Hi,I'm having some trouble understanding the implications of the assumption that stock returns are lognormally distributed. Playing around with the probability calculator at optionistics (calculator), and putting in the following parameters:current price: $5.0days to expiration: 2volatility: 1.0interest rate: 0.00001and setting the upper and lower bounds both to $5, it reports that the probability that the stock will be less than $5 is 51.47%, and the probability that it will be above $5 is 48.53%. This effect is exaggerated if the days into the future are increased, e.g. 100 days into the future the probability of being less than $5 is reported as 59.7%.How is it that the probability of any decrease in price is computed as being so much higher than the probability of an increase? Intuitively, the probability of being below or above the current price should be 50%. Any clarifications on why that isn't the case would be greatly appreciated.Many thanks

QuoteOriginally posted by: notNotDanielHi,Intuitively, the probability of being below or above the current price should be 50%. Any clarifications on why that isn't the case would be greatly appreciated.This is not true for a lognormal distr as it is skewed; up move is more likely than a down move; imagine the extreme of a price that is =0 and can't go negative - the prob of above 0 would be 100% and below 0 would be 0%.The results you show don't seem correct, I agree. But I do not know what it means for the upper and lower bounds to be $5.Cheers,Alk

looks like optionistics is returning the delta for the call and put as probs. i have no idea why ...

I believe that using GBM, which many option pricing formulas assume, the distribution of log returns will be:~Normal ( log S + (r - sigma^2/2)*T, sigma^2 * T)This is assuming your drift is set to the risk-free rate of return. So if you have a zero interest rate and a positive volatility (positive sigma), you will end up with a negative expected return.

you are correct about the GBM .. however, the probability that a GBM is above a level K is given by N(d2) where d2 is the usual BS parameterplugging the OPs parameters (stock =5, level = 5, r = 0.00001, vol = 100%, t = 2/365) gives me .485241however using a vol of 1% gives .500148hth

QuoteOriginally posted by: wpgabrielI believe that using GBM, which many option pricing formulas assume, the distribution of log returns will be:~Normal ( log S + (r - sigma^2/2)*T, sigma^2 * T)This is assuming your drift is set to the risk-free rate of return. So if you have a zero interest rate and a positive volatility (positive sigma), you will end up with a negative expected return.Might be slightly confusing but the log returns are irrelevant and are difficult to interpret - however, the expected price will be S*exp(rT) from the properties of the lognormal expectation being (r+sig^2/2). That's why the assumption is soooo smooooth because assuming this particular GBM is consistent with the no arbitrage argument for Forward prices being S*ecp(rT). Why do you think that the expected return is negative?Alk