Questions tagged [random-variables]
The random-variables tag has no usage guidance.
41
questions
1
vote
1
answer
249
views
On first and last zeros before t in a Brownian Motion
Suppose we have the following random variables, given a fixed $t$ we define the last zero before $t$ and the first zero after $t$:
\begin{align*}
\alpha_t &= \sup\left\{ s\leq t: B(s) = 0 \...
3
votes
1
answer
201
views
Given $\mathbb{E}[X]$, $\mathbb{E}[\max(0,X)]$, and $\mathbb{E}[\min(0,X)]$, what is $\mathbb{E}[f(X)]$
Let $X$ be any random variable with any distribution. Given that we know $\mathbb{E}[X]$, $\mathbb{E}[\max(0,X)]$, and $\mathbb{E}[\min(0,X)]$, can you write a formula for $\mathbb{E}[f(X)]$ where $f$ ...
- 143
1
vote
0
answers
61
views
Sharpe ratio and uniformly distributed random portfolio
I am currently working on this paper which derives the Sharpe ratio distribution of uniformly random porfolios:
https://www.researchgate.net/publication/...
- 23
5
votes
2
answers
196
views
Expectation of integral where one of limits of integration is a random variable
Is it correct to write
\begin{equation}
E_t \int_0^{X_T} f(z) dz = \int_0^\infty \left(\int_0^x f(z) dz \right) p(x)dx \,\,?
\end{equation}
Here $X_T$ is a positive random variable with density $p(x)...
user34971
1
vote
2
answers
241
views
Taleb's Black-Swan: interpretation of the exponent
I am reading Taleb's "Black Swan" (revised 2020th edition). In chapter 16 "The Aesthetics of Randomness" he describes the meaning of the exponent in the context of extrapolation. ...
- 183
2
votes
1
answer
206
views
Covariance matrix for multiple assets - Second attempt
Ok, on the advice of administration I open a new question, hoping that in this way it becomes clearer.
Like I said before, I am trying to understand how the authors of this (page 76) and this (page ...
- 21
1
vote
0
answers
84
views
Concentration of measure phenomena in financial mathematics
Concentration of measure is a small area of statistics and probability theory that proved inequalities regarding the statistical properties of sets of random variables that exclude one of those random ...
- 2,935
4
votes
1
answer
286
views
sub-Gaussian random variables in financial economics
Unlike financial time series that typically possess fat tails, sub-Gaussian random variables have strong decay in the tails of their distribution. do sub-Gaussian random variables or processes appear ...
- 2,935
1
vote
2
answers
242
views
Exchangeability of random vector
I hope you can help me with this rather basic question that I asked myself. A random vector $(X_1,...,X_n)$ is said to be exchangeable if it has the same distribution as the permuted random vector $(...
- 181
0
votes
3
answers
219
views
Are mean-variance efficient portfolio weights random variables with probability distributions?
The mean-variance model outputs a portfolio weight vector whose elements are individual asset weights that sum to 1. Regardless of which portfolio along the efficient frontier is being solved, the ...
- 2,935
1
vote
0
answers
85
views
Full Copula View using Meucci's Full Flexible View
I'm currently setting up an "Investment Framework" that should allow the following steps:
Investment Committee (IC) has to decide on probabilities for 4 different market states. I have historical ...
- 245
1
vote
0
answers
125
views
Simulate correlated Brownian motions conditioned on future state(s)
Consider a model defined by 2 geometric Brownian motions
$$dY_{1}(t) = \sigma_{2} Y_{1}(t)dW_{1}(t)$$
$$dY_{2}(t) = \sigma_{2} Y_{2}(t)dW_{2}(t)$$
with $Y_{1}(0) = y_{1}$, $Y_{2}=y_{2}$ and $dW_{1}(...
user40884
1
vote
0
answers
254
views
Process Transforms (Fractional Difference)
Let's say I have a process $X_t$ with unknown variance process $V_t$.
Then, I write $\mathrm{EMA}[X_t]$ to be the 5 sec exponential moving average of $X_t$.
Consider the transformation $$\sum (X_t-\...
- 21
0
votes
1
answer
149
views
Generate Random Variable Using Acceptance Rejection Method
I have a question about acceptance rejection method and really appreciate your advice:
Suppose we want to generate random variable that has probability density function $f(x)$, since we're using ...
- 607
1
vote
0
answers
31
views
is it possible to make changes to use the affine property of Normal random variables, rather than the Central Limit Theorem?
I have proven the distribution of a discrete time model, evolving over a uniform mesh with $\delta t = T/L$ is given by
$$S(t_{i+1}) = S(t_i) + \mu \delta t S(t_i) + \sigma\sqrt{\delta t}S(t_i)Y_i,$$
...
- 25
2
votes
1
answer
26
views
How would I develop confidence bounds for a function of 3 random variables, 2 of which are correlated?
I am tasked with developing confidence intervals for the function x = 1 - |(a+b)/c|
where a, b and c are random variables. a and b are normally distributed, but c is heavily skewed left. further ...
- 21
1
vote
2
answers
121
views
How to create a volatile market, by combining less volatile markets?
This might be against the law of gravity, but I'll give a try 🙂
Is there a way to combine two financial products $p_1$ and $p_2$, into a single product $p_c$ that is more volatile than its ...
- 450
3
votes
3
answers
284
views
Getting sets of random correlated variables
For the training of a machine learning model I need to add additional features (macro variables), and these features are correlated. I need to run the model N times, and for each time I have to add ...
- 40
0
votes
0
answers
44
views
Can we generate(in high dimension) uniformly distributed variables in a finite volume other than a cube?
I'd like to know if there is in the literature a (computationally cheap) algorithm to generate uniformly distributed variables in high dimension for a volume other than a cube and without using ...
