5

Yes, this technique is called moment matching variance reduction and it may indeed lead to a form of variance reduction. The first and second order moments correspond to the mean and the variance of the distribution. You can extend to higher order moments, which is of course more difficult to implement and creates some extra overhead. The mean can be adjust ...


4

I finally got the idea behind the example. To illustrate it in a more general setting I will present a rigorous proof: Let $x_k$ denote the salary and $b_k$ the number of persons that earn $x_k$ or more. Following the proposed power law by Taleb, we have: $x_{k}:=x_02^k$ and $b_k:=b_0\left(\frac{1}{2}\right)^{ka}$, where $a\geq1$ and $k\in \mathbb{N}$. The ...


3

Actually there are many different approaches to prove randomness (academic) or disprove randomness (fund managers to persuade their clients or their bosses ;-) in financial markets. One approach I find especially interesting is based on algorithmic information theory. Basically what that does is to find an algorithm to compress financial data. The fewer ...


3

You have to distinguish (at least) two approaches: 1) derivatives pricing: Here you assume that there is a probability measure other than (but somehow tied to ) the real world measure - say $Q$. Under $Q$ your underlying is a martingale. Then pricing derivatives is calculating expectations. This measure $Q$ is linked to the principle of no-arbitrage. If ...


3

Here is one recipe, in case you can live with Spearman rank correlation. (Which you should: linear correlation is often not appropriate in the non-normal case. And in the normal case, there is almost no difference between the two correlation types.) Generate samples of your $k$ features with all the desired attributes. These samples may be random or ...


3

This looks to me like a range accrual. Let $t_1, \ldots, t_n$, where $0 < t_1 < \cdots < t_n$ be business days that are being considered. We compute \begin{align*} E\left(\sum_{i=1}^n \pmb{1}_{b_1 < S_{t_i} < b_2} \right) &=\sum_{i=1}^n E\left(\pmb{1}_{b_1 < S_{t_i} < b_2} \right)\\ &=\sum_{i=1}^n \left[E\big(\pmb{1}_{S_{t_i} >...


3

First I provide a brief description of Halton sequences. A Halton sequence is a deterministic sequence of numbers that provides well-spaced 'draws' from an interval and provides negative correlation between simulated probability for individuals. Generation is based on a prime number Sequence is constructed based on finer and finer prime-based divisions of ...


3

Mersenne Twister is currently the most used PRNG in the quant world. It was even incorporated in C++11 so it can be considered standard nowadays. Any PRNG with reasonable statistical quality shall perform well (equivalently) for pricing, so that differences relate more to convenience (speed, parallelizability etc..). If the statistical quality is poor then ...


3

A useful decomposition is, in R's matrix notation, V = S %*% C %*% S, in which S is a matrix with the standard deviations on the main diagonal and zeros elsewhere, and C is the correlation matrix. (See this note on Matrix Multiplication with Diagonal Indices.) To get a meaningful V, you need to have C positive (semi)-definit. A simple way to achieve this is ...


2

I haven't heard of a method to do it your way. Usually, you start with covariance matrix and do Cholesky in order to be able to generate random draws from the multinomial normal distribution with given covariance matrix. Maybe that is what you want? In any case, if you need precisely specify the variances, see: Hirschberger, M., Y. Qi, & R. E. Steuer ...


2

It can be seen that $Y_1^2+Y_2^2=-2\log{X_2}$ and that $Y_2 \over Y_1$ $=\tan(2\pi X_1)$. Therefore $X_1={1 \over{2 \pi}}{\arctan{Y_2 \over Y_1}}$ and $X_2=\exp{-(Y_1^2+Y_2^2) \over 2}$. Taking differential to get $dX_1= {1 \over{2\pi}}{{-Y_2dY_1+Y_1dY_2} \over{Y_1^2+Y_2^2}}$. Similarly, $ dX_2= {\exp {-{Y_1^2+Y_2^2} \over 2}(Y_1 dY_1 + Y_2dY_2 )}$. ...


2

Much of what follows can be found in Glasserman (2003), Chapter 5, Monte Carlo methods in financial engineering. The reason for using low discrepancy numbers is because they are somewhat "equidistributed", meaning that you can guarantee that they fill the unit interval in a regular fashion without having large gaps. (the same is true for the unit square, or ...


2

Cholesky (or SVD or any other approach based on matrix multiplication) only works for normal distributions, which your features cannot be, given that they have values within finite intervals. To see why Cholesky does not work, assume two additional features, which are independent uniform $(U_1,U_2)$. Now you want to create features with correlation $\rho$ ...


