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These equations gives the probability of a successful trade for a European put finishing in the money (that is, the probability that the strike price is above the market price at maturity). European call finishing in the money (that is, the probability that the strike price is below the market price at maturity) Probability of a Successful Option Trade ...

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Okay, this is a bit of an involved question, but the intuition is as follows: As Ross (1976) truly conceived it, being risk-neutral means being indifferent between any gamble and its mean payoff. This is equivalent to linear Von-Neumann Morgenstern preferences over all wealth levels, not just positive ones. A classic experiment to distinguish between ...

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It sounds to me like you have a Markov model that is not "lumped", it's just that certain transitions don't provide you with any payout. I would model the true transition probabilities. Now, let's ask what the probability of getting a one is, assuming that we won't stay at zero, i.e. $P(X_1=1 | X_0=0, X_1 \not = 0)$. We recall that $$P(A|B)=P(A,B) / P(B)$$ ...

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