What Even is an Option?

Options are financial instruments that allow someone to make a bet on something without having to own that thing, which is also called the underlying. That underlying could be a stock, an index, a currency, or even a futures contract. An option is a type of derivative, since its value depends on that underlying.

I’ve already put together a small option pricing exercise using Delta and Gamma. However, being able to quickly calculate a price change doesn’t really tell us what the thing we’re pricing is. So, let’s go back a couple steps and define the actual instruments, and then see what makes them quite fun.

What are we actually buying?

An option is a contract that gives its buyer the right, but not the obligation, to buy or sell an underlying at a specified price, according to the contract’s exercise rules.¹ That ability to choose is the important part. If using the right would be disadvantageous, we can just let it expire.

There are two basic types:

• A call gives us the right to buy the underlying.
• A put gives us the right to sell the underlying.

Using that right is called exercising the option. As with pretty much anything, these options are different in the EU and in the US. Specifically, European-style options allow exercise only at expiry; American-style options allow it on or before expiry. Either can be traded before expiry, so exercising and selling the option are two different things.

To describe one of these contracts, we need a couple more definitions:

• Spot price, $S$: the current market price of the underlying.
• Strike price, $k$: the fixed price at which the option lets us buy or sell it.
• Expiry, $T$: when the contract expires. The time remaining until then is usually written as $\tau$.
• Premium: the price we pay to buy the option itself. This is separate from the strike price!

For the examples, we’ll use a stock as our underlying, quote everything per share, and ignore fees and financing costs.² We’ll look at what happens at expiry first; we’ll get to prices before expiry afterwards.

A call, with actual numbers

Suppose a stock is currently trading at £100. We buy a call with a strike of £100, expiring in a month, and pay a premium of £5.

What happens if the stock reaches £120 at expiry? Our contract allows us to buy something worth £120 for only £100. That right is worth £20. After subtracting the £5 we paid for it, our profit is £15.

What if the stock falls to £80? Buying it for £100 would be pretty silly when we could just buy it in the market for £80. So, we don’t exercise the option, and it expires worthless. We still paid £5 for it, though, so our loss is £5.

We can write the call’s payoff at expiry as:

Here, $S_T$ is the stock price at expiry. The $\max$ is the mathematical version of our ability to choose: we take the difference if it’s positive, and zero otherwise.

But, the payoff isn’t our profit. If $C_0$ is the premium we paid:

For our £100-strike call costing £5:

One thing to notice is that the stock can go up, and we can still lose money. At £103, we were right about the direction, but the move wasn’t enough to cover what we paid. Our break-even at expiry is $k + C_0$, which here is £105.

That qualification, at expiry, matters. If we sell the option before then, our profit depends on its sale price, which can include value from the time still remaining.

And a put?

A put works in the other direction. Suppose we pay £5 for a put with the same £100 strike. If the stock falls to £80, we have the right to sell something worth £80 for £100. That right is worth £20, giving us a £15 profit after the premium.

Its payoff and profit are:

The put’s break-even at expiry is $k - P_0$, or £95 in this example.

Plotting both makes this quite easy to see: the premium shifts each payoff curve down by £5.

This also shows why options aren’t only used to speculate. If we already own the stock, buying a put gives us a way to protect ourselves against a fall below the strike. We are paying for the ability to sell at a fixed price even if the market price collapses. This is called hedging, and the put behaves quite a bit like insurance.

Being “in the money”

You’ll often see options described by their moneyness, which just compares the current spot with the strike:

• In the money, or ITM: exercising would give a positive payoff. For a call, $S > k$; for a put, $S < k$.
• At the money, or ATM: spot is at, or very close to, the strike.
• Out of the money, or OTM: exercising would give no positive payoff. For a call, $S < k$; for a put, $S > k$.

Being in the money doesn’t mean we’re making money! Our call at £103 was in the money, with £3 of payoff, but we still lost £2 after paying the premium. Moneyness describes the contract, not the result of our trade.

We are specifically interested in at the money contracts, because that is the point where we have the most optionality, which is exactly where the fun is at!

Someone is on the other side

So far, we’ve only been buying options. For us to buy one, someone else has to sell one. Selling an option to open a short position is called writing it. The buyer has the choice; the writer takes on the obligation to fulfil the contract if exercised against them.

We call buying and holding an option being long, and writing one being short. These words describe our position in the option, as you might have seen in trading spot as well, rather than the direction we want the stock to move. A long put generally benefits from the stock falling, even though we’re “long”!

For a single option held to expiry, the writer’s profit is the reverse of the buyer’s:

The buyer can lose the premium they paid. The writer can earn at most the premium received, but their loss can be much larger. For an uncovered short call, there is no finite maximum loss, since the stock price can keep rising. For a short put on a stock, the worst case is the stock going to zero, giving a loss of $k - P_0$ per share.

So, the choice we bought has a cost, and someone is being paid to take the other side of it — as is the case with everything in finance.

Why isn’t the price just the payoff?

Here’s an interesting question: our call has a £100 strike, and the stock is currently £100. Its payoff if it expired right now would be zero. Why would we pay £5 for it?

Because it doesn’t expire right now. There is still a month in which the stock could move above £100, while our payoff can’t fall below zero. We’re paying for that remaining possibility.

The payoff calculated using today’s spot is called intrinsic value. The difference between the option’s price and its intrinsic value is called time value, or extrinsic value.¹ For our at-the-money call, the £5 premium is entirely extrinsic value. At expiry, there is no time left, and the option’s value is just its payoff.

