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I am reading through Hull's book asking myself this question to understand exotics.

I currently believe that theta should equal 0 until the forward start time, $t_*$, if the call pays no dividends. This is because the call's value would be defined as:

$C = S_0e^{-q(t_*-t_0)} (e^{-q(t_f-t_*)}N(d_1)-e^{-r(t_f-t_*)}N(d_2))$

where $t_f$ = expiration date of option, $t_0$ = time of valuation, and $S_0$ = current spot price. So $t_0 < t_* < t_f$. If the call pays no dividends, then $q=0$ and $e^{-q(t_*-t_0)} = 1$, thus the change in the call price w.r.t. $t_0$ is 0. Is my understanding correct?

Now, regarding vega, I'm not sure how a change in vega would affect the price of the forward starting call option above. Can someone help me go through the steps to understand this?

Thank you.

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