Given two standard normal distributed variables $Z_1$ and $Z_2$ then $N$ is defined as $$N=\alpha Z_1 + \beta Z_2$$ How do I derive $\alpha$ and $\beta$ such that $N$ is standard normally distributed with correlation $\rho$ to $Z_2$?
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2$\begingroup$ I couldn't find a proper headline for this question. So feel free to suggest a new heaadline in the comment section or just edit it. A sidenote: This is part of a bigger Monte Carlo Simulation scheme to calibrate a so called SABR model. $\endgroup$– financegradCommented Apr 11, 2018 at 9:12
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1 Answer
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Let $m=cov(Z_1,Z_2$). Then you have the following:
$$var(N)=var(\alpha Z_1)+var(\beta Z_2)+2cov(\alpha Z_1, \beta Z_2)=\alpha^2+\beta^2+2\alpha \beta m$$ Hence the standard deviation of $N$ is such that $\sigma_N=\sqrt{\alpha^2+\beta^2+2\alpha \beta m}$ and since you want $N$ to be a standard normal than you have that $\sigma_N=1$ $(1)$
Note how you have that $\mathbb{E}[N]=0$ by definition.
Now you have:
$$cov(N,Z_2)=cov(\alpha Z_1+\beta Z_2, Z_2)=\alpha m+\beta var(Z_2)=\alpha m+\beta$$
Hence you have the equality:
$$\rho=\frac{cov(N,Z_2)}{\sigma_N\sigma_{Z_2}}=\alpha m+\beta \text{ }(2)$$
Now solve $(1)$ and $(2)$for $\alpha$ and $\beta$
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2$\begingroup$ Things I believe need to be made clear: 1. for the above scheme to be sure to have $N$ normal for any $\rho$, $Z_1$ and $Z_2$ will need to be jointly normal. 2. that the mean is already $0$, leaving you with two equations in three unknowns ($\alpha$, $\beta$ and $m$) to solve: (i) making the variance equal to 1 and (ii) making the correlation equal to $\rho$. The formula for the variance is given above but the need to set it to $1$ is not. 3. The third unknown, $m$ (along with my point 1) may be most easily dealt with by making $Z_1$ and $Z_2$ independent (giving $m=0$). $\endgroup$– Glen_bCommented Apr 11, 2018 at 22:50
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$\begingroup$ I just saw that the problem need $N$ to be a standard normal and will edit it. Thanks! $\endgroup$– asdfCommented Apr 12, 2018 at 19:53