European Put–Call Parity with Continuous Dividends

Summary

The note explains how dividends affect put–call parity for European options. The relationship between a call and put with the same strike and expiry is expressed using the underlying asset’s forward price and the discount factor. Dividends change the forward price; higher dividends lower it, which in turn changes the relative prices of the call and put.

The proof starts with the payoff identity at expiry: the underlying price minus the strike equals the call payoff minus the put payoff. The answer then applies no-arbitrage reasoning: portfolios with identical expiry payoffs must have the same value beforehand, or buying the cheaper portfolio and selling the more expensive one would create an arbitrage. The note gives an intuitive proof rather than a book citation or a detailed treatment of continuous dividend yields. Its argument assumes European options and the ability to trade the relevant portfolios under no-arbitrage conditions.

Key ideas

• At expiry, the underlying asset minus the strike equals the call payoff minus the put payoff.
• European put–call parity relates option prices to the forward price and the discount factor.
• Dividends affect parity through their effect on the forward price.
• No-arbitrage pricing implies that portfolios with identical future payoffs must have equal current values.

Tags

# Put-Call Parity with dividends


# Put-Call Parity with dividends












In which book will I find the exact proof of put-call parity in the case when asset pays continuous dividend? I need a book to cite this result

## Answer by Soumirai (score 5, accepted)

https://quant.stackexchange.com/a/60289

Diviends or not, the put-call parity (with European options) always hold:

$ C(S,K) - P(S,K) = F - K*DF $

In the RHS, dividends will impact the forward $F$ (higher dividends imply lower forward). So the LHS should be lower as well: the Call costs less and the Put costs more.

The proof is straightforward, you just notice that at maturity $T$ you have:

$S_T - K = (S_T - K)\mathbb{I}_{S_T>K} + (S_T - K)\mathbb{I}_{S_T<K} \\= (S_T - K)\mathbb{I}_{S_T>K} - (K - S_T)\mathbb{I}_{S_T<K} = (S_T-K)^+ - (K-S_T)^+$

By non-arbitrage arguments, if this equality holds at $T$ it must hold at any time $t<T$ (i.e. the RHS and LHS portfolios must have same value at any time $t<T$). Otherwise you can construct an arbitrage by buying the cheap one and selling the expensive one.

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