American Call Pricing and Early Exercise for Non-Dividend Stocks

# How to and What is the price of an American call option for non-dividend stock?


# How to and What is the price of an American call option for non-dividend stock?












I want to know how to price an American call option for non-dividend stock (with concrete and simple binomial pricing model, with risk neutral assumption).

I understand that for an European call option, (in Binomial pricing model), the price is simply:

$$V_n(\omega) = \frac{1}{1+r} (PV_{n+1}(\omega H) + QV_{n+1}(\omega T) )\tag1$$

- $P,Q$ are risk neutral probability for the stock price to go up (denoted $H$) or down (denoted $T$)

- $\omega$ represent the current state of the stock price (i.e. what has happened from $t=0$ up until now $t=n$

- $V_{n+1}$ is the value of the corresponding European option at the next time step in a Binomial Pricing Model

From Early execise of American Call on Non-Dividend paying stock. and many other materials also state that early exercise of American call options is not optimal compared with selling it.

However, many materials also state or imply that European calls and American calls are identical when the underlying stock pays no dividends, which means (1) should apply right? (for anytime before maturity)

This confuses me as I thought the pricing of an American call option should be:

$$V_n(\omega) = \text{max} \Big( S(\omega) - K, \frac{1}{1+r} (PV_{n+1}(\omega H) + QV_{n+1}(\omega T)) \Big) \tag2$$

So let's say if I have a deep in-the-money American call option for non-dividend stock (not expired), what is the price for this option? Because since early exercise is not optimal, which add no time value for the American call option, why should one pay for the option according to equation (2) instead of (1)? Especially, when many seems to agree that European calls and American calls are identical when the underlying stock pays no dividends

## Answer by KT8 (score 0)

https://quant.stackexchange.com/a/78257

Your formulas seem correct, and what you have to do is to build the tree you have written in (2).

Now regarding the american vs european question, let me state a question: For a dividend paying underlying, what is the the advantage of an american call w.r.t its european counterpart?

... and there you have the answer. An american option gives you the right to exercise anytime. Lets say you are sitting at $t=0$, that the option matures at $T$ and that the stock pays a dividend at $0< t_d< T$. If the dividend wasn't there, keeping the option to expiry or selling right at the beginning is sort of equivalent, as in the european call option case., i.e. as you mentioned

> [...] exercise of American call options is not optimal [...]

But now let's think about the dividend, the american option allows to execute up to $t_d$ (after $t_d$ it will always be less convenient). Therefore, in comparison to the european option, the possibility to exercise in $t \in [0, t_d)$ makes the american option have a premium over the european one.

Hope this helps!