# 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

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)
• @P.Diaz For $rT<<1$, you have $e^{rT} \approx 1+rT$ (first order Taylor expansion), hence the RHS in the first formula is equivalent to $P−C+S−K(1−rT)$, which can seen as a discrete risk-free rate compounding version of $P−C+S−Ke^{−rT}$ – Quantuple May 1 '16 at 8:58