Hi Everybody,I have a quick question. How do you calculate the probability of an event occuring using the lognormal distribution ? also what are the inputs needed ?this is stuff i should remember from school, but it is gone from my head. Thanksjaiman

A normal distribution, so far as I can tell, measures the possible sum of an infinite number of infinitesimal errors, along a single axis, after they have been accumulating at a certain rate for a certain amount of time.Like suppose you put a bead on a wire and stretched the wire across the threshold of the main entry gate to a Metallica concert (and did it in such a way that the bead could move along the wire, but the wire was too tight to the ground for anyone to trip over).Assuming the bead started far enough from either wall to not reflect any bounday artifacts, and there were no effects of people entering on the right and exiting on the left, the location of the bead after 50,000 concertgoers pass through should be normally distributed.As you have fewer and fewer people with larger and larger clumsy feet, the distribution should become fat-tailed if they jolt the bead varying lengths, and binomial if they all jolt it the same distance.As such, anything that measures, or any sample that reflects, all these errors, should be normally distributed.Suppose these same concert-goers trudged through the mud, leaving a pattern of foot traffic in the dirt. You could recover the normal distribution from their errors by rolling a golf ball across the uneven expanse of dirt.Assuming some errors cause the outcome to go one way along the axis, and some errors the other, the net of offsetting errors will be normally distributed around this mean.The volatility, then, is the total number of infinitesimal errors. Meaning, how far forward do you let the golf ball roll, before you have measured the left-right deviation of its course.Of course, the rolling golf ball would not have its volatility increase by the squareroot of distance, as a normal distribution does with the squareroot of time. Since its speed and axis would decay, it might even roll backwards.And then, the lognormal distribution is just the natural log of the normal distribution, which means that errors near zero will be smaller than errors which occur at 100.To go back to our concert-gate metaphor, the lognormal distribution is like if the doorway was only bounded on one side, and the closer you got to that side, the smaller people with smaller feet were tripping over the bead, causing it to get stuck on the side since they tended not to kick it very far.So the normal distribution calculates the probable net number of errors, or left-right accidents, that will manifest in a particular sample.MP

I'd probably answer that question slightly differently (although I'm not saying Mobpsycho's answer is wrong).A variable may be lognormally distributed if it can take any positive value. So for starters, if the variable can be a negative number, or if there are positive values which it can't take, it isn't lognormal. Also, if there are discrete values that have a positive probability of occurring, it isn't lognormal - lognormal is a continuous distribution. [Although discrete or truncated distributions may be approximated with lognormal distributions.]The inputs needed are the mean and the standard deviation of the underlying normal distribution. Note that given the exponential power, the mean of the lognormal distribution will be higher than e to the power of the mean of the underlying normal distribution.Being a continuous distribution, if the even you talk about is like "what is the probability that the stock price equals 5", then the probability will be zero. To work out the probability of it being between a and b, take the cumulative distribution that it is less than b, and subtract the cumulative prob that it is less than a.Excel can work out cumulative probabilities. Alternatively, you can take the logs of all the values, and then just use a normal distribution.Let me know if you need further info.Regards,Guy

Thanks for the info Guy.Would you happen to have the formula for the cdf of the lognormal and normal distributions. I cannot seem to find my trusty stats text book that i've used in the past.thanksjaiman

The cdf of the normal dist is erf((x-mu)/sigma)the cdf of the lognormal dist is erf((ln x -mu)/sigma)where erf is the error function - see http://www.efunda.com/math/error_zeta/error_zeta.cfm for more info (and definition and approximations).mu is the mean of the underlying normal distribution, and sigma is the s.d. of the underlying normal dist.

Perfect, Thanks Guy.

Here is an issue that keeps bugging me,Seems to me that the disctinction between normal and log normal distributions are not very clear when returns are analysed in literature.I've found the same approach in a two different VaR calculations (which leads me to some unresolved questions):Let R denote rate of return of a stock such as R = Price_i/Price_i-1Now let r = ln(R) and assume that r is normal distributed. This makes R Log-normal distributed.Say we calculate r for a number of days and we estimate mean and variance of observed r (the normally distributed variable). Here comes the logical jump: r_variance is then used to denote volatility of the stockposition, so that we estimate VaR (relative) to: Price*shares*alpha*sqrt( delta_t * r_variance)alpha = confidence factor for normal distributiondelta_t = time scalingHow can the variance of the normal distribution ( ln(R) ), be transferred to an estimation of the volatility of the stock price?For me, if I wanted the volatility of the stock price I would resort back to R and use the linking between the normal / lognormal distribution so thatI could get R_variance..Am I missing something out?Best regards,

QuoteOriginally posted by: gjlipmanThe cdf of the normal dist is erf((x-mu)/sigma)the cdf of the lognormal dist is erf((ln x -mu)/sigma)where erf is the error function - see http://www.efunda.com/math/error_zeta/error_zeta.cfm for more info (and definition and approximations).mu is the mean of the underlying normal distribution, and sigma is the s.d. of the underlying normal dist.You may also find this description of error functions very useful Error function on mathworld

No, you're on the right track.Firstly, the volatility of a lognormally distributed variable (eg stock price) is defined as the standard deviation of the log of the stock price, so as far as terminology goes, this is all ok. This is reasonably given the transformation under Ito's lemma:dS=mu S dt + sigma S dzdx=(mu-0.5 * sigma^2) dt + sigma dz(where x=ln S)so the volatility coefficient stays the same.The problem is that while variance of x grows with square root of time, the variance of S only grows with the square root of time over very short periods of time. As an example, if you use the formula:Price*shares*alpha*sqrt( delta_t * r_variance)you will get a real probability that the stock will go below 0. But if the volatility is low or the time period is low, it is a close enough approximation.If you want to be more accurate, use a lognormal approximation for VaR purposes, rather than a normal approximation (but few practitioners bother).