Study of Last Passage Times up to a Finite Horizon

In Chapter 1, we have expressed the European put and call quantities in terms of the last passage time $\mathcal {G}_{K}^{(\mathcal{E})}$ . However, since $\mathcal {G}_{K}^{(\mathcal{E})}$ is not a stopping time, formulae (1.20) and (1.21) are not very convenient for simulation purposes. To counter this drawback, we introduce in Section 5.1 of the present Chapter the ℱ t -measurable random time: $$\mathcal {G}_K^{(\mathcal{E})}(t)=\sup\{s\leq t;\;\mathcal{E}_s=K\}$$ and write the analogues of formulae (1.20) and (1.21) for these times $\mathcal {G}_{K}^{(\mathcal{E})}(t)$ . This will lead us to the interesting notion of past-future martingales, which we shall study in details in Section 5.2.