I understand that a lower exercise price and increased underlying asset/risk free rate/volatility and time will increase a call option.
I am just wondering for a put option is it the opposite for each factor in terms of increasing the put option? Other than it would also want a low exercise price?
There is no reason why a put option would necessarily have a low price. What determines the price for the option is the exercise price and the likelihood of the actual price ending up below the exercise price. (Just as the price of a call option is determined by the exercise price and the likelihood of the actual price ending up above the exercise price)