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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

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  • $\begingroup$ To understand why -assuming zero dividends- the American call is not worth more than the European call you don't need a tree. It is almost a one-line-proof. Beware of the sign of the interest rates. $\endgroup$
    – Kurt G.
    Commented Feb 12 at 9:51
  • $\begingroup$ @KurtG. Thanks for your reply! I followed your answer and other proof. My main main confusion comes from the below: 1. When I was introduced to American call option as a beginner (no dividend), the pricing of it follows (2) above and one should exercise if $S(\omega) - K > \frac{1}{1+r} (PV_{n+1}(\omega H) + QV_{n+1}(\omega T))$, isn't this contradict that "American call is not worth more than European call" that worth $\frac{1}{1+r} (PV_{n+1}(\omega H) + QV_{n+1}(\omega T))$ at the time of exercise? $\endgroup$
    – TJT
    Commented Feb 14 at 7:24
  • $\begingroup$ @KurtG. 2. if I have a portfolio with an American call with strike $K$ and exactly $K$ cash at time t, I exercise it (in-the-money) and sell the stock at $S_t$. At maturity (time $T$), consider the stock price now is $S_T < K$, Wouldn't the exercised portfolio value $S_te^{r(T-t)}$ > $Ke^{r(T-t)}$? Please point out the mistake in my thinking. When I asked my lecturer, she basically told me to F-off, just study and remember the proof in Shreve's book, which really doesn't address my question as I want to know WHY/Where I am thinking this incorrectly? $\endgroup$
    – TJT
    Commented Feb 14 at 7:37
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    $\begingroup$ What has the exercise decision at $t$ to do with the stock price $S_T$ that the stock will later have at maturity? At $t$ you can at best base your decision on the conditional expectation. Your other questions I cannot answer because you have never explained the notation $PV,QV$ and so on. But as I say: we don't need tree notation to understand American options. Better to use conditional expectations. $\endgroup$
    – Kurt G.
    Commented Feb 14 at 8:06
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    $\begingroup$ Some progress. Why does your expression (2) confuse you? Nobel laureate Robert Merton has shown in the one-line-proof to which I linked above that when $r\ge 0$ and the stock pays no dividends that your (2) simplifies to (1). I do not want to give further explanations. You should spell out all the details for yourself to learn them. $\endgroup$
    – Kurt G.
    Commented Feb 14 at 10:53

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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!

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