Call-put parity.

a) Show that the relation \((x-K)^{+}=x-K+(K-x)^{+}\)holds for any \(K, x \in \mathbb{R}\).

b) From part (a), find a relation between the prices of call and put options with strike price \(K>0\) and maturity \(N \geqslant 1\) in a market with risk-free rate \(r>0\).

Hints:
i) Recall that an option with payoff \(\phi\left(S_{N}ight)\) and maturity \(N \geqslant 1\) is priced at times \(k=0,1, \ldots, N\) as \((1+r)^{-(N-k)} \mathbb{E}^{*}\left[\phi\left(S_{N}ight) \mid \mathcal{F}_{k}ight]\) under the risk-neutral measure \(\mathbb{P}^{*}\).
ii) The payoff at maturity of a European call (resp. put) option with strike price \(K\) is \(\left(S_{N}-Kight)^{+}\), resp. \(\left(K-S_{N}ight)^{+}\).