# Why is the correlation that Futures Options have with interest rates opposite to that of the one Stock Options have?

The Put-Call Parity relationship for Stock Options is the following:

Call Price - Put Price = Stock Price - Exercise Price + Carrying Costs - Dividends

But for Options on Futures where the Options have Stock-type (cash) settlement as is true in the U.S. futures markets:

Call Price - Put Price = Futures Price - Exercise Price - Carrying Costs

For the Put-Call Parity relationship of Stock Options, it both intuitively and mathematically makes sense. Because Call prices increase (decrease) when interest rates increase (decrease), and Put prices decrease (increase) when interest rates increase (decrease).

You can re-arrange the relationship of the Stock Options Put-Call Parity relationship and prove the following for example:

Call Price = Stock Price - Exercise Price + Put Price + Carrying Costs - Dividends

As assumed previously, the carrying costs (risk-free rate usually) gets added to the Call price and thus increases it, while dividends decrease it. This intuitively makes sense.

By buying a Call you avoid paying carrying costs that you would if you were long stock (basically interest on the notional value of the stock), so if you want to avoid that by buying a Call (replicating what a long stock position would do) you need to pay MORE for that call, this is a reflection of you not having to carry the stock until expiration.

And in the case of Puts, cost of carry is something you FORGO when you use puts (puts replace selling or selling short). By using Puts you are neglecting to earn the cost of carry you would get by selling or selling short (getting cash into your account which would grow at the risk-free interest rate). This is why Puts are CHEAPER. You leave money on the table (interest earned) if you use a Put to replicate a straight sale.

And this also makes more sense when the relationship is placed as: $$C_0+K*e^{-rt} = P_0 + S_0$$

The Call already has interest rate value added to it under Black-Scholes, but the Put has the interest rate value and or carrying costs subtracted from it. So when you add the Call and the Strike together, it should be the Present Value of the Strike, which is discounted. That's why the Stock + Put is equal to Call + PV(Strike).

So my question(s) are how come for the Put-Call Parity relationship of Options on Futures, carrying costs are being subtracted instead of added like they are for normal Stock Options? And how come an increase (decrease) in rates causes Calls on Futures to decrease (increase) while Puts on Future to increase (decrease)?

Thank you.