- 291
2
votes
0
answers
42
views
Convolution of generalized hyperbolic distribution
I have a question concerning the convolution of generalized hyperbolic distributions.
Proposition 6.13 of McNeil, Embrechts, Frey states the following:
If $X$ has a $d$-dimensional generalized ...
- 1,406
3
votes
2
answers
55
views
What are the underlying events that the random variables map to the real line in the derivation of the Black-Scholes PDE?
When we first try and set up a model for the evolution of S, the value of the underlying stock, I have seen in a lot of textbooks that they model the evolution by the formula $$\frac{dS_t}{S_t}=\mu dt+...
- 235
1
vote
1
answer
124
views
Computing Montecarlo VaR for a single asset
I'm trying to understand the procedure to compute the Value-at-Risk for a single asset by implementing the Montecarlo technique.
Here it follows the procedure step-by-step in 5 points:
selecting the ...
- 2,466
1
vote
1
answer
409
views
Probability density function of the sum of two independent Levy-distributed random variables?
I posted the following questions in math stack exchange
https://math.stackexchange.com/posts/2762047/edit
Here's the text:
Prove that the sum of two independent Levy-distributed (having parameter $c$)...
- 1,012
2
votes
2
answers
204
views
Evidence that supports the assumption that prices are random processes
I have heard that the price of stock or future changing over time is a random process, namely, a martingale, and no one can have an edge. Is there any evidence supporting this assumption?
Why do so ...
4
votes
2
answers
204
views
Optimal number of iterations for quasi-Monte Carlo
I'm quoting from Peter Jäckel's book "Monte Carlo Methods in Finance", page 96:
...For low-discrepancy numbers, the situation is different. Sobol numbers and other number generators based on ...
- 316
1
vote
3
answers
1k
views
Generating a random covariance matrix with variances in range
I would like to generate a random covariance matrix with variances in certain range.
How can it be done? (In R if possible)
I tried to generate a lower triangular matrix $L$ where the diagonal $D = ...
- 131
0
votes
1
answer
89
views
Tools to measure the randomness of database
I was working on historical data looking for anomalous patterns that we would not expect to occur at random. I'd like to create a scheme to analyze data and markets to test for statistical ...
- 175
-1
votes
3
answers
164
views
Is there a stochastic equation which can model returns according to its four moments?
The normal stochastic equation only models mean and standard deviation.
For now, I'm randomly picking returns from a historical CDF of the returns. I'd like to have some flexibility when it comes to ...
2
votes
2
answers
6k
views
Box-Muller Method Proof
Here we want to show that the Box-Muller method generates a pair of independent standard Gaussian random variables. But I don't understand why we use the determinant? For me when you have two ...
- 23
7
votes
1
answer
9k
views
How to simulate a jump-diffusion process?
I would like to price Asian and Digital options under Merton's jump-diffusion model. To that end, I will have to simulate from a jump diffusion process.
In general, the stock price process is given ...
- 431
2
votes
1
answer
73
views
Expected number of days inside a corridor
Is there a simple (ish) approximation for the expected number of steps a random walk is within a set of bounds over a given time period? - in particular if i presume log normal and constant vol.
If i ...
- 2,521
1
vote
0
answers
167
views
GARCH model why we take assumption that returns arei.i.d. random variable? [closed]
In GARCH model why we take assumption that returns are i.i.d.?how can we explain it to a layman?
- 11
2
votes
1
answer
4k
views
How to use Halton sequence in monte carlo simulation
Does anybody know how to use the Halton pseudo random technique in monte carlo simulation. I'm able to generate the sequences and I know they are correct. I checked a couple of numbers from different ...
- 345
6
votes
1
answer
368
views
(Re) normalisation of random variable in Monte-Carlo simulations
I have a very simple model (CIR) with a very simple discretisation scheme (Euler) and I use it to do Monte-Carlo Simulations. It is working.
Someone insisted that renormalization of my random ...
- 1,179
1
vote
1
answer
263
views
Effects of random-generator-choice on derivative's price
There is a plethora of pseudo-random-generators out there. Some of them are definetly better and some of them severily underperform.
My standard tool is Mersenne Twister - when I need to generate ...
- 3,357
11
votes
3
answers
345
views
Are there any standard techniques for adding realistic synthetic microstructure noise to a price series?
This may seem like a strange question, but for my particular application we need to actually add synthetic microstructure noise to real time charts. The signal should still be representative of the ...
- 241
18
votes
6
answers
3k
views
How to generate a random price series with a specified range and correlation with an actual price?
I want to generate a mock price series. I want it to be within a certain range and have a defined correlation with the original price series.
If I choose, say, oil, I want as many time series which ...
11
votes
2
answers
5k
views
How can I compare distributions using only mean and standard deviation?
I only have means and standard deviations of samples of two random variables. What technique can I use to determine how similar the distributions these describe are? Assume that the values are built ...
- 113
9
votes
2
answers
4k
views
Why doesn't Black-Scholes work in discrete time?
I have a question considering Financial markets in discrete Time.
One of the main theorems in discrete time is the following.
In finite discrete Time with trading times t={1,...,T} the following are ...
- 91
12
votes
1
answer
539
views
Do people use unbounded interest rate models, and what alternatives exist?
A simple interest rate model in discrete time is the autoregressive model,
$$
I_{n+1} = \alpha I_n+w_n
$$
where $\alpha\in [0,1)$ and $w_n\geq 0$ are i.i.d. random variables. When working with ruin ...
- 2,701
4
votes
1
answer
2k
views
Linear combination of gaussian random variables
I know what random variables are but I don't understand what a linear combination of gaussian random variables is. Can anyone please give me an explanation or clues?
Thanks in advance,
Julien.
- 121