My spellcheck won't recognise it and if there is indeed no such word, what should I use instead? I'm writing an article which contains the sentence "it takes a special kind of positivity to..." etc etc.
Spellcheck programs aren't dictionaries. Always check with a dictionary if you are uncertain about the spellcheck's query of a word. Positivity is fine, although the sentence you're writing does seem to be a bit - I'm sorry new agey bollocks. Is there not some way to reword it? "It takes an especially positive outlook to ..." or something. Your wording means there are many different kinds of positivity, but only one of them is suitable for whatever is that follows "to".