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1. If we assume that returns(defined as the log of price relatives) are normally distributed (with zero mean), we are required to accept that the probability of a return of .0953 is the same as the probability of a return of - .0953. We are also required to accept that the probability of a .0953 return does not equal the probability of a return of - .1053.2. The Geometric Brownian Motion requires that small percentage changes of the stock are normally distributed (at least that is what is claimed by many). This means that the probability of a .1% increase in the stock is equal to the probability of a .1% decrease.3. 1. above contradicts 2. above4. If the probability of percentage changes are equally (with zero mean) , then there will be a tendacy for the stock to go zero. With the higher volatile stocks going to zero faster. The stock will proceed to zero at a pace of vol^2/2 each year on average.5. Is there someone who will comment on how these options models incorporate two contradictory assumptions.Peace :Worldoptions

For small asset moves (changes over a small period of time), returns on an asset & percentage changes of an asset are nearly equal. Quote4. If the probability of percentage changes are equally (with zero mean) , then there will be a tendacy for the stock to go zero. With the higher volatile stocks going to zero faster. The stock will proceed to zero at a pace of vol^2/2 each year on average.Under a zero mean or drift scenario, but subject to Brownian randomness, the future value of the asset will be its price today. Recall, that if m=drift rate of the asset then the expected value of the asset at time T is S(0)*exp(m*T), and note that it contains no volatility factor. It seems that you are looking at either the SDE for ln[asset] or the closed form expression for the asset process (GBM) and inferring that if I set the asset drift to zero then the drift rate is -vol^2 / 2. It's not see the discussion on the Drift Thread.