Yes, your sentence needs the preposition 'at'. The adverbial phrase 'at home' consists of the preposition 'at' and the noun 'home'. This phrase is similar to 'at work' , 'in the office' or 'in my house'.
You are right that 'home' can often be used without a preposition, but this is only when it is functioning as an adverb of place/direction in its own right - like 'away', 'abroad', 'up', 'down', 'back', 'over', 'here' or 'there'. We use 'home' as an adverb with the verb 'to be' : "I am home", similar to "He is away" or "She is here"; we also use it in combination with verbs of movement: "I'm driving home" "She's walking home" "When are you coming home?".
I hope that helps.