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What does 'simulate a covariance matrix' mean? If the question means, generate an arbitrary correlation matrix for 1000 stocks, then we can choose any symmetric matrix with all 1s down the diagonal, so long as every element is between -1 and 1 and the matrix is positive semi-definite. The large size of the matrix means that putting random values in every ...


5

If you assume that a financial asset price has a change that is a wiener process then you can view the future value of that asset as the initial value plus the sum of the independent daily changes (for equity or returns based then you would need log version of this): $$ S_t = S_0 + \sum \Delta S_i $$ where $\Delta S_i = S_i - S_{i-1} $ is a wiener process. ...


4

Just to expand on Alex answer. Empirically it is simply not true. Focusing on the diagonal of the variance-covariance matrix, we know that there is a large variance risk premium. Take a look at table 3 from Carr and Wu (2009). Regarding covariances we do not have much evidence, because there are no options on every single pair of stocks. However, we do know ...


3

I'm in no way a portfolio theory expert, but the negative of a convex function is concave and vice versa. You can look at minimizing a concave function as maximizing a convex function and vice versa. Also, the optimization problem is over the weights, and not over densities (which variance is concave in as your link shows). Portfolio variance is convex in ...


3

OK, so think of it this way... Your standard (Markowitz) covariance matrix is a sample observation. That may or not be close to the population sigmas and correlations of your sampled markets. Even if close, the sample-vs-population errors will create asset allocation errors. The identity matrix here is the "complete strategic ignorance" covariance ...


3

You can think of it in Bayesian terms. To start with, knowing nothing at all about stocks, you might assume that stock returns are i.i.d with unit variance. This would be your prior. It is very simple and is well behaved because the identity is invertible. Then you would gather some empirical data on stock returns and measure the actual variances and ...


3

I assume you found these weights by Markowitz Optimization? It is quite common that MVO will deliver extreme weights with some weights well above 100 percent (implying leverage, i.e. buying the stock on borrowed money) and others massively negative meaning a leveraged short position. These weights are not usable in a real portfolio. Let's examine the first ...


3

Note that \begin{align*} \left\langle \int_0^t \sigma_s^1 dW_s^1, \int_0^t \sigma_s^2 dW_s^2\right\rangle &= \int_0^t \sigma_s^1 \sigma_s^2 d\langle W_s^1, W_s^2 \rangle\\ &=\int_0^t \rho_s\sigma_s^1 \sigma_s^2 ds. \end{align*}


3

I will just clarify Point 2 in StackG excellent answer. (It's really a comment, but it's too long and has too much math symbols to fit in the comment field.) Suppose you're given a covariance matrix $C$ for the returns of $n$ assets. (1000 $\times$ 1000 is 1 million entries - should not be too large for modern computers to work with, but do be mindful of ...


3

I think what you are effectively looking at is $$\ \begin{align} \log(S_{AUDCAD})&=\log(S_{AUDUSD})\pm\log(S_{USDCAD})\\ \Rightarrow z&=x\pm y \end{align} $$ Thus, $$ \sigma_z^2=\mathrm{E}\left(\left(x\pm y\right)^2\right)- [\mathrm{E}(x\pm y)]^2 =\sigma_x^2+\sigma_y^2\pm 2\sigma_{xy} $$ Hence, $$ \tag{1} \sigma_{xy}=\frac{\sigma_z^2-\sigma_x^2-\...


2

As an addition to the already rich answers, I would suggest you to read the following paper by Marcos L. De Prado on the computation of Forward-Looking Correlation Matrices. Estimation of Theory-Implied Correlation Matrices https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3484152


2

The Ledoit-Wolf estimate cited by @develarist can be quite good, but as you say you already knew about "shrinking". It takes the population of correlations observed as an effective Bayesian prior for any given correlation, so it sort of inherently assumes that all pairs are similar an some sense. It would not work well, say, with known block sets of highly ...


