Proof of Put-Call Parity with/without dividend

1. Without Dividend

$\displaystyle C-P=S-d \cdot K$

Put-Call Parity is a relation between put price and call price shown above.

The meaning of alphabets are:

C : Call option's price (now)
P : Put option's price (now)
S : Stock price (now)
d : Discount factor. It takes value between 0~1, exclusive.
K : Strike price of call and put option (we assume that call option and put option have the same strike price)

Assume $S_{T}$ is a stock price at the end

Then create a portfolio which includes:

Then,

$\displaystyle \max (0, S_{T} - K)$

$\displaystyle - \max (0, K - S_{T}) + K$

$\displaystyle = S_{T}$

Here, the stock and the portfolio has the same payoff at the end.

Therefore, based on the no-arbitrage assumption, they should have the same cost.

Hence, $S = C - P + d \cdot K$

The equation we want to prove is:

$\displaystyle C-P+D =S - d \cdot K$

Where D is the dividend's present value.

Assume $S_{T}$ is a stock price at the end

Then create a portfolio which includes:

Then,

$\displaystyle \max (0, S_{T} - K)$

$\displaystyle - \max (0, K - S_{T}) + K$

$\displaystyle + \frac{D}{d}$

$\displaystyle = S_{T} + \frac{D}{d}$

Here, the stock and the portfolio has the same payoff at the end.

Therefore, based on the no-arbitrage assumption, they should have the same cost.

Hence, $\displaystyle S = C - P + D + d \cdot K$

$\displaystyle \iff C-P+D =S - d \cdot K$