Put–Call Parity and the Relationship Between Option Time Values

Summary

The note derives how call and put time values relate under put–call parity. Starting with the usual decomposition of each option into intrinsic value and time value, substitution into parity shows that the call’s time value exceeds the put’s by the strike price minus the present value of that strike. Thus, with zero interest rates, the two time values are equal; with nonzero rates, they generally are not under this intrinsic-value convention.

The answer also gives an alternative decomposition that discounts the strike when defining intrinsic value. Under that convention, the modified call and put time values are equal at any interest rate. This is an algebraic no-arbitrage implication, not a claim that the ordinary time values always match. The result depends on the definitions used and assumes the stated put–call parity setup; the short explanation does not discuss complications such as dividends or other contract details.

Key ideas

• Substituting standard intrinsic-value decompositions into put–call parity yields a time-value difference equal to the strike less its present value.
• With zero interest rates, the standard call and put time values are equal.
• With nonzero interest rates, standard time values generally differ under the stated definitions.
• Discounting the strike in the intrinsic-value definitions makes the modified call and put time values equal at any interest rate.

Tags

# Put-Call parity arbitrage relationship


# 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 $P$, $TV_P$?

## Answer by nbbo2 (score 3)

https://quant.stackexchange.com/a/54109

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.