Regarding your particular sentence, native speakers will usually make a definite reference by using a definite noun phrase; this is based purely on the sentence structure.
I am currently reading the CNN article "Australia's top stock exchange is having an awful Monday".
But this does not mean that other, very similar sentences cannot use an indefinite noun phrase, even with the name of the specific news article provided in the sentence.
One reason you might be confused is that perhaps you have fallen into the common error (made by both native and non-native speakers of English) of thinking that an indefinite noun phrase cannot be specific. However this is not true.
An indefinite noun phrase can be either specific or nonspecific.
In I want to marry a woman from Ohio (and I don't care which one) the noun phrase 'a woman from Ohio' is not specific. It refers to any woman from Ohio.
In I married a woman from Ohio (and her name is Katie), the noun phrase 'a woman from Ohio' is specific. It refers to the specific woman from Ohio that the speaker married.
Thus, even if you are referring to a specific news article on CNN, you can use either a or the. In fact, if you compose it correctly, the sentence can be exactly the same except for the article (a or the). The following sentence differs from the sentence in your question by one word (the inclusion of called):
I am currently reading a/the CNN article called "Australia's top stock exchange is having an awful Monday".
Sometimes a specific noun phrase can use either a or the. And I rarely use bold for entire sentences, but this bears making a point very strongly: The English articles are not called the specific and 'inspecific' articles, but the definite and indefinite articles. Thus, the difference in using an indefinite noun phrase and a definite noun phrase in the sentence above is not that of specificity (being specific or not specific) but of definiteness (being definite or indefinite).
Finally, as I said above, an indefinite noun phrase can make a nonspecific reference:
I would like to read a CNN article about how Australia's top stock exchange is having an awful Monday. I don't care which one it is. Can you recommend one to me?