Ultimate Guide to Put-Call Parity with Continuous Dividends

Put-call parity with continuous dividends is different from put-call parity without dividends. Dividends increase the value of the underlying asset. Thus, if you want to maintain put-call parity, you need to eliminate this effect by discounting the underlying asset’s price using the dividend yield.

To understand better put-call parity with continuous dividends, you first need to understand what is put-call parity, and what is put-call parity with discrete dividends (non-continuous).

What is Put-Call Parity

Put-call parity is a relationship that European calls and puts with the same characteristics (same underlying asset, strike price, and expiration date) obey.

Put-call parity states these two sums are equal:

1. Price of the call + the present value of a deposit worth the strike price.
2. Price of the put + the current price of the underlying asset.

To demonstrate this, consider both items above as portfolios 1 and 2:

	Today	Expiration: Stock Price < Strike Price	Expiration: Stock Price > Strike Price
Long Call	Pay Call Price (c)	Receive 0 (Call is OTM)	Receive S – K
Deposit Strike	Pay Strike (K)	Receive K	Receive K
Payoff 1	– c – K	+ K	+ S
Long Put	Pay Put Price (p)	Receive K – S	Receive 0 (Put is OTM)
Long Stock	Pay Stock Price (S)	S	S
Payoff 2	– p – S	+ K	+ S

This table uses simplified notations, as it doesn’t differentiate between the stock price today and tomorrow, and doesn’t take into account the interest on the deposit.

But it illustrates the main point:

Since the payoff in every scenario at expiration is the same, the components you need to construct the portfolios must cost the same as well. Otherwise, there’s an arbitrage opportunity.

Put–call parity does not hold for American options because you can exercise them before the expiration date.

Put-Call Parity Formula

Now that you know how does put-call parity work, let’s look at its formula:

\[c_0+Ke^{-rT}=p_0+S_0\]

c₀ is the price of the European call option.

Ke^–rT is the present value of a deposit worth the strike price and paying the risk-free interest rate (r). Deposit for how long? Until the expiration of the options (T). This means you deposit slightly less than the value of the strike price, so that at expiration you receive exactly the value of the strike price.

p₀ is the price of the European put option.

S₀ is the current market price of the underlying asset, which generally is a stock.

You can use this formula when the underlying asset does not pay dividends. If it does, things change:

Put-Call Parity with Dividend

There are two types of dividends you must take into consideration:

• Discrete Dividends: Paid once during the life of the option.
• Continuous Dividends: Instead of receiving cash, you receive additional fragmental units of the underlying asset.

Let’s understand how a discrete dividend affects put-call parity for now.

A discrete dividend increases the value of holding the underlying asset.

The S₀ we saw above is actually the dividend-adjusted price. The thing is that we were considering no dividend.

Now we’re considering there is one, so we need to remove it. Why? Because the left-hand side of the put-call parity equation does not benefit from a dividend, as opposed to the right-hand side as it includes the stock.

Assuming only one dividend is paid during the life of the option, the put-call parity with dividends formula is the following:

\[c_0+Ke^{-rT}=p_0+S_0-De^{-r\tau}\]

Where De^–rτ is the present value of a borrowing worth the discrete dividend discounted at the risk-free rate. Again, the goal is to borrow slightly less than the worth of the dividend, so that when the dividend is paid, you pay back the bank exactly the value of the dividend when accounting for interest.

To eliminate the effect of the dividend, you ask for a loan worth the dividend so that when the stock pays it, you use the money received to pay back the loan:

	Today	Dividend Payment Date (τ)	Expiration: Stock Price < Strike Price	Expiration: Stock Price > Strike Price
Long Call	Pay Call Price (c)	0	Receive 0 (Call is OTM)	Receive S – K
Deposit Strike	Pay Strike (K)	0	Receive K	Receive K
Payoff 1	– c – K	0	+ K	+ S
Long Put	Pay Put Price (p)	0	Receive K – S	Receive 0 (Put is OTM)
Long Stock	Pay Stock Price (S)	Receive Dividend	S	S
Borrow Dividend	Receive Dividend (D)	Pay Dividend	0	0
Payoff 2	– p – S + D	– D + D = 0	+ K	+ S

Again, payoff 1 and payoff 2 are equal, therefore they must cost the same. This means the put-call parity with a discrete dividend is guaranteed.

Now, what happens when you have a continuous dividend yield as opposed to a single dividend payment before the option’s expiration?

Put-Call Parity with Continuous Dividend Yield

When the underlying asset pays a continuous dividend yield (δ), instead of deducting the dividend from the underlying asset’s price, the only difference is that you need to buy just Se^−δT units of the underlying asset today in order to have 1 unit at the option’s expiration (T).

Therefore, the formula for the put-call parity with continuous dividends is:

\[c_0+Ke^{-rT}=p_0+S_0^{-\delta T}\]

This put-call parity with a dividend yield assumes you’re reinvesting the dividends in the underlying asset immediately after receiving them. It is as if you’re receiving additional fragmental units of the underlying asset.

This way, the option’s put-call parity when the underlying security pays dividends is maintained.

Put-Call Parity with Dividends FAQs