2

Quantile regression is considered a robust procedure but lacks the quality of being fully differentiable. There are also regularized regression models like ridge regression, lasso regression and elastic net regression that implicitly consider the covariance of the data like OLS, but additionally reduce volatility in estimates through the introduction of bias....


2

For the same reason you can't meaningfully measure covariance/correlation using price of individual assets...correlation (covariance by extension) represents the comovement in deviations from individual means. You can't represent that if the mean continues to change (ie, series considered aren't stationary). Same goes for multiple assets as is represented ...


2

In theory, the Ledoit and Wolf shrinkage estimator is supposed to guarantee a positive-definite matrix, given that it adds a positive-definite matrix (the target) to a semi-positive one (the sample covariance). I can see four reasons why you didn't get a positive-definite matrix: Your true covariance is effectively not full rank, i..e you have perfect ...


2

Note that the function $f$ only depends on $|t-u|$, meaning it is actually symmetric: $f(x)=f(-x)$. Doing the change of variable $\tau:=t-u$: $$\begin{align} \int_0^Tdu\int_0^Tf(t-u)dt &=\int_0^Tdu\int_{-u}^{T-u}f(\tau)d\tau \\ &=\int_0^Tdu\left(\int_0^{T-u}f(\tau)d\tau+\int_0^uf(\tau)d\tau\right) \\ &=\int_0^T{du \left(\int_0^T{f(\tau) \textbf{1}...


2

The interpretation and units problem, ie the lack of an easily intuitive answer, is precisely why quants/econometricians etc. tend to shy away from talking too much about covariances [even if they are absolutely necessary; and frequently used]. Thus if anything involving covariances has to interpreted, let alone explained, the default is usually to express ...


1

Hi: Exponential smoothing weights observations by taking a weighted combination of the old estimate and the new. So, if you denote your original matrix ( or current covariance matrix ) as $R_t$ and your new one as $R^{*}_t$, then exponential smoothing does $R_{t+1} = \lambda R_{t} + (1- \lambda) R^{*}_t $. But there are two issues with doing this update. ...


1

First, a correction is in order: the math question you cite is the variance for a Bernoulli random variable as a function of the parameter $p$. That is, indeed, concave in $p$. However, the variance of a portfolio, $w^T\Sigma w$, is not concave in $w$. So your initial presumption of concavity is not correct. For a Bernoulli random variable, the uncertainty ...


1

Generally weights larger than 1 do correspond to leverage https://www.investopedia.com/terms/1/130-30_strategy.asp Without knowing the context, my guess is that those numbers don't represent the total portfolio weights and are instead just a mathematical weight in which case yes, you can convert them. You could break out the long and short components so that ...


1

I don’t see how just calculation of Portfolio variance would need an invertible var-covar matrix, I mean you don’t even have to use the matrix notation to calculate it. It may be so that lower time frames would output unstable values of the individual variances and pairwise correlations. However there are certain methods to achieve a stable covariance ...


1

It does not matter whether you measure covariance of two portfolios or two securities, the formula is the same. Simply instead of returns and expected values for securities, put those for portfolios.


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I have actually considered the problem that you are working on, though configured somewhat differently. There isn't going to be a universal answer to your question. See, in particular, Holland, Paul W. Covariance Stabilizing Transformations. Ann. Statist. 1 (1973), no. 1, 84--92. Nonetheless, there are answers, some already mentioned. I would argue ...


1

This is not a complete answer, more a different perspective to the answers already given. If you have some a-priori knowledge about the covariance structure and about the factors influencing it, you should try to reflect this in your statistical model. Three ideas: Divide your sample into subpopulations with identical factor values and estimate separately. ...


1

Portfolio beta is a function of market vol, portfolio vol and correlation between market and portfolio; so correlation is indeed the only free variable. (But if you have complete control over the portfolio-construction process, you might as well target beta and then scale the weights so that you meet your volatility target.) To control correlation, you'll ...


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