Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 39 / 2 / a / Solution

The square root is concave on $[0,\infty )$, and $\sqrt{S_T}$ is integrable since $\mathbb{E}\sqrt{S_T}\le \sqrt{\mathbb{E}{S}_T}$. For $t\le T_1\le T_2$, the conditional Jensen inequality and the martingale property giveTaking conditional expectations with respect to ${\mathcal{F}}_t$ and using the tower property of conditional expectation yieldsThus the maturity curve for a square-root stock claim is nonincreasing; it need not be strictly decreasing, as a constant stock gives equality. The inequality holds for every ordered pair of maturities. Under the usual right-continuous versions of the market processes, it gives the corresponding nonincreasing version of the maturity curve.