One of the key differences between options and linear derivatives, such as forwards futures, or swaps, is the option premium.

This is the non-refundable payments that the option buyer makes to the option seller.

Its purpose to compensate the seller for taking on the asymmetrical payoff profile.

That options create meaning that the option buyer will gain if things move in their favor more than they will lose if things go against them.

Now, one of the most important questions for both traders and investors is what should this option premium be? Let's approach this intuitively.

The reason that the option premium exists is simple.

The risks and opportunities in options differ significantly for buyers and sellers.

For the option buyer, the worst case scenario is easy.

If the option is out of the money, they just let it expire worthless.

But if the option is in the money, the buyer can exercise the option and potentially earn a large payouts, especially in the case of a long call, where the upside is theoretically unlimited.

For the option seller, it's the opposite.

Their best case scenario is that the option expires worthless, but the worst case, they could face substantial losses if the market moves significantly against them.

Consequently, the purpose of the option premium is to compensate the seller for taking on this less favorable risk profile.

After all, who would willingly sell an option at zero cost when the best case scenario is zero profit, but their worst case outcome is a potentially large loss? So conceptually, the option premium should equal the expected loss of the option seller.

Expected loss is generally calculated as the probability of losing multiplied by the anticipated amount lost.

To make this clearer, let's look at a simple example.

Imagine we're playing a fair coin toss game with a $1 coin.

There are two possible outcomes, heads or tails, each with a 50% probability.

Now, if I invite you to play this game and you get to keep the coin, if you guess correctly, how much should I charge you to play? Well, the answer is I should charge you my expected loss.

I have a 50% chance of losing, and if I lose, I lose a dollar.

So my expected loss is 50 cents, 50% of $1.

If we played this game an infinite number of times, neither of us would come out ahead half the time, You'd guess correctly, and I'd lose $1.

But I'd have collected 50 cents as the premium.

So my net loss is 50 cents.

The other half of the time you'd guess incorrectly, and I'd keep the 50 cents premium making a net profit of 50 cents on average over time, it would balance out.

When pricing an option, we're essentially doing the same thing.

We calculate the expected loss, and to do that we need to answer two key questions.

What is the probability that the option will be exercised? And if it is exercised, how much will a seller have to pay out? But while the coin toss is pretty simple, financial markets are far more complex.

For example, the future price of a stock can theoretically be anywhere between zero and infinity.

This makes calculating the expected loss and therefore the option premium in real life, not trivial.

To handle this complexity, we rely on option pricing models like the black skulls model, binomial trees, or Montecarlo simulations.

These models help answer those two questions using various assumptions about price movements and probability distributions.