Why when interest rates go up, put price goes down?

Not clear why. With calls, it makes sense. If interest rates go up, you have paid a small amount for the call, and are investing the remaining funds to earn on the high interest. It seems the same thing happens with puts. You pay a small amount to buy the put, deposit the remaining funds. At expiration, you buy the stock and exercise the put. What am I missing?

I always look at all of these questions with Put-Call Parity in the eyes of the synthetic instruments P=C+X/(1+R)^T-S C, X, S remain constant R goes up X/(1+R) goes down… so P goes down.

if r goes up, c also increases so that argument isnt necessarily valid think of the binomial model when r increases the uptick probability is higher and therefore the downtick prob is lower if i am long a put, high probabilty of a downward price move is in my favour so i want high down probability, hence a lower r, so i am willing to pay more for a put if r drops, and conversely, if r goes up i am willing to pay less for a put

A Putable bond is issued at a lower coupon then a non-callable bond. So if interest rates go up, while the Put gains in value, the high interest rate pushes it downwards ( Inv relation between Interest & Price). So, in effect, the putable bond might lose in value. Just my 2 cents, I haven’t checked the book yet for this answer. Just applied some logic here.

I think the put-call parity relation in this question will hold. Correct me if am wrong.

Thanks everyone…using the put-call parity is a good way of seeing how that holds…I overlooked that, but I still would like to see this at some intuitive level, like with calls: if interest rates go up, and since you have paid a small amount for the call, you are investing the remaining funds to earn on the high interest… can’t seem to see it that way with a put.

rate up = forward price of the asset up rate up = long forward position value up long call ~ long forward long put ~ short forward rate up = short forward position value down

That’s also a good way of seeing the effect of a rate rise on a forward contract because F(0,1 year) = S0 (1+r)^t - CF (1+r)^t, so when r goes up, first term goes up by a larger amount than the second term (both go up). But with a put option, you pay say $2000 for the put, and you deposit $98k at the high rate (assuming you have $100k). At expiration, because rates are up, you still earn more on the $98k. So, why does the put price go down…this is what’s confusing from an intuitive standpoint.