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Isn't the option's delta a close approximation for the probability the option will be in the money?


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Using the $T$-forward measure $Q^T$, where the numeraire is the price of the zero-coupon bond $p(t, T)$ maturing at time $T$, we can see that the forward rate is the expectation of the future short rate $r_T$: $$ f(t,T) = \mathbb{E}^T \left[ r_T \mid \mathcal{F}_t \right] \, . $$ See chapter 26 of Tomas' book ...


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Actually, the probabilities in the first case will not sum to exactly 1, since you are truncating the distribution (S' is unbounded above), but will be arbitrarily close. To get the 'right' cumulative probability, you have to adjust for the step size; so, in the first case, you were assigning a weight of 1. In the second case, using a weight of 0.1 will ...



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