already reflects that low chance, see Expected Value; it is not free money.

Why options exist

Three reasons people pay for that bent payoff. Leverage: a small premium controls a large position, so a modest price move becomes a big percentage return. Hedging: a put is insurance on something you own, it pays out exactly when you're hurting. Defined risk: a buyer knows the most they can lose, the premium, before they ever put the trade on. Pricing all of that fairly, turning a view about how much the price will move into a dollar premium, is the job of The Black-Scholes Model and the volatility that feeds it.

Key terms

• Call / put — the right to buy / to sell at the strike.
• Strike ($K$) — the fixed price the option lets you trade at.
• Premium — what the option costs to buy.
• Expiry — the date the right runs out.
• Payoff — what the option is worth at expiry.
• Profit — payoff minus the premium paid.
• Breakeven — the price where profit crosses zero (strike ± premium).
• Moneyness — whether exercising now would pay (ITM), break even (ATM), or pay nothing (OTM).
• Intrinsic / time value — the in-the-money amount now, versus the extra paid for future potential.
• Writer — the seller of the option, who collects the premium and takes the opposite payoff.