Let's explore the difference between the premium of European options and otherwise identical American options.

Intuitively, American options should be more expensive given their greater flexibility around exercise.

But is this always the case? Let's take a closer look.

Conceptually, we can think of an American option as two components and otherwise identical European option.

Since the American option can be exercised at maturity like the European option, plus the option to exercise this European option early, therefore, the premium of an American option should be the sum of the premium of the European option and an additional amount that reflects the value of the option to exercise early.

This means that the price differential between otherwise identical American and European options depends on the value of this early exercise feature.

So far so good, but what's the practical relevance of this? Well, let's start with outta the money or OTM options since outta the money options wouldn't be exercised under current market conditions.

The early exercise feature doesn't provide any additional value.

After all, exercising, an option that's outta the money would simply lock in a loss, which no rational investor would do.

So in these cases, the premium of the American option is essentially the same as that of the European option because the added flexibility isn't worth anything.

But what about in the money or ITM options? Does the ability to exercise early always have significant value? Here? The answer is not necessarily exercising an American option. Early generally means realizing its current intrinsic value.

However, it also means giving up any remaining time value.

The option still holds.

In many cases, selling the option onto someone else is the optimal decision because the market price of the option, the premium reflects both intrinsic and time value.

So selling the option allows the holder to capture both components while early exercise only captures intrinsic value.

This trade off often diminishes the The actual value of the early exercise feature.

And that's why in many cases, the premium of an American option is not significantly higher than that of its European counterparts.

However, there are situations where the rights to exercise early can have meaningful value.

For example, consider an American put option in a high interest rate environment.

Early exercise allows the holder to sell the underlying asset sooner and invest the proceeds at the prevailing high interest rates.

In this case, the financial benefits from reinvesting the cash can outweigh the lost time value making early exercise, and therefore the early exercise option valuable.

Another common scenario involves American call options on dividend paying stocks just before the X dividend date.

It might be optimal to exercise the call early to capture the dividend payments.

While this does mean forfeiting any remaining time value, the dividend received can more than compensate for that loss making early exercise, the better choice to sum up the premium of an American option is always at least as high as the premium of an otherwise identical European option because of the added flexibility.

But in many cases, the difference isn't substantial.

It's only under specific conditions that the value of early exercise and thus the premium difference becomes meaningful.