Questions tagged [distribution]
The distribution tag has no usage guidance.
156
questions
0
votes
0
answers
34
views
Probability Theory: Maximizing the difference between distribution functions
Given a sample of observations $X$, by changing a parameter $p$ we can divide $X$ into two subsamples $X_1$ and $X_2$ (this division is done in a non-trivial way which is nonetheless irrelevant to ...
- 21
4
votes
0
answers
97
views
Modeling orderbook shapes as distribution
What are different distribution models typically used for generating orderbooks under high volatility, illiquidity, and multiple exchanges with different fees?
- 61
1
vote
1
answer
63
views
How to find out if Asymmetric Laplace Distribution is having Finite/Infinite Variance?
I was fitting the NIFTY 50 Daily Log Returns (To be more precise Returns in this case refers to the Log of 1+Returns rather than Log of Returns as Log cannot be taken of negative values which returns ...
- 91
0
votes
1
answer
133
views
Student-t measure of return volatility and time scaling
I have a series of price returns of an asset (4 days worth of data). They are relatively high-frequency.
My ultimate goal is to calculate realized volatility, but using a student's t-distribution.
I ...
1
vote
2
answers
223
views
Can I apply the Kelly criterion directly, without fitting any distributions?
Problem
I want to apply the Kelly criterion to asset returns, so that I know how much to hold of each, ideally (and how much I should keep as a cash reserve).
As far as I understand the Kelly ...
- 123
0
votes
1
answer
77
views
Pareto comparison of return distributions
In making a choice among financial strategies, each of which has some estimated return distribution, some strategies will clearly be better than others. But many times, the choice is a question of ...
1
vote
1
answer
384
views
Integral of brownian motion wrt. time over [t;T]
From the post Integral of Brownian motion w.r.t. time we have an argument for
$$\int_0^t W_sds \sim N\left(0,\frac{1}{3}t^3\right).$$
However, how does this generalise for the interval $[t;T]$? I.e. ...
- 528
0
votes
0
answers
46
views
How to compute the combined probability of loss for 2 time series (consisting of historical stock prices)?
May I please ask the community's support with the following problem?
I have 2 time series, with approximately 1000 observations each (same number of observations for both). They represent the daily ...
0
votes
1
answer
127
views
Terminal wealth distribution from dollar cost averaging
If monthly stock market returns follow an IID lognormal distribution, the terminal wealth distribution of investing a lump sum for many years is also lognormal. What is the terminal wealth ...
- 594
1
vote
0
answers
44
views
Assymetric Rate Distribution
The pandemic has disavowed any notion of nominal rate distributions to being truncated at 0%. However, if Central Banks at Debtor nations are conflicted in that they are incented to suppress interest ...
- 5,652
0
votes
1
answer
146
views
Why stock prices changes don't follow Pareto Distribution?
I calculated the distribution of the stock price changes (diffs). The diffs are multiplicative, $d_t=p_{t} / p_{t-1}$.
As far as I know the distribution should look like Power law distribution (Pareto ...
- 241
0
votes
0
answers
102
views
basic numerical integration question related to case of high positive volatility skew
is the below equation true irrespective of if that 2nd derivative turns out to be negative or >1 , (ie even if theres an arbitrage) ?
the reason i ask is that i am writing a single asset montecarlo ...
- 766
1
vote
1
answer
244
views
Fat tailed can be estimated through a t-distributions?
I have a simple question that makes me doubt a bit.
In a multiple choise exam I ecountered this question:
"if the stocks returns are not normally distributed, the fat tail effect can be estimated ...
- 11
9
votes
0
answers
321
views
On a time integral of Brownian motion up to the hitting time
Just come up with a 'simple' and interesting problem that I've been struggling to deal with for some time. Consider a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t\in[0,T]},\...
- 91
0
votes
2
answers
73
views
Creating a set of histories that satisfies certain statistics
I'm looking at a download of BlackRock's capital market assumptions, which gives a bunch of statistics, such as expected and quartiles for asset classes' returns for different timeframes, volatilities ...
- 175
4
votes
1
answer
171
views
Reconciling Two Claims About Volatility Under Fat Tails
I have read the Wikipedia article on volatility, and Nassim N. Taleb's Incerto, and found two statements attributed to Mandelbrot's views, which appear to be in contradiction.
Taleb (who was mentored ...
- 143
0
votes
2
answers
235
views
Estimating distribution of rate of return
Let $f[t]$ be the price of a stock at time $t$. We can calculate the rolling rate of return of the stock in a window of length $n$ by computing:
$$r[t] = \frac{f[t] - f[t-n]}{f[t-n]}$$
$r[t]$ is ...
