Black-Scholes for American Call Options: Why It Works

Imagine you’re holding a ticket to a fantastic concert, let’s say for your favorite band. This ticket gives you the option to attend the concert. Now, there are two main types of these concert tickets, think of them like this: one is a ‘European-style’ ticket and the other is an ‘American-style’ ticket.

The European-style ticket is a bit strict. It says you can only decide if you’re going to the concert, and therefore use the ticket, on the exact day of the concert itself. You can’t use it before, and you can’t use it after. It’s use it on concert day, or lose it.

The American-style ticket is much more flexible. It says you can decide to go to the concert and use your ticket anytime between now and the day of the concert itself. You have a window of opportunity to use it.

Now, let’s talk about pricing these tickets fairly. Suppose there’s a famous formula, let’s call it the ‘Concert Ticket Pricing Formula’, which was originally designed to figure out the fair price for those strict European-style tickets. This formula, in the world of finance, is similar to the Black-Scholes formula. It’s a mathematical way to estimate what a European option is worth.

European options, like our European tickets, can only be exercised at a specific date in the future. American options, like our American tickets, can be exercised at any time up to a specific date. The Black-Scholes formula was technically created for European options. So, you might wonder, how can we use this formula for American options, especially when they seem so much more flexible and valuable because you can use them earlier?

The fascinating thing is that for a specific type of American option, namely an American call option on a stock that doesn’t pay dividends, the Black-Scholes formula works surprisingly well, even though it was designed for European options. Why is this the case?

The key lies in understanding when it actually makes sense to use that extra flexibility of the American-style ticket, or in financial terms, when it makes sense to exercise an American call option early.

Think about our concert ticket again. If you have an American-style ticket, when would you actually choose to use it before the concert day? Perhaps if something really amazing came up and you absolutely couldn’t go to the concert anymore, and you found someone willing to buy your ticket right now. But generally, you’d probably hold onto it until closer to the concert date, hoping the excitement builds up and maybe the ticket becomes even more valuable.

Now consider a stock that doesn’t pay dividends. Dividends are like little cash payouts companies sometimes give to their stockholders. If a stock does pay dividends, and you hold an American call option on it, there might be a reason to exercise it early. Why? Because if you exercise early, you get to own the stock and receive those upcoming dividend payments. You wouldn’t get the dividends if you just held onto the option.

However, if the stock doesn’t pay dividends, there’s usually no good financial reason to exercise an American call option early. Think about it. An option’s value comes from two main sources: its intrinsic value and its time value. The intrinsic value is the immediate profit you’d make if you exercised the option right now. The time value is the added value because there’s still time left until the option expires, and the stock price could potentially move even more in your favor.

When you exercise an American call option early on a non-dividend-paying stock, you give up the time value. You are essentially trading the potential for future gains for immediate ownership of the stock. But because the stock doesn’t pay dividends, you aren’t gaining any extra cash flow by owning the stock right now compared to just holding the option. In fact, you might be losing out because the option itself might still increase in value if the stock price goes up further.

Because there’s typically no financial advantage to exercising an American call option early on a non-dividend-paying stock, its behavior, in terms of when it’s optimal to exercise, starts to look a lot like a European option. You’re essentially going to wait until the very end, or close to it, to decide whether to exercise.

This is why the Black-Scholes formula, even though technically designed for European options, can be a reasonable approximation for valuing American call options on non-dividend-paying stocks. The underlying principle is that in the absence of dividends, the early exercise feature of the American call option becomes less relevant in practice. The optimal strategy for exercising an American call option in this specific scenario closely mirrors the exercise strategy of a European call option, which is just at expiration. Therefore, the pricing model designed for the latter can be applied, with reasonable accuracy, to the former. It’s a useful simplification that works because of the specific conditions we are considering: American call options and non-dividend-paying stocks.