In LOS63.o they ask you to explain the effect of an interest rate change on the option’s price.
In the Schweser notes, they explain it via rearranging the put call parity formula, which clearly shows interest up -> call price up or put price down.
The reasoning for the call price increase is IMO that when you expect an increase in a stock, it’s cheaper to buy the call option and invest the difference between the call price and the stock price against a higher interest rate. Hence it becomes more attractive to buy a call and the price goes up.
Now, I can’t really see the logic behind the price decrease for the put option. Does it become more attractive to buy the underlying?
So a call is effectively a deferred purchase. Look at it as an alternative to owning and carrying the stock: when rates are high i would prefer to buy the call because buying the stock and carrying it would be dear. Likewise, a put is a deferred sale, as an alternative i could short the stock and receive interest on the cash balance (hold the borrow fees etc. constant). So if rates rise, then it becomes relatively cheaper to sell the stock and recieve slightly higher interest rather than purchasing the put, so accordingly the put is priced lower for a rise in rates.
Another way of thinking about it is that when we model it, the underlying is really the forward price (apologies, this is not a precise way of expressing it) hence S^erT. If r rises then the forward price is going to be higher, so the call will be higher, the put lower. That’s a shabby way of expressing it, but i like it.
In option pricing the interest rate has 2 effects - on the forward price, and on the cost of carrying the option premium. I’m not sure if the syllabus covers it, but futures options with stock-type settlement (most US FOPs) the relationship should be different: increased rates should cause both calls and puts to be lower.