Difference between "have had" and "having"

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.

Solution 1:

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,
starting at 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.

Solution 2:

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.