2

You approach sounds good, but there is no need to compute the inverse Fourier transform. The characteristic function of a Levy-distributed random variable with parameter $c$ is given by \begin{equation} \phi(\omega; c) = \exp \left\{ -\sqrt{-2 \mathrm{i} c t} \right\}. \end{equation} Thus, the characteristic function of the sum of two i.i.d. Levy-...


2

Let $X_1$ and $X_2$ be your two assets and $C$ your financial product. For now we only assume products which are a linear combination of $X_1$ and $X_2$ with no shorting allowed, hence: $$\begin{align} & C=\alpha X_1 + (1-\alpha)X_2 \\ & 0\leq \alpha \leq 1 \end{align}$$ Letting $\rho$ be the correlation between returns $r_1$ and $r_2$, we have: $$\...


2

Throw in correlation as the additional variable. Similarly, volofvol could be another candidate to play with. A spread option could have a larger volatility than either of the two underliers.


2

The original mean-variance model was static and assumed that the mean vector $\mu$ and covariance matrix $\Sigma $ are known. These determine the optimal portfolio weights that in this case are deterministic as well. However, in practice people do two types of modifications. First because these means and covariances are generally time-varying, we instead use ...


2

They are independent. The point is that $y$ is derived from your easily sampled distribution $g$ randomly. Now you have a random test (via $v$) that decides whether to accept $y$ or not as part of the random sample of the harder to sample $f$. The procedure uses $M$ in the accept-reject method and whilst you can derive conservative estimates with $M$ quite ...


2

@Wombat: I think it's best to think of it this way. Suppose you have a time series and the joint distribution of $n$ elements ( in a certain, specific order) of the series is normal with mean zero and some non-diagonal covariance matrix $\Sigma$. Then, in this case, exchangeability means that you could take the $n$ elements in any order and the joint ...


2

Here, I try to help a bit with the matrix norm question. Assume an $M$-dimensional multivariate normally distributed return vector $\widetilde{R}$. The covariance of these returns is $\Sigma$, and the expected returns are $\check{R}$. The probability density function of $R$ is $$ f(R;\check{R},\Sigma)=\left(2\pi\right)^{-\frac{M}{2}}\left|\Sigma\right|^{-\...


2

I don't know how to interpret the above example, but wealth distribution, of which inequality is one of the measures, is frequently described by the Pareto distribution. Also, the IRS publishes annual income distribution statistics which is a good source to check the narrative. Top 1% earns some 20% of all income.


1

let's take X1=1 and X2=2: The covariance matrix is zero (X1 and X2 are independents) and X1, X2 have different distributions!!!


1

In the original Markowitz papers, no. In the so called 'resampled efficiency' or 'resampling frontier' method by Michaud, the weights are recalculated over and over from perturbed versions of the covariance matrix, to account for the fact that the covariance matrix is not known exactly (estimation error). In this case yes, the weights are random variables.


1

There are ways that you might think of portfolio weights as estimates and thus random variables. If you are working with the optimizer, you may be able to get the inverse Hessian out of it. If so, that can be used to get you estimates of the standard errors of your portfolio weights. One big caveat though: any weight with a binding constraint will likely ...


1

Confidence intervals are applied to estimates to give a sense for potential error. If a, b, and c are R.V. as you described, they're independent by nature, making confidence intervals irrelevant. Either you're misunderstanding what was asked of you or your problem is misspecified (or you just misspecified it here).


1

I posted a free self contained excerpt of my book Modern Computational Finance that explains Sobol's sequence and in particular its Latin Hypercube property, meaning that each axis is sampled evenly but in a different order for different axes, as long as the number of samples is a power of 2 minus 1. I hope it helps: https://medium.com/@antoine_savine/sobol-...


1

Since the correlation matrix $C=LL^{\top}$ is also $C=U^{\top}U$, where $U$ is the upper triangular matrix, rather than $L$ the lower triangular matrix, you can transform an uncorrelated features matrix $F$ containing features 1, 2 and 3 in its columns by multiplying this $F$ matrix with $U$, giving the correlated features matrix $F_c$: $$F_c = F U $$ In ...


1

Think of it this way, $\Omega$ elements are the states of the world. This means the set of {macroeconomic data, geopolitic situation, weather, fundamentals of the firms, sentiment of market participants, etc.} and anything else that might have any influence on the stocks prices... As Ezy explained, what makes probability theory useful is that it allows us ...


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