Consider an asset price process \(\left(S_{t}ight)_{t \in \mathbb{R}_{+}}\)which is a martingale under the risk-neutral probability measure \(\mathbb{P}^{*}\) in a market with interest rate \(r=0\), and let \(\phi\) be a convex payoff function. Show that, for any two maturities \(T_{1}
\[
\mathbb{E}^{*}\left[\phi\left(p S_{T_{1}}+q S_{T_{2}}ight)ight] \leqslant \mathbb{E}^{*}\left[\phi\left(S_{T_{2}}ight)ight]
\]

i.e. the price of the basket option with payoff \(\phi\left(p S_{T_{1}}+q S_{T_{2}}ight)\) is upper bounded by the price of the option with payoff \(\phi\left(S_{T_{2}}ight)\).

i) For \(\phi\) a convex function we have \(\phi(p x+q y) \leqslant p \phi(x)+q \phi(y)\) for any \(x, y \in \mathbb{R}\) and \(p, q \in[0,1]\) such that \(p+q=1\).

ii) Any convex function \(\left(\phi\left(S_{t}ight)ight)_{t \in \mathbb{R}_{+}}\)of a martingale \(\left(S_{t}ight)_{t \in \mathbb{R}_{+}}\)is a submartingale.