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
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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.
