Fast method of pricing an American call option for a non-dividend paying stock

[sridhar]

David:

Please clarify something. I came across a problem asking to price an American call option for a non-dividend stock using a 2-period model. Since this is non-dividend, we can use the exact same procedure as if we are valuing a European call option, correct?

Since it is never optimal to EARLY-exercise such an American call, we need not worry about comparing the payoff at the end of Period 1 vs the expected value of the call at the end of period 1. I understand we'd do this for an American put -- dividend or no dividend.

So, can I just charge ahead with the method for pricing a European call in such a case?

--sridhar

 

[David Harper CFA FRM]

Hi sridhar,

Yes, it is still worth checking at the node. No you shouldn't charge ahead ; Hull has examples of where the intrinsic value > expected PV and the exercise would be justified. Therefore, it does make a difference in some cases (in the upper branches of the tree, occasionally). I realize it seems inconsistent with the no exercise rule but that rule is not universally true; it is conditional on holding the stock. Whereas the nodes are simulating immediate (cashless) exercises: exercise then sell the stock immediately to pocket the gain. The binomial tree method is (IMO) quite sound; I prefer to treat the other don't-exercise-Americian-early as somewhat conditional and highly situational (i.e., not particularly useful, nobody acts this way)...it is really only saying, if i know at maturity i will be holding the stock (either way), I may as well defer the non-inflating strike price which is always cheaper to pay in the future. Note the slight difference from a maximize-current-value rule. David

 

[sridhar]

David:

I am a little puzzled at your answer. Also puzzled at your "not particularly useful, nobody acts this way" comment.

Hull gives a detailed explanation for why an American call option (on non-div stock) is never optimal to exercise. If I buy a CISCO Jan 2010 call at a strike of 20, I can safely not exercise it until Jan 2010 -- if the price of CISCO goes above the 20 BEFORE Jan 2010, I can always choose to SELL the call, instead of exercising it. There is no benefit (from a payoff standpoint) to actually exercise it.

In fact, if I do exercise it say, on July 1, 2009 -- then if on Jan 2010 (on the expiry date), if the CISCO stock is at 15, I am holding CISCO shares with a $5/share loss. On the other hand, if I don't exercise it, on the day of expirty in Jan 2010, I can either let it expire worthless or (for some reason), actually exercise it, buy it for $20/share and nurse a $5 loss. I don't think anyone would actually exercise it. I would rather let it expire and buy shares at $15.

It seems to me that this is one case, where the mathematical rationale and the practical rationale are in sync. My question was not meant to be theoretical. It is more of a FRM exam time-saving tactic

--sridhar

 

[sridhar]

In my previous post, when I said "I can always SELL the call", I meant I can always SELL back the call i bought. In fact my discount broker allows a "SELL to CLOSE" option. Closing off my long call.

 

[David Harper CFA FRM]

Sridhar - But it's not useful as an time-saving tactic b/c there are possible trees where your shortcut will give incorrect answer. I'd like it *if* intrinsic always < PV of expected, then it would be a great shortcut. Regarding the "nobody act that way," I just meant that study after study after study shows that executive/employee option holders exercise way before maturity (on average 1 to 2 years after vesting). Because that no-exercise rule implicity does not consider risk aversion (and for that matter, liquidity). If people are risk averse and they "prefer a bird in the hand today" (as they often do), then the immediate cashless exercise is more realistic (90%+ are cashless exercise). So i don't disagree with your analysis above, i just meant that is the behavior that (i) makes assumptions about risk aversion and (ii) ignores liquidity preference (cash today versus uncertain gain in the future). David

 

[David Harper CFA FRM]

...let my try your patience another way. Your analysis above makes more sense to me "in the middle" of the binomial tree, where the intrinsic value will be less than exp PV anyway. But this shortcut is only really violated in the upper nodes of the tree (e.g., up then up then up), and here, the intuition behind max (intrinsic, expected PV) I find quite appealing over the theoretical "don't early exercise:" if as the option holder you think "right now i am in the money, and i can exercise for this gain, i will do this opportunistically because i can avoid the risk of the stock going down in the future. The more that an individual is risk-averse (i.e., certain gain now over uncertain future even if greater) and the more he/she prefers liquidity, the more he/she will want to seize on the current intrinsic value. David