Call vs. Put Option

I have two interrelated questions that have been bothering me for some time. I have read all the stuff online and it still doesn't make sense to me:

Let us assume:

• 0% interest rate (both hedge funding and discounting rate)


• 0% dividends

Hence, Forward=Spot.

1. 
   

   Why is the price of a ATM put equal to the price of a ATM call? What is happening to the log-normal distribution here, shouldn't the call be more expensive? (two 50% increases lead to a greater payoff(225% Spot) than 2 50% decreases (25% Spot).

   
   



2. 
   

   Very related.. why is then a 110% Call Option worth more than a 90% Put option (under the same conditions as above)?