Hull 09 18

[trabala38]

Hello David,

While doing Hull ex. 09 18, I encountered an issue:

In your solution (cf http://www.bionicturtle.com/wiki/Hull.09.18/), you state:

"P>=c + K*EXP[-rT] - S0
and since c = C,
P >= C + K*EXP[-rT] - S0, or
C - P >= S0 - K*EXP[-rT]"

I don't understand why c=C. For me, C>=c due to the fact that American options have more "optionality", hence, more value and thus, C>=c (cf. Hull Chapt 9, p212). The only thing that makes me think that C could equal c is the fact that an American call options with no dividend should never be exercised early. In this specific case (=no dividend), an American option is thus equal to an European option. Am I right, or is it something flawed in my "reasoning" ?

Also, please notice that the sign of the last equality is in the wrong direction : it should be C - P <= S0 - K*EXP(-rT).

You also use c=C in the second part of the proof, but here, given the direction of the inequality c + K >= P + S0 => C + K >= P + So, so it is not a problem (both C=c or C>=c keeps the inequality true).

Thanks in advance !

Best regards,

trabala38

 

[David Harper CFA FRM]

Hi trabala38,

This is Hull's question and answer, FWIW (that's why we label it "Hull") ... and, honestly, i never fully understood the step that assumes C = c. First, this does assume non-dividend-paying stock because it refers to equation 9.4 which is put-call parity without dividends. Second, as far as I can tell, he may be leaning on his argument that:

• It is never optimal to early exercise an an American call option on a non-dividend paying stock. This is the best I can do to find his justification for c = C, but I totally agree that it does not match C >= c, and
• It can be optimal to early exercise an American put option a non-dividend paying stock, justifying P >= p

With respect to: P >= C + K*EXP[-rT] - S0, I get:
P >= C + K*EXP[-rT] - S0,
--> P - K*EXP[-rT] >= C - S0,
--> P - K*EXP[-rT] + S0 >= C,
--> - K*EXP[-rT] + S0 >= C - P,
--> S0 - K*EXP[-rT] >= C - P, so I agree with you, that looks like a typo. Nice catch.

Thanks, David

 

[trabala38]

OK,

Thanks, very clear. We are on the same page ;-)

Regards,

trabala38