# Put-Call parity arbitrage relationship

I would like to know what the relationship is between the time value of call/puts. From the put call parity formula

$$C-P = S_{t} - PV(K)$$

and that value of call/put options is simply the sum of the intrinsic and time values $$C=(S-K)^++TV_C$$ $$P=(K-S)^++TV_P$$

Does that then imply that the no-arbitrage relationship is that the time value of $$C$$, $$TV_C$$, is equal to the time value of $$P$$, $$TV_P$$?

• How are your time value defined? Note that, the conditional expectation is linear. May 14, 2020 at 15:00
• Yes, they are the same. it is easier to see if you put your equation in terms of forward value. May 14, 2020 at 15:27

Into the first equation we can substitute $$C$$ and $$P$$ as given by the other two equations, we get:

$$(S-K)^+ -(K-S)^+ +TV_C - TV_P = S-PV(K)$$

$$S-K+TV_C-TV_P=S-PV(K)$$

$$TV_C-TV_P=K-PV(K)$$

If interest rates are zero then $$PV(K)=K$$ and then we indeed have

$$TV_C=TV_P$$

Note: as suggested in the comments above a slightly different definition of Intrinsic Value and Time Value might be preferable here. If we define the "modified time values" in the following way

$$C=\underbrace{(S-PV(K))^+}_{IV^{'}_C}+TV^{'}_C$$

$$P=\underbrace{(PV(K)-S)^+}_{IV^{'}_P}+TV^{'}_P$$

then we would have equality of the (modified) time values for any level of interest rates:

$$TV^{'}_C=TV^{'}_P$$