How do I approach this proof to show that P (T, K) is non-decreasing function of T without making any model assumption

Question

Let P (T, K) be the price of a Put option with maturity T and strike K, and assume that the interest rate is zero, i.e., r = 0. By no-arbitrage pricing rule, show that P (T, K) is non-decreasing function of T without making any model assumption, i.e., show that for any K > 0, P (T1, K) ≤ P (T2, K), ∀ 0 < T1 < T2, in any model that does not have arbitrage

Answer 1

One possible way to go about this question, is to prove the following Lemma

Lemma

If $X_t$ is a martingale, $K < \infty$ a real constant, $0 < T_1 < T_2$ then

$$ \mathbb{E}[(K - X_{T_2})^{\text{+}}|\mathcal{F}_0] \ge \mathbb{E}[(K - X_{T_1})^{\text{+}}|\mathcal{F}_0] $$

We can argue the proof like this:

$$ \begin{equation} \begin{split} P(T_2, K) &= \mathbb{E}[(K - X_{T_2})^{\text{+}}|\mathcal{F}_0] \\ &= \mathbb{E}[\mathbb{E}[(K - X_{T_2})^{\text{+}}|\mathcal{F}_1] \;| \mathcal{F}_0] \\ &\ge \mathbb{E}[(\mathbb{E}[(K - X_{T_2}|\mathcal{F}_1])^{\text{+}} \;| \mathcal{F}_0] \\ &= \mathbb{E}[(K - X_{T_1})^{\text{+}}|\mathcal{F}_0] = P(T_1, K) \end{split} \end{equation} $$

where we used the tower property of the conditional expectation, Jensen inequality on the positive part function (which is convex) and the martingality of $X_t$.

There is also another possible solution, which is to emply the usual strategy "buy low, sell high" and look at the PnL at eah date, but I thought this derivation is a bit more elegant in my opinion.

Please note that this proof does not require a model and the No-Arbitrage condition is respected since the conditional expectation of the discounted payoff is the (fair) no arbitrage price of a put option (here since $r = 0$ you don't really see the discount, but it should be there)