Generalized Black-Scholes Formulae for Martingales, in Terms of Last Passage Times

Let (M t ,t≥0) be a positive, continuous local martingale such that $M_{t}\xrightarrow[t\rightarrow\infty]{}M_{\infty}=0$ a.s. In Section 2.1, we express the European put $\Pi(K,t):=\mathbb{E}\left[\left(K-M_{t}\right)^{+}\right]$ in terms of the last passage time $\mathcal {G}_{K}^{(M)}:=\sup\{t\geq0;M_{t}=K\}$ . In Section 2.2, under the extra assumption that (M t ,t≥0) is a true martingale, we express the European call $C(K,t):=\mathbb{E}\left[\left(M_{t}-K\right)^{+}\right]$ still in terms of the last passage time $\mathcal {G}_{K}^{(M)}$ . In Section 2.3, we shall give several examples of explicit computations of the law of $\mathcal {G}_{K}^{(M)}$ , and Section 2.4 will be devoted to the proof of a more general formula for this law. In Section 2.5, we recover, using the results of Section 2.1, Pitman-Yor’s formula for the law of $\mathcal {G}_{K}$ in the framework of transient diffusions. The next sections shall extend these results in different ways: In Section 2.6, we present an example where (M t ,t≥0) is no longer continuous, but only càdlàg without positive jumps, In Section 2.7, we remove the assumption M ∞=0, Finally, in Section 2.8, we consider the framework of several orthogonal local martingales.