FRM 2008 Practice PII question 5 - Option with dividend paying stock

[fullofquestions]

QUESTION
Which of the following factors will not necessarily increase the price of a European call option on a
dividend paying stock as this factor increases in value?
a. The risk free rate.
b. The stock price.
c. The time to expiration.
d. The volatility of the stock price.
Answer: c
a. Incorrect. An increase in the risk free rate will decrease PV(X) and necessarily increase the price
of the European call.
b. Incorrect. An increase in the stock price will necessarily increase the price of the European call.
c. Correct. Because dividends paid before the expiration of the option might decrease the value of
the stock price, it is possible that the value of the call option will decrease as the time to expiration
is increased passed scheduled dividend payout dates.
d. Incorrect. An increase in the underlying stock price will necessarily increase the price of the
European call.

Basing my thoughts on the formula: European Call = Max[S - Ke^-(rt), 0]
An increase in volatility could just as easily translate in a decrease in stock price therefore increasing the value of the option so dismissing choice d is very tenuous. In any event, increasing time to expiration directly affects the discount function so that Ke^-(rt) is smaller, therefore making the value of the European Call larger. I get that the time to expiration can be increased to such a point where a dividend is paid out on the stock so that the option decreases in value. In my opinion, this explanation can just as easily be countered by saying that increasing the volatility of the stock price may result in the stock price to decrease, therefore, reducing the price of the option. Any thoughts?

 

[David Harper CFA FRM]

Hi FoQ,

I think this question is wrong (caveat: my first impression only, okay, I would like to research): theta is always negative for Euro call options unless (I think) you have a currency option with sufficiently high FX rate such that your dividend yield effectively exceeds your riskless rate...which will not happen for a call option on a *stock*. Ergo, I think increase in all of these imply increases in Euro call option on dividend paying stock. Put simply, Euro call option on stock:

rho is always positive
delta is always positive
theta is always negative (ie. to increase time is to at least increase value)
vega is always positive

i.e., there is no answer to the question ... requires dividend to exceed rf rate which is *unfair* requirement

Re: "An increase in volatility could just as easily translate in a decrease in stock price"
This is very tempting...and you could be right about the cause-effect here ... but this concerns the possible impact of one input on another: the BSM does not handle such complexity...it is very temping to introduce outside/empirical logic and observations... but the question is about the contents of the model; i.e., value = f[six inputs] and the fact that the model is unrealistic is sort of beside the point..this is about the Greeks, you want to focus on the Greeks...

David