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Primal and dual pricing of multiple exercise options in continuous time. (English) Zbl 1270.91090
This paper focuses on the pricing of options with multiple exercise times with a finite maturity, and in a continuous-time setting. It thus extends the works of R. Carmona and S. Dayanik [Math. Oper. Res. 33, No. 2, 446–460 (2008; Zbl 1221.60061)], R. Carmona and N. Touzi [Math. Finance 18, No. 2, 239–268 (2008; Zbl 1133.91499)] and A. Zeghal and M. Mnif [Int. J. Theoret. Appl. Finance 9, 1267–1297 (2006; doi:10.1142/S0219024906004037)]. The first paper considered the same problem with an infinite time horizon whereas the latter two studied it with a finite time horizon but with more restrictive assumptions. The finiteness of the time horizon (and the refraction period of such options) may break the right-continuity of the value process, which is the issue tackled here. The main results of this paper are twofold: Theorem 2.2 proves the existence of a Snell envelope and its Doob-Meyer decomposition, and a reduction principle of the multiple stopping problem as a nested sequence of single stopping problems. Theorem 2.3 proves a dual minimisation problem. These two theorems rely on the standing assumptions that the discounted cash-flow process $$Z$$ has right-continuous non-negative paths satisfying $\mathbb{E}\left(\sup_{t\in [0,T]}|Z_t|^p\right)<\infty,$ for some $$p\geq 1$$, where $$T$$ represents the maturity of the option.

##### MSC:
 91G20 Derivative securities (option pricing, hedging, etc.) 60G40 Stopping times; optimal stopping problems; gambling theory 65C05 Monte Carlo methods 60G44 Martingales with continuous parameter
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