As the lognormal distribution imposes bounds of attainable correlations as discussed in Attainable correlations for lognormal random variables my question would be what happens if say we want to do a monte carlo simulation of two stocks, where the historical correlation coefficient is out of the bounds for two log-normal distributed variables (i.e. it is found to be 0.4, whereas the theory only admits 0.1). What would happen if I use the historical inputs to make a monte-carlo simualation using the brownian motion to simulate if the correlation bounds are violated.
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$\begingroup$ Your linked question deals with the specific case of $\log(X_1)\sim \mathcal{N}(0,1)$, and $\log(X_2)\sim \mathcal{N}(0,\sigma^2)$; other bounds on the correlation happen for different parameters. Sample correlations from log-normals can exceed the bounds of population correlation, often substantially. And perhaps your stock prices are not actually log-normally distributed $\endgroup$– HenryCommented May 9, 2020 at 21:41
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$\begingroup$ @Henry So say if I want to do a monte carlo simulation for two series, whose returns are distributed normally i.e. N(mean = 0, volatility = 1.95) and N(mean = 0, volatility =0.4) and the correlation to be 0.1, and I use these returns to move to price level by exponentiating them (as the prices are log-normal), what would this imply for my distribution of prices (as the 0.1 correlation can technically not be reached at the price level or respectively how could I calculate the new bounds as I thought the plot in the link represents the ratio of volatilities on the x axis). $\endgroup$– macro123Commented May 9, 2020 at 22:49
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$\begingroup$ Are you talking about returns or stock prices? Stock prices are the product of many returns $\endgroup$– HenryCommented May 9, 2020 at 22:57
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$\begingroup$ @Henry to Simulate the two Stock prices via Monte Carlo I first need to simulate two correlated (via Cholesky decomposition) bivariate normal random variables. Following the geometric brownian motion approach I can then generate returns i.e. for one period by using the formula mean + sigma*dW_t . To get the series of prices instead of returns I would then use S0*exp(r), where r is the simulated return. $\endgroup$– macro123Commented May 9, 2020 at 23:05
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