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I am currently working on a project at an investment bank regarding new accounting regulations on financial instruments. The task at hand it to understand the connection between a large array of variables. In its simplest form, my question can be presented as follows:

Let $\tilde{X}$ and $\tilde{Y}$ be two random variables with sample means $\hat{\mu_{\tilde{X}}}$, $\hat{\mu_{\tilde{Y}}}$, sample variances $\hat{\sigma_{\tilde{X}}}^2$, $\hat{\sigma_{\tilde{X}}}^2$ and sample covariance $\hat{\sigma_{\tilde{X}\tilde{Y}}}^2$.

I have ran 1000 Monte-Carlo simulations for $\tilde{X}$ under a geometric Brownian motion given by

$$d{\tilde{X_t}}=\hat{\mu_{\tilde{X}}}\tilde{X_t}dt+\tilde{\sigma_{\tilde{X}}}\tilde{X_t}dW_{t},$$

where $W_{t}$ is the Wiener process; normally distributed - $\mathcal{N}(0,t)$

One particular simulation, say $i$ of 1000 is of peculiar interest. I am now interested in the possible values for $\tilde{Y}$.

I have tried to take an econometric approach. That is, run a regression on $\tilde{X}$ with $\tilde{Y}$ and using the regression equation

$$\tilde{X_i} = \tilde{\beta_0} + \tilde{\beta_1}{\tilde{Y_{i}}},$$ $$\implies \tilde{Y_i} = \frac{\tilde{X} - \tilde{\beta_0}}{\tilde{\beta_1}},$$

try and back out the values of $\tilde{Y_i}$.

I am not quite pleased with the results. I am therefore hoping for an alternative method. I have other things in mind - that is, creating a new distribution for the sample but was hoping to gather the thoughts of the experts on this forum.

Thank you in advanced to everyone who takes the time to read and/or provide their thoughts on the matter.

Gus

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    $\begingroup$ My 2 cents. Specifying the individual means + covariance matrix unequivocally characterises the joint distribution of $X$ and $Y$ for elliptical distributions only. If you are comfortable with this assumption, then in your case the two-log returns will be jointly Gaussian and the conditional distribution of $\ln(Y) \vert \ln(X)$ will also be Gaussian and computable in closed form (stats.stackexchange.com/questions/30588/…). $\endgroup$ – Quantuple Nov 3 '17 at 9:37
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Quantuple's comment is completely relevant here. You have defined the relationship between X and Y through the covariance matrix. The traditional way of generating a set of paired variables (or multiple variables) using the monte carlo method is to use Cholesky decompisition on the matrix. You have: $$ \Sigma = \begin{bmatrix} \sigma_x \sigma_x & \rho_{xy} \sigma_x \sigma_y \\ \rho_{xy} \sigma_x \sigma_y & \sigma_y \sigma_y \end{bmatrix} $$ Define the Cholesky decomposition: $\Sigma = \mathbf{LL^T}$ where $\mathbf{L}$ is a lower diagonal matrix. For the 2x2 case we have: $$\mathbf{L}= \begin{bmatrix} \sigma_x & 0 \\ \rho_{xy} \sigma_y & \sigma_y\sqrt{(1-\rho_{xy}^2)} \end{bmatrix} $$ Now if, for each iteration you define $\mathbf{Z}=[Z_1, Z_2]$ two independent standard normal variables, the combination $\mathbf{LZ}+\mu$ will define paired samples whose covariance matrix tends to what you want for large samples. I believe (although have not looked for a long time) that the distribution of the covariance matrix for n samples is defined by the Wishart distribution. You can also use PCA decomposition for this but is more computationally expensive I believe.

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