From Kaplan:

“An increase in the risk-free rate increases the values of call options on equities and decreases the values of put options on equities.”

My assumption was if interest rates went up (lowers equity values), it would decrease the value of a call option; hypothetically due to some stocks dropping to the point of reversing a call option from in the money to out of the money.

Help?

They’re talking about put-call parity. They’re assuming that the stock price is unchanged.

Note, too, that this doesn’t have anything to do with the volatility of interest rates, only the level of interest rates.

May I ask for your concise and easy to understand brief description of the put-call parity? It obviously rings a bell, but I have too many bells ringing in my head as of lately.

stock plus put equals call plus bond

S_0 + p_0 = c_0 + \dfrac{X}{\left(1 + r\right)^T}

A protective put has the same payoff as a call plus a bond (whose par value is the strike price of the options).