How does the put option values fall and rise while call options values rise and fall as the rerlevant stock prices rises?

The Payoff i.e. profit for a Call Option is St-X where St is the market price at time t and X is the exercise price. Assuming that it is an American Style option where it can be exercised at any time, If St is significantly greater than the exercise price,X, (the agreed price to buy an option at) then if the option holder exercises it immediately they will be 'in-the-money.' This means it has a high intrinsic value which causes a rise in value for the option.

The Payoff for a Put Option is X-St where X=exercise price and St equals market price at time t. If the market price increases the gap between X and St (Payoff or Profit) reduces or if X<St then they will be making a loss. This will mean it will have a low intrinsic value (value if exercised immediately) therefore the value of the option will fall.