put option and interest rates

In the formula for the price of a call option, the final term (with 'e' in it) is discounting the exercise price at the interest rate over the period of the option. If interest rates increase then we are discounting at a higher rate. So the term with 'e' in it reduces and so the price of a call option increases. The price of call and put options move in opposite directions and so the price of a put option will increase.