There is a question that says a value of a call option will increase if risk-free rate goes up. Why?

My understanding was that when risk-free rates go up, the inverse relationship with the prices indicate that the prices will go down. If so, the call option expires worthless and the value goes down, not up. What am I missing here?

If the risk-free rate increases then the present value of the strike price (X) decreases; to maintain equality, either the price of a call option must increase or the price of a put must decrease.

Be careful that you are not misunderstanding the relationship of calls to bonds with embedded options and interest rates. where did you read this?

In an embedded option, yes if rates rise, the price of the bond falls, making the call less valuable (negative relationship)

If you take a call on an interest rate , then if the interest rate rises, you can call and lock in a lower rate therefore there is a positive relationship.

From another perspective: If you hold a call option on a stock and the risk-free rate increases, the present value of the strike price you pay to buy the stock will decrease, which is a benefit, thus the value of the call option increases.

Put option vice versa, the present value of the strike price you receive decreases, lower option value.

Thanks for the comment. I’m going to reply to the third comment which is contrary what you said. It makes sense whether you buy a call on the rate vs. the price itself.

Are you implying short-position on the call option? Then it makes sense. However, if you are long-position on the call option, then the value expires worthless. This is correct??