What is the difference between the following two sentences?
- I have had a headache since this morning.
- I am having a headache since this morning.
I've had a headache since
past.time.point is normal, idiomatic, and grammatical.
*I'm having a headache since
past.time.point, however, is ungrammatical,
because the present progressive construction ('m having) refers to the present moment,
while the prepositional phrase (since
past.time.point) refers to a length of time in the past,
past.time.point and continuing to the present,
rather than directly to
present.time.point, which is what the present progressive needs. Thus,
I'm having a headache right now/today/at the moment are all OK, for instance.
As Barrie points out, today morning is not grammatical in British or American English;
the idiom there is this morning instead.
Neither is grammatical in British Standard English, where what you want is I've had a headache since this morning. However, I believe today morning is found in Indian English.