Here’s how that looks as expiry gets closer. It’s not the cleanest graph, but it shows how the optionality increases the premium of an option.

Notice how the smooth curves approach the payoff’s sharp corner at the strike as time runs out. Each curve shows the call’s price at different possible current stock prices; it isn’t a prediction of where the stock will go.

This is why we can’t price an option from the stock price alone. Strike, time remaining, expected volatility, interest rates, and dividends all matter too. A call expiring tomorrow and a call expiring next year are very different instruments, even if they have the same strike and underlying.

Comparing options

Let’s hold everything else fixed and change one input at a time:

These comparisons are for ordinary, or vanilla, calls and puts. The time comparison needs a little care: interest rates and dividends can complicate it for European options.³

The strike comparison is particularly easy to see. The right to buy a stock for £90 must be worth at least as much as the right to buy it for £110, with everything else the same. Wherever the stock ends up, the £90-strike call has at least as much payoff. For puts, the order reverses: the right to sell for £110 is more valuable than the right to sell for £90.

This gives us a useful exercise before we do any actual pricing: if two options have the same underlying and expiry, can we immediately tell which should cost more? If their strikes and expiries differ, however, we may have competing effects, and the answer isn’t always obvious.

Volatility is the interesting one

Volatility describes the size of price fluctuations. It doesn’t tell us whether the stock goes up or down. Realised volatility measures movements that actually happened; implied volatility is the volatility input that makes a pricing model match the option’s observed market price.

In other words, implied volatility is backed out of the premium. It’s a way to express how the market is pricing uncertainty, rather than a promise about how much the stock will actually move. Why does higher implied volatility increase the value of both calls and puts? Their payoffs are convex. Larger favourable moves can give us larger payoffs, while unfavourable moves stop hurting the payoff once it reaches zero. More dispersion can therefore make either right more valuable. Of course, the premium we pay increases too.

And this is where being right about direction can still fail us. We could buy a call, see the stock rise, and still see the call’s price fall because time passed or implied volatility dropped enough to outweigh that rise.

A bet on movement

What if we don’t have an opinion on direction, but think the stock will move a lot?

We could buy a call and a put with the same strike and expiry. This is called a long straddle. Adding their payoffs gives us:

That’s quite a nice result! At expiry, the payoff depends only on how far the stock finishes from the strike, in either direction.

Using our £100-strike options, each costing £5, the total premium is £10. Our profit at expiry is therefore $|S_T - 100| - 10$, and our break-even prices are £90 and £110.

So, “the stock moved” isn’t enough. It has to finish far enough from the strike to cover both premiums. A stock that moves wildly during the month but ends at £100 gives this position zero payoff at expiry. Trading the position along the way would be a different question.

Back to Delta and Gamma

Now that we know what changes an option’s price, we can ask how much it changes. This is what the Greeks describe: sensitivities of the option price to different inputs.

If we write the option price as $V$, the two Greeks used in the pricing exercise are:

Delta tells us approximately how much the option price changes for a small change in spot, with the other inputs fixed. A call with a Delta of $0.50$ gains approximately £0.50 when the stock rises by £1. A put with a Delta of $-0.50$ loses approximately £0.50 on that same rise.

Gamma tells us how Delta itself changes as spot moves. For ordinary long vanilla options, Gamma is generally positive before expiry: as spot rises, Delta increases. A call becomes more positively exposed; a put’s negative Delta becomes less negative. As spot falls, the reverse happens.

Thus, Delta isn’t a constant that we can use for every possible move. Accounting for its change gives us the second-order approximation:

For an option worth £5, with $\Delta = 0.50$ and $\Gamma = 0.08$, a £2 rise in the stock gives:

If the stock falls by £2 instead, the estimate is £4.16. The Gamma term is positive for either non-zero move, but it doesn’t guarantee a profit: the Delta effect can outweigh it, and time and volatility can change too.

Two other Greeks follow directly from what we’ve discussed: Theta measures the effect of time passing, and Vega measures sensitivity to implied volatility. Long vanilla options usually have negative Theta and positive Vega. Rho measures sensitivity to interest rates.⁴ When we write an option, the position’s Greek exposures have the opposite signs to those of the buyer.

The Delta-Gamma exercise isolates a small spot move while holding those other inputs fixed. It’s an approximation around the current price, and large moves or changes in time and volatility require us to reconsider it. You can find the derivation and practise the mental maths in Option Pricing!.

One more interesting relationship

Calls and puts also can’t be priced independently of each other. Consider buying a call and writing a put with the same strike and expiry. At expiry, their combined payoff is:

That’s exactly the payoff from owning the stock and owing an amount $k$ at expiry. Therefore, for European options on a non-dividend-paying stock, the prices must satisfy the put-call parity:³

Here, $r$ is the continuously compounded risk-free interest rate, and $\tau$ is the time remaining in years. The term $ke^{-r\tau}$ is the amount of money today that grows to $k$ at expiry.

If we take interest rates to be zero, this simplifies to:

For example, with spot at £102 and a £100 strike, a £6 call implies a £4 put with the same expiry.

The reason is no arbitrage: portfolios with identical future payoffs should have the same price today. If they don’t, in the idealised setting we could buy the cheaper one and sell the more expensive one, locking in the difference. Real trading adds spreads, fees, and funding costs, but this relationship is still a very useful consistency check.

Anyhow, the weirdness of options doesn’t end there either (see exotics and other combos); even so, you can have quite a lot of fun with these simple things we’ve described here.