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The average of the exponentials is not the exponential of the average. It is always higher due to convexity (Jensen inequality). So there is no contradiction between the average of $X_T$ being negative and the average of $S_T$ being $S_0$. So the question is: are your results really significantly different from what you would expect? Have you tried ...


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What you do is: You simulate a Brownian path - with the correct standard deviation. Then you simulate the Compound Poisson process. In each time step you sample a jump or no jump and the jump size if there was one. In each time step you draw from a Bernoulli distribution - which is to my knowledge just an approximation. For the compound Poisson process I ...



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