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The intrinsic value of a call option is found by subtracting the discounted strike price from the current share price:

$IV = S - X/e^{rT}$

Put-Call parity:

$S + p = c + X/e^{rT}$

$c = p + (S - X/e^{rT})$

Since the second term is literally the definition of the intrinsic value of a call, should the time value of the call option $(c-IV)$ be equal to the value of a put with the same strike price???

What is the economic intuition?

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    $\begingroup$ Your definition of $IV$ in somewhat non standard, usually it is $IV=\max(0,S-X)$. No discounting, and a restriction to non-negative values. With your revised definition the result you state is correct, I believe. $\endgroup$
    – nbbo2
    Nov 23, 2023 at 15:51

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Mathematically (not economically) speaking, an ITM (OTM) call option should be worth more (less) than the OTM (ITM) put option by a value equal to your "IV" (spot minus strike). Also, IV in quant finance is more commonly used to abbreviate implied volatility IMO.

Edit. My point is that it is not possible for a call and put with same option characteristics to be both ITM or OTM, they must differ by an amount, which is the "IV" in this case.

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