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Maybe it is better to use martingale theory to characterise whether it is an equality or not. Let $S_t$ be a (right)-continuous positive martingale with $S_0 < H$. Let $\tau = \inf \{ t > 0| S_t = H \}$. The option pays 1 unit of cash at $\tau$, and there is no maturity (perpetual option). What is the price of the option? I.e. compute $$P_0 = ...



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