Consider an asset price \(\left(S_{n}ight)_{n=0,1, \ldots, N}\) which is a martingale under the riskneutral probability measure \(\mathbb{P}^{*}\), with respect to the filtration \(\left(\mathcal{F}_{n}ight)_{n=0,1, \ldots, N}\). Given the (convex) function \(\phi(x):=(x-K)^{+}\), show that the price of an Asian option with payoff

\[ \phi\left(\frac{S_{1}+\cdots+S_{N}}{N}ight)\]

and maturity \(N \geqslant 1\) is always lower than the price of the corresponding European call option, i.e. show that

\[ \mathbb{E}^{*}\left[\phi\left(\frac{S_{1}+S_{2}+\cdots+S_{N}}{N}ight)ight] \leqslant \mathbb{E}^{*}\left[\phi\left(S_{N}ight)ight] \]

Use in the following order:

(i) the convexity inequality \(\phi\left(x_{1} / N+\cdots+x_{N} / Night) \leqslant \phi\left(x_{1}ight) / N+\cdots+\phi\left(x_{N}ight) / N\),

(ii) the martingale property \(S_{k}=\mathbb{E}^{*}\left[S_{N} \mid \mathcal{F}_{k}ight], k=1,2, \ldots, N\).

(iii) Jensen's inequality \[
\phi\left(\mathbb{E}^{*}\left[S_{N} \mid \mathcal{F}_{k}ight]ight) \leqslant \mathbb{E}^{*}\left[\phi\left(S_{N}ight) \mid \mathcal{F}_{k}ight], \quad k=1,2, \ldots, N \]
(iv) the tower property \(\mathbb{E}^{*}\left[\mathbb{E}^{*}\left[\phi\left(S_{N}ight) \mid \mathcal{F}_{k}ight]ight]=\mathbb{E}^{*}\left[\phi\left(S_{N}ight)ight]\) of conditional expectations, \(k=1,2, \ldots, N\).