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# Option questions

#### ajsa

##### New Member
Hi David,

1. could you explain why the difference between an American call and an American put (C–P) is bounded by the inequality on p86:
S - K <= C - P <= S - K*exp(-rT) ?
Also does this inequality imply C>=P for American option?

2. is it true that An American option on a dividend paying stock may be worth less than its European analogue?

3, on p87, "An American option should never be exercised early in the absence of dividends. In the case of a
dividend-paying stock, it would only be optimal to exercise immediately before the stock goes exdividend.
Specifically, early exercise would remain sub-optimal if the following inequality applied:"
Does this paragraph only refer to call, but not put?

4. Could you also pls explain the dividend effects on lower/upper bounds of European and American put/call options?

Many thanks.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi asja,

(sorry for delay...was immersed in study notes for valuation)

1. I frankly would not have an intuition for this. I needed to find that Hull has a question on this specifically (9.18) which I just added to the forum. Please see blog post here and derivation here (please note: someone asked about this last year and I could not figure it out then or now on my own and I am still puzzled Hull's assumptiont in the middle that c=C. Probably I am just not getting it...)
1a. Yes, for either Euro or American, both c-p and C-P (assuming same strike and maturity on same underlying!) must be > 0.

2. Re: "is it true that An American option on a dividend paying stock may be worth less than its European analogue?"
No, I do not think this is true for either call or put. C >= c and P >= p. Do our notes say this? I don't see where? I see we have:
"All other things being equal, the value of an Amerian style option must be at least as great as a European option with the same features; Value [American option] >= Value [European option]"

3. Yes, this refers to a call option (I see the notes do not specify, I have noted the imprecision....)
...because the put has a less restrictive condition: it may be optimal to exercise an American put prior to expiration on a non-dividend paying stock. This condition continue to apply to a put on dividend paying stocks. (and, as the idea of "immediately before the stock goes ex-dividend" assumes the stock price will drop by the dividend, this "urgency" applies to the holder of the call option not the holder of a put who would *benefit* from a stock drop)

4. In the case of a call, the lower bound for Euro call, changes from c >= S-K*EXP[-rT] to c>=S-D-K*EXP[-rT], where D is PV of dividends (Hull 9.5)
Similarly, for Euro put, p >= D+K*EXP[-rT] (Hull 9.6)
Intuitively, i imagine dividends the same as in Black-Scholes: they effectively reduce the stock price...
you can think of total return = capital appreciation (ie.., price gain) + dividends
Then, assume say total appreciation = 10%: if no dividends, call holder enjoys all the benefit of the total return,
but if dividends= 3%, then call option holder forgoes dividends and enjoys benefit of only 7% appreciation.
In a loose sort of way, similarly here, the call holder is "getting reduced" for the dividends *because* he/she forgoes them (unlike the owner of the underlying)

Hope this helps...David