What is the difference between assuming the underlying is lognormal and assuming it is normal?When lognormal the underlying becomes returns where as when normal the underlying becomes asset prices--is this similar to the idea of arithmetic vs. geometric brownian motion?Thanks

One practical difference is that lognormality means the underlying process never has a negative number. It's illustrative to plot the normal and lognormal distributions to get a feel for their range and domain. Do you have access to matlab?

Zakduka, your point on ABM and GBM is exactly right. GBM is in terms of returns: dS/S = mu dt + sigma dWABM is in terms of price: dS = mu dt + sigma dWOf course you can generalise either of these to make mu and/or sigma functions of S and t.Bachelier assumed ABM in his 1900 paper, but since Samuelson made the suggestion in the 1960's it has been very much more common in finance to assume lognormality. The main reason for this (as Eredhuin points out) is that share prices can't become negative or zero under the lognormal/GBM assumption. Of course, if you ever want to allow for the possibility of a zero share price, you could use ABM. Where corporate credit risk is modelled in terms of share price, this assumption is often desirable. For example, www.creditgrades.com make the assumption that equity follows a process that is partly ABM and partly GBM for exactly this reason.

Thanks guys!Dont have access to Matlab. But i'll look up the graphs in a statistics book. Currently going through my past schoolwork and was afraid of getting lost in the math without having the right intuition. Thank you

A very visual way to convince anybody to use log-normal distribution for stock returns instead of normal is to look at charts:Check out this chart of Dow Jones using log axishttp://finance.yahoo.com/q?s=^DJI&d=c&t=my&l=on&z=b&q=land compare it to regular charthttp://finance.yahoo.com/q?s=^DJI&d=c&k=c1&a=v&p=s&t=my&l=off&z=m&q=lZ

1) dS/S = u dt + sigma dz ==> this assumes that returns are normally distributed and so prices will be lognormally distributed.Could some one tell me the rationale behind WHY in this case the returns are normally disributed (something to do with generalized Weiner processes) ? Also, WHY will the mean be equal to (u)dt and std deviation equal to (sigma)dz ?? 2) If we rearrangedS= uS dt + (sigma)S dz ==> then this assumes that prices are normally distributed and so returns will be lognormally distributed ? Thanks in advance

The first equation is GBM, geometric brownian motion.dS/S = u dt + sigma dzIn this equation the only source of uncertainty is the "dz" term, which - by definition - has normal distribution with mean zero and standard deviation 1. The "sigma" term is a scaling device so the term "sigma dz" is normally distributed still with mean zero, but now with standard deviation sigma times 1 = sigma. As this is the only source of uncertainty, both sides of the equation are normally distributed with standard deviation of sigma. The term on the left hand side of the equation is a term for returns, i.e. the small change in the share price divided by the share price. Putting everything together, this means that the returns are normally distributed with standard deviation sigma.In your second equation, all you've done is multiply both sides by S, so you've still got the same equation. So this still assumes that returns are normally distributed. If you want to assume that arithmetic price changes are normally distributed, you need to go with arithmetic brownian motion, ABM:dS = u dt + sigma dzYou can see this is exactly the same as your first equation (meaning that both sides are normally distributed with st dev of sigma) but now you have price on the left hand side, instead of return. So in this equation you embody the assumption of normally distributed absolute price changes, rather than returns. Using numbers, GBM might assume a standard deviation of 1% price return each day, whereas ABM might assume a standard deviation of $1 price change each day.

QuoteOriginally posted by: ZiggyA very visual way to convince anybody to use log-normal distribution for stock returns instead of normal is to look at charts:Check out this chart of Dow Jones using log axis<a target=new class=ftalternatingbarlinklarge href="http://finance.yahoo.com/q?s=^DJI&d=c&t ... q=l</a>and compare it to regular chart<a target=new class=ftalternatingbarlinklarge href="http://finance.yahoo.com/q?s=^DJI&d=c&k ... m&q=l</a>I do not see how: the log of the price (DJ index is a price-index) does not represent the log of the returns. Actually, you should have graphed log (Pt/Pt-1) instead.

@ JohnnyWhile the GBM is the most accepted model for Stocks, Index and Currencies, what are the most easiest to understand stochastic models for <b>Interest Rates</b> ? I heard about Vasicek, and other that I found in Hull's and Wilmott's respective Bible but may be it could be explained by someone there to make me understand clearly.thanks

This is really Pat's area. Try a search under "interest", "models" and "Pat". I'm pretty sure he gave a thorough answer to this question only a few days ago.

Just as equity is not supposed to go to zero (excepting for the fact that it does), interest rates aren't supposed to become unbounded or negative, so some kind of mean-reversion in the short rate needs to appear. Because the end result is pretty much bounded, any detail about how you stay between those bounds is extra... hence mean-reverting lognormals vs. CIR vs. other power laws etc. And, just like equity, to understand what you should be doing at the extremes, you have to go where you can observe them... i.e. observe Japan on the low side (i.e. does lognormal crawl out of recession too slowly) or Brazil on the high side (how strong is the force that ends hyperinflation).

Zaduka,Have a look at www.mathworks.com, in their online documentation/statistics toolbox. They have notes and plots on all the distributions you will ever need (probably).RegardsVeegan

FinkbartonQuote do not see how: the log of the price (DJ index is a price-index) does not represent the log of the returns. Actually, you should have graphed log (Pt/Pt-1) instead. Actually I was referring to price changes, but those charts give visual emperical grounds for log normal returns. Drawing chart on lognormal charts show price changes show similar characteristics for any level of the underlying. When displaying normal chart you would have to assume that volatility is consistantly increasing with higher levels of the underlying in order for the chart to fit any model.This was not a scientific attemt, just a simple laymans presentation on the underlying distribution Z