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Judging from the oscillations near $S=0$, it looks like the payoff function is causing these problems. Your payoff should go towards -1 as $S$ goes towards zero, but your computer might just evaluate it at $S=0$, producing nonsense as a result. Depending on the exact implementation, this will then spread through the neighborhood of that point, causing ...


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You don't need any assumption about the distributional properties of $S_t$. What matters for the FTAP is the drift only. By definition, the risk neutral measure $Q$ is the measure, equivalent to the natural measure $P$ (*), under which the local rate of return (i.e. the instanteneous drift of the SDE of $S_t$ per unit of $S_t$) of "any" traded asset $S_t$ (...



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