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Short and sweet: I'd like to model $n$ random variables representing price changes of individual assets. Each of these should be distributed as a log normal variable with a median of 1. Is there a way to generate these variables with a given correlation matrix?

Longer explanation: I'd like to model a price change from time $t_0$ to $t_1$ assuming price grows with probability 50%. For a single asset, we could model the growth as log normal.

$$G \sim \text{lognormal}(\mu, \sigma)$$

Setting $\mu = 0$, the median will be $\text{exp}(\mu) = 1$. Meaning price will increase ($G > 1$) with probability 50%.

We could also get any variance we would like by changing $\sigma$.

Now lets assume we have $n$ such assets that can grow and they are correlated according to a correlation matrix A. The traditional way to generate random variables that are correlated would be to generate independent variables and then adjust them using Cholesky decomposition.

In this case that wouldn't work as it would disrupt our goal of having the median be equal to 1. Moreover, the linear sum of two lognormal distributions is not lognormal.

Questions:

  • Is there an algorithm to generate such numbers?
  • Is this a reasonable model for and a reasonable set of constraints to model stock price movement from $t_0$ to $t_1$ or is a different set of assumptions commonly used?
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    $\begingroup$ Use stats.stackexchange.com/questions/6853 to relate the correlations to correlations of the logarithms -- then generate the logarithms using any of the many standard methods for multivariate Normal variables. $\endgroup$
    – whuber
    Jul 9 at 14:56
  • $\begingroup$ Where does this correlation matrix come from? If it comes directly from observed asset prices, you could get the correlations of the log-prices instead, then create a normal model with that correlation matrix, and then take exponentials of that normal model. $\endgroup$
    – Matt F.
    Jul 10 at 23:01
  • $\begingroup$ Hey @MattF, thanks, yeah that's exactly what I'm doing in practice right now, and worst case will have to find the raw data feeds, but would like to leave the question as written. $\endgroup$
    – Peteris
    Jul 11 at 22:07
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Using copulas: Use a Gaussian copula with a correlation matrix the one desired to combine the lognormal marginals. An algorithm to generate the joint distribution with the required properties is described Ross, S. Simulation book. Other types of copulas may be more appropriate even when they do not satisfy the strict correlation requirement.

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    $\begingroup$ Again, the interest is in the details: exactly how does one determine the copula from the specified correlation matrix of the lognormal variables? Your link is dead, so that's of no help. $\endgroup$
    – whuber
    Jul 13 at 14:30
  • $\begingroup$ sciencedirect.com/topics/mathematics/gaussian-copula. The colon was missing in Stavros Tsalides's answer $\endgroup$
    – nikos
    Jul 13 at 15:55
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"the linear sum of two lognormal distributions is not lognormal": so first generate normal distributions, then finally take logs.

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    $\begingroup$ You mean to exponentiate the Normal variates, not take their logs. But you still beg the question, because its resolution is in the details: namely, exactly what parameters must be used for generating the (multivariate) Normal values? $\endgroup$
    – whuber
    Jul 9 at 15:59

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