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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 $P$, $TV_P$?

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

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

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