I am simulating Geometric Brownian MotionI am using the following processS(t) = S(t-1)*Exp((u - .5 * sigma^2)dt + sigma * nrand * dt^.5) where nrand~N(0,1)I am trying then to fit a lognormal curve to the final price outcomes of the price pathsWhat mean and variance should I use for the lognormal distribution to get it to fit the output?The empirical distribution seems to be slightly offset compared to theoretical distribution. Indeed if I use a shift parameter in the log-normal distribution then I can get it to fit. But I have no theoretical reason to use thisCan anyone help here?

The mean is S(0).exp [ meu.T ] Don't remember off the top of my head what the variance is.. but both these concepts are discussed very well and concisely in Hull (chapter 10 or 11 I think)... well work a look!Sam

Why don't you just calculate the variance directly ?E[{ S*exp(r-0.5*vol^2)dt+vol*sqrt(dt)*z)-S(r*dt) }^2] = (S*exp(r*dt))^2 * { exp(vol^2*dt) - 1 }Here I use the mean is S*exp(r*dt), as sam stated.Useful relationship is E[ exp(-0.5*a^2*dt+a*sqrt(dt)*z) ] = 1, and you may find you need to treat a as 2*vol somewhere in your calculation.Hope it helps.

Suppose you try to represent the dynamic of monthly stocks returns with a GBM.Interesting question is : how much data do I need to get a correct estimation of the variance (such as + or - 0,5% per example)and how much monthly data do I need to estimate the mean ?I've done a theoretical computation using the postulate that the data comes "really" from a geometric brownian motionand I get rather depressing answers, such as 4,976 years for the mean.And that's not counting with the obvious fact that the geometric brownian motion is a rather poor fit.Do I go wrong ?

I got it to workthanks, sam and monk.Palsky, Interesting point. Why not use daily observations then? or even tick by tick data? (I do realise you then have some problems with stationarity variance, but at least you will have enough data points). And yeah I realise GBM, isn' t a perfect fit, but I am only trying to do an assignment, not price actual deals of my simulated valuations.

Well, if the dynamic of stock indexes was "really" the effect of a unknown GB generator, it would be equivalentto measure volatility on a monthly or daily (or anythingly) basis.But in fact, real world dynamics are rather different at different time scales. I work on asset liability Monte Carlo models forlife insurance and pension funds (historical probability). This domain is very far from derivatives pricing applications. One problemis that short term risk is primarily linked to short term volatility, while long term pension fund risk is more dependent on mean long termreal stock yields.The short term pricing problem with GBM come from leptokurtism, but the ALM risk comes from complex (perhaps chaotic) long termdynamics and possible mean reversion of PER or dividend yields.

QuoteOriginally posted by: palskySuppose you try to represent the dynamic of monthly stocks returns with a GBM. Interesting question is : how much data do I need to get a correct estimation of the variance (such as + or - 0,5% per example) and how much monthly data do I need to estimate the mean? I've done a theoretical computation using the postulate that the data comes "really" from a geometric brownian motion and I get rather depressing answers, such as 4,976 years for the mean.And that's not counting with the obvious fact that the geometric brownian motion is a rather poor fit.Do I go wrong?One of the properties of geometric brownian motion is that you can (in theory) estimate the variance exactly with any sample, however short, but estimating the mean has an error inversely proportional to the standard deviation of the length of the interval. For stock returns the monthly standard deviation is an order of magnitude higher than the mean, so you need 100 months just to get down to 100% error. 0.5% means 40,000 times that or 4 million months.So my estimate is almost 100 times yours. But both are absurd, obviously we do not expect parameters to remain constant over more than a few years, even if data were available. The only practical way to estimate mean stock returns is to use the average of many stocks over long periods of time, and to accept errors of significant fractions of the mean. No extra sampling or clever math is going to help you.For most purposes, knowing the mean is not very important. It is what it is. That's what the market gives us. Like it or hate it, you either take that or the risk-free rate. Knowing the distribution is important for planning purposes, but the mean is of mostly academic interest.Estimating the variance is much easier. As troywilson suggested, you can use more frequent data. Using the monthly high/low prices adds most of the efficiency of tick-by-tick data, and it's easier to get and work with. You also have implied option volatility to help you.The fact that GBM is not a good model doesn't matter much for mean estimation. On one hand, it has a significant effect on volatility estimation. On the other hand, you're probably not interested in mathematical variance, but on some generalized measure of distribution spread. This can be pinned down pretty accurately.

Thanks, Aaron,I am sure there is no such thing as a long term stocks mean return inscripted on stone in a secret part of the universe, and thatmy question is at best a naive one.But you cannot do strategic asset allocation in a risk neutral framework, because if any asset gives the same return, it follows thatyou can invest 100 % short term, which my pension funds customers object to.Thats my first problem, I need a mean return for various assets to compute long term financial risk and return, and my ignorance of thecorrect mean parameters just wheigh more than the historical or implied volatility of a stock index.Note that the fact I am ignorant does not exonerate me to make long term allocation decisions, and I have not only return objectives,but also real return objectives.Second problem is that, for accounting reasons, I am exposed to short term volatility, even if my liabilities wont mature before 20 years.Therefore GBM is not enough good even to cover this sort of risk.I use combined Poisson process (multinomial jumps) to model this part of risk. But that doesnt gives me a solution for the other, long term part.I have worst problems, because I also need to evaluate complex liabilities within a non arbitrage framework, but thats would be a subjectfor another thread.

When your model calls for inputs you can't get, it's time to get a new model.For asset allocation, you have to use means based on long-run averages of a large universe of securities, with judgmental adjustments. If you try to use the last three months return in the trucking sector, you'll get silly allocations. But with 75 years of S&P500 returns you can start to guess a value, and you can refine it by looking at some fundamental analyses.