1990 Toyota Celica Battery Discharges overnight

I just bought a 1990 Toyota Celica to save on my gas bill. My GMC truck has a 454 in it. Anyway the car seems to run good for having 224000 miles on it. It doesn't even smoke. The last couple of days the battery has been discharging overnight and I need to jump it in the morning. It will run fine and start even after a couple of hours. I'm not sure what it could be but I did notice that the Positive Cable on the battery stays fairly warm even after two hours of being shut down as if there is a load on it after the car is turned off. Not sure what would cause this and if changing the Alternater, regulater and battery all at once will fix it? The starter works fine also, but could that cause a draw on the batterey with the ignition off. Will just a bad battery stay warm? The problem is I've only had the car a week and don't know any of the history of it but it runs great. I hope someone can shed some light on this for me.

Thanks,
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