7 votes
Accepted

Given $\mathbb{E}[X]$, $\mathbb{E}[\max(0,X)]$, and $\mathbb{E}[\min(0,X)]$, what is $\mathbb{E}[f(X)]$

I don't think one can answer your question. Suppose $X=e^{\mu+\sigma Z}$ is log-normal, i.e. positive. Thus, $\mathbb{E}[\max\{0,X\}]=\mathbb{E}[X] $ and $\mathbb{E}[\min\{0,X\}]=0$. From just knowing ...
Kevin's user avatar
  • 15.9k
4 votes
Accepted

Taleb's Black-Swan: interpretation of the exponent

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 ...
Philipp's user avatar
  • 183
4 votes

On first and last zeros before t in a Brownian Motion

Intuitively speaking, you generally have an event for which you do not know when it occurs (the time of the occurrence of the event is random), but you do know that it will occur at some point in the ...
Hans-Peter Schrei's user avatar
3 votes

Evidence that supports the assumption that prices are random processes

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 ...
vonjd's user avatar
  • 27.4k
3 votes

Evidence that supports the assumption that prices are random processes

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 $Q$. ...
Richi Wa's user avatar
  • 13.7k
3 votes

Getting sets of random correlated variables

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 ...
Enrico Schumann's user avatar
3 votes

Generating a random covariance matrix with variances in range

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 ...
Enrico Schumann's user avatar
3 votes

Expected number of days inside a corridor

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}...
Gordon's user avatar
  • 21.1k
2 votes
Accepted

Is there a stochastic equation which can model returns according to its four moments?

I suggest you have a look at the paper: Schloegel, Erik (2010) "Option Pricing Where the Underlying Assets Follow a Gram/Charlier Density of Arbitrary Order", Journal of Economic Dynamics and Control,...
LocalVolatility's user avatar
2 votes

Box-Muller Method Proof

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 ...
Vu Nguyen's user avatar
2 votes

Generating a random covariance matrix with variances in range

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 ...
Jan Sila's user avatar
  • 732
2 votes

Optimal number of iterations for quasi-Monte Carlo

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 "...
oliversm's user avatar
  • 1,389
2 votes
Accepted

Getting sets of random correlated variables

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 ...
g g's user avatar
  • 1,973
2 votes

How to create a volatile market, by combining less volatile markets?

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}...
Daneel Olivaw's user avatar
2 votes

How to create a volatile market, by combining less volatile markets?

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.
bhutes's user avatar
  • 996
2 votes

Taleb's Black-Swan: interpretation of the exponent

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 ...
Sergei Rodionov's user avatar
2 votes

Covariance matrix for multiple assets - Second attempt

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 ...
Kermittfrog's user avatar
  • 6,554
2 votes
Accepted

Exchangeability of random vector

@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 ...
mark leeds's user avatar
  • 1,102
2 votes
Accepted

Probability density function of the sum of two independent Levy-distributed random variables?

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{...
LocalVolatility's user avatar
2 votes

Are mean-variance efficient portfolio weights random variables with probability distributions?

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 ...
fes's user avatar
  • 1,727
2 votes
Accepted

Generate Random Variable Using Acceptance Rejection Method

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 ...
Attack68's user avatar
  • 10.1k
2 votes

Expectation of integral where one of limits of integration is a random variable

This is related to the integrated tail probability expectation formula: $$ X= \int_0^X dx = \int_0^\infty 1_{X>x} dx,$$ followed by $$ E[X] = E \left[ \int_0^\infty 1_{X>x} dx \right] = \int_0^...
ir7's user avatar
  • 5,053
1 vote

How would I develop confidence bounds for a function of 3 random variables, 2 of which are correlated?

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. ...
Chris's user avatar
  • 1,643
1 vote

Exchangeability of random vector

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

Are mean-variance efficient portfolio weights random variables with probability distributions?

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 ...
nbbo2's user avatar
  • 11.2k
1 vote

Are mean-variance efficient portfolio weights random variables with probability distributions?

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, ...
kurtosis's user avatar
  • 2,880
1 vote
Accepted

Optimal number of iterations for quasi-Monte Carlo

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 ...
Antoine Savine's user avatar
1 vote

Getting sets of random correlated variables

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 ...
develarist's user avatar
  • 3,000

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