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I am trying to understand the reason why the author of my book resort to Cholesky decomposition in this particular case.

He starts assuming that we know the historical prices of each asset, so (for the relationship between price and return $R_t:=\frac{S_t}{S_0}$) the conditions

  • (2.16) $\rightarrow \mu-\Gamma_t\sigma\leq \widetilde{r}_t^S \leq \mu+\Gamma_t\sigma$
  • (2.17) $\rightarrow e^{t\mu_{\operatorname{log}}-\Gamma \sqrt{t}\sigma_{\operatorname{log}}}\leq \widetilde{R}_t^S\leq e^{t\mu_{\operatorname{log}}+\Gamma \sqrt{t}\sigma_{\operatorname{log}}}$

that represent respectively the uncertainty set for single-period returns and cumulative returns (with $\mu,\mu_{\operatorname{log}},\sigma, \sigma_{\operatorname{log}},\Gamma \in \mathbb{R}^+$) allow us, like he says, to estimate the upper bound and lower bound for $\widetilde{S}_t^m$ (with $\sim$ that denotes the randomness of variable and $m$ is the number of assets included in the underlying basket of derivative). In fact, knowing that

$U:=\begin{Bmatrix} \widetilde{R}_t^S: \underline{R}_t^S \leq \widetilde{R}_t^S \leq \overline{R}_t^S \end{Bmatrix}=\begin{Bmatrix} \widetilde{R}_t^S: (1+\underline{r}_{t-1}^S)\widetilde{R}_{t-1}^S \leq \widetilde{R}_t^S \leq (1+\overline{r}_{t-1}^S)\widetilde{R}_t^S \end{Bmatrix}=\begin{Bmatrix} \frac{\widetilde{S}_t^S}{S_0}: \frac{\underline{S}_t^S}{S_0} \leq \frac{\widetilde{S}_t^S}{S_0} \leq \frac{\overline{S}_t^S}{S_0}\end{Bmatrix}=\begin{Bmatrix} \frac{\widetilde{S}_t^S}{S_0}: (1+\underline{r}_{t-1}^S)\frac{\widetilde{S}_{t-1}^S}{S_0} \leq \frac{\widetilde{S}_t^S}{S_0} \leq (1+\overline{r}_{t-1}^S)\frac{\widetilde{S}_{t-1}^S}{S_0} \end{Bmatrix}$

for $S$ the single asset, we can extend this consideration to $m$ assets and obtain

$\Rightarrow\left\{\begin{matrix} \frac{\widetilde{S}_{t}^m}{S_0}\leq \frac{\overline{S}_{t}^m}{S_0}\Rightarrow \widetilde{S}_{t}^m\leq \overline{S}_{t}^m,\forall t=1,...,T,\forall m=1...,M\\ \frac{\widetilde{S}_{t}^m}{S_0}\geq \frac{\underline{S}_{t}^m}{S_0}\Rightarrow \widetilde{S}_{t}^m\geq \underline{S}_{t}^m,\forall t=1,...,T,\forall m=1...,M\\ \frac{\widetilde{S}_{t}^m}{S_0}\leq \frac{\widetilde{S}_{t-1}^m}{S_0}(1+\overline{r}_{t-1}^S)\Rightarrow \widetilde{S}_{t}^m\leq \widetilde{S}_{t-1}^m(1+\overline{r}_{t-1}^S),\forall t=1,...,T,\forall m=1...,M\\ \frac{\widetilde{S}_{t}^m}{S_0}\geq \frac{\widetilde{S}_{t-1}^m}{S_0}(1+\underline{r}_{t-1}^S)\Rightarrow \widetilde{S}_{t}^m\geq \widetilde{S}_{t-1}^m(1+\underline{r}_{t-1}^S),\forall t=1,...,T,\forall m=1...,M \end{matrix}\right.$

for $m=1,...,M$ the basket of assets. Thus we can estimate $\underline{S}_t^m$ and $\overline{S}_t^m$ precisely because of those conditions. In particular we extract samples from time series of prices of $m$ assets, i.e. to say from time series of log-returns, and under a condition of invariance of statistic properties of samples (and hence of parameters of mean and variance) due to second order stationarity of processes related to the single time series we proceed to estimate their respective sample means and standard deviations. This condition, obviously, is based on the assumptions of normality and independence of log-returns that apply per every singly series. (We are in a Black and Scholes environment). Then he says that we have to consider the correlation between the assets and to do it we have to consider the correlation between the asset returns. So we'll use the covariance matrix of the single period returns. Therefore we consider, one by one, all possible couple of assets, and for any $j$-th realization of returns of assets (for $j=1,...,n$ and $n$ the sample size) we'll calculate the sum of cross products of deviations in order to complete the covariance matrix. We also know that $\sum$ is symmetric and positive definite. Well.

At this point, author introduces Cholesky decomposition. You can also see here (pag. 31-33).

My question is: why the Cholesky decomposition? To what end?

Thanks in advance just for reading.

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  • $\begingroup$ The link you provided does not have a "page 31" $\endgroup$ Dec 4, 2020 at 14:33
  • $\begingroup$ mathoverflow.net/questions/93119/… $\endgroup$
    – Ryan Howe
    Dec 4, 2020 at 14:37
  • $\begingroup$ @BenGrossmann Sorry, I edited. Please, see the correct link in the post. $\endgroup$
    – user51121
    Dec 4, 2020 at 14:40
  • $\begingroup$ @RyanHowe Thanks for your link, I'll read it. $\endgroup$
    – user51121
    Dec 4, 2020 at 14:41
  • $\begingroup$ The inequality set up is actually explained below that "is a general norm of a factor, depending on the modeler's preference, it can be" and it shows how they work analytically below it. It links to another paper if you look at that reference. References 4-6 dl.acm.org/doi/10.1016/j.cor.2006.02.011 $\endgroup$
    – Ryan Howe
    Dec 4, 2020 at 14:45

1 Answer 1

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It says in the paper you linked. If a matrix is symmetric positive definite then it has a Cholesky decomposition. It is the LU decomposition but for symmetric positive definite matrices.

It says on page 848

...Let $\Sigma$ be the covariance matrix of the single period returns. Given the $\Sigma$ is asymmetric and positive definite, it has a Cholesky decomposition and we can compute matrix $C = (\Sigma^{-1})$

It then sets up a relationship for the correlation between the asset returns

$$ \bigg\| \mathbf{C} \bigg( \tilde{\mathbf{R}}_{1} - \hat{\mathbf{R}}_{2} \bigg)\bigg\| \leqslant \mathbf{\Gamma} $$

It is quicker than the standard LU decomposition and there is a connection between it and other decompositions.

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  • $\begingroup$ Thanks again for your answer. What I don't understand is the reason why he uses the decomposition. Maybe to convert $\sum$ in a linear system? $\endgroup$
    – user51121
    Dec 4, 2020 at 14:47
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    $\begingroup$ There is a system of inequalities below that and they show analytically how that inequality looks when you use different norms. Further below it says they used linear programming. $\endgroup$
    – Ryan Howe
    Dec 4, 2020 at 14:54

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