Put-Call Parity for Estimating Present Value of Dividends

Summary

The document reconciles two expressions for estimating the present value of dividends from option prices. It states European put-call parity in terms of the spot price, strike, discount factor, call price, and put price, then rearranges the relationship to isolate the net dividends’ present value. The explanation derives parity from the difference between call and put payoffs at expiry and a discounted risk-neutral expectation.

The answer also expresses parity through the fair forward price, with that price adjusted for dividends. Its main caveat is contract style: the stated relationship holds for European options. For American options, early exercise means the equality does not strictly apply, though the answer describes it as an approximation near at-the-money strikes. The document provides an algebraic explanation but no market data or worked numerical example.

Key ideas

• European put-call parity relates call and put prices to spot, strike, discounting, and dividends.
• Rearranging parity isolates the present value of net dividends.
• The relationship follows by discounting the difference between call and put payoffs under risk-neutral valuation.
• American options do not strictly satisfy the same equality because of early exercise.

Tags

# Is this representation of the put-call parity correct? (Implied dividend estimation)


# Is this representation of the put-call parity correct? (Implied dividend estimation)












I am looking at implied dividend yields to be obtained from the put-call parity and have come across the following answer:

Implied dividend estimation

It states that $$ PV(div) = P - C + (S - K) + K(e^{rT} - 1), $$ however the put-call parity as I know it states

$$ C - P + PV(div) = S-K(e^{-rT}) $$

I have looked at it for a while and cannot match these two expressions. Do you have any input in what I might be missing?

Thanks, Diaz

## Answer by Quantuple (score 2, accepted)

https://quant.stackexchange.com/a/25722

Call-put parity writes (to see this, notice that $(S_T-K)^+ - (K-S_T)^+ = S_T - K $ and take the discounted risk-neutral expectation $E^{\mathbb {Q}} [. \vert \mathcal {F}_0 ]$ on both sides): $$ C(K,T) - P(K,T) = DF ( F(0,T) - K ) $$ with $DF = e^{-rT} $ the discount factor, and $F(0,T)$ the fair forward price given by $$ F(0,T) = (S_0 - D^*)e^{rT} $$ with $D^*$ the net dividends' present value and $S_0$ the spot price. So indeed $$ D^* = S_0 - Ke^{-rT} - C(K,T) + P(K,T) $$ Careful though that this relationship only holds for European options (for American options this does not strictly hold, although close to atm it is not a bad approximation)

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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