0
votes
0
answers
572
views
Probability Distribution at each Simulation Period using Geometric Brownian Motion
I am using the equation $S_t = S_0e^{(\mu-\frac{\sigma^2}{2})t+\sigma\epsilon\sqrt{t}} $ to simulate a financial metric at each $t$, where $t=1$ and $T=5$. Stated in plain English, I am trying to ...
- 75
2
votes
0
answers
75
views
non gaussian distributions with higher moments and time scaling properties?
If we assume a portfolio comprised of n asset classes, whose log returns can be modeled with a distribution. I am interested in finding a distribution that:
incorporates higher moments (skewness and ...
- 21
1
vote
0
answers
40
views
How can I find the distribution function of the following random variables?
Suppose that the random variables $Z_i$ are defined as follows:
\begin{equation}
Z_i = D(0, t_i)(R_{i-1} +c)\Delta N,
\end{equation}
where $D(0, t_i)= \exp\{-\int_{0}^{t_i} r_u du\}$ for which $r_u$ ...
- 409
1
vote
2
answers
259
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
0
answers
114
views
The distribution of the jump diffusion process
In the Merton jump diffusion model the process of the share price can be expressed as $$S_{t}=S_{0}\cdot\exp\left\{ X_{t}\right\} ,$$ where $$X_{t}=\mu t+\sigma W_{t}+\sum_{i=1}^{N_{t}}Y_{i}.$$
Here $...
- 183
0
votes
1
answer
121
views
Calculating the Value-at-Risk when changing the confidence level
If I have a VaR estimate at a 95% confidence interval is 10, how do I calculate the approximate level of the VaR if the confidence level was raised to 99%, assuming a one-tailed normal distribution?
- 15
-2
votes
1
answer
94
views
Should stock return series be modeled with a parametric distribution, or an autoregressive function? [closed]
If I have prior knowledg that a stock return series follows a parametric distribution, such as a Student t-distribution with 4 degrees of freedom, without actively looking for prior knowledge of ...
- 2,970
2
votes
1
answer
179
views
What does this absolute return distribution chart show?
I was reading some pages in Professional Automated Trading by Eugene Durenard when I came across this chart:
The caption says: "S&P Absolute Return Distribution: Log-Log Scale".
The ...
- 521
0
votes
1
answer
88
views
FX spot distribution with student-t returns
If I am modelling my returns as $\sim N(0, \sigma^2)$, then I can evolve my spot distribution as: $$S_{t} = S_{0}e^{(\mu - \frac{1}{2}\sigma^{2})t + \sigma dW_{t}}$$
where $S_{0}$ is the spot, $\mu$ ...
- 141
0
votes
0
answers
402
views
What should degrees of freedom $\nu$ be set to when modeling financial returns that follow the t-distribution?
The closer the t-distribution degrees of freedom ($\nu$) is to 0, the more heavy are the tails, whereas high degrees of freedom recovers the normal distribution.
In finance, what value is usually used ...
- 2,970
3
votes
0
answers
283
views
Large deviations theory in finance
In probability theory, the theory of large deviations concerns the asymptotic behavior of remote tails of sequences of probability distributions.
A related post says:
Large deviations theory is ...
- 2,970
1
vote
3
answers
181
views
Do portfolio mean and portfolio variance have probability distributions?
If $X$ is a $T\times N$ matrix of multivariate asset returns,
and $w$ is some optimal portfolio weight vector,
then the portfolio return series is $r_p = X w \in\mathbb{R}^{T}$. This return series ...
- 2,970
2
votes
0
answers
682
views
law of absolute of max of brownian motion
What is the law of $\max\left(|B_t|\right)$ for $t$ in $[0,T]$ and $B_t$ is a Brownian motion?
Any references for properties of this process?
- 131
1
vote
1
answer
125
views
Does Value-at-Risk have any mathematical equivalence to copulas?
Portfolio Value-at-Risk estimated using the copula approach often just means generating artificial data sampled from a parametric copula('s joint multivariate distribution) as a model fit over the ...
- 2,970
5
votes
3
answers
472
views
What is the distribution of the risk-free asset?
If the risk-free asset has a volatility of $0$, therefore making its mean equal to the risk-free rate, $r_f$, does this mean that it has no probability distribution, and therefore there is no reason ...
- 2,970
0
votes
1
answer
131
views
How important is the chronological ordering of historical returns?
The returns of asset $A$ in chronological order are
0.03
0.01
-0.04
0.02
0.05
-0.10
0.02
The expected return, or sample mean, is $-0.00143$ while its sample ...
- 2,970
3
votes
2
answers
248
views
Interpretation of a uniform asset return distribution
Typically asset return distributions are bell-shaped with most mass occurring in and around the center, 0% returns, and less so in the tails, with the left tail representing the probability of large ...
- 2,970
0
votes
1
answer
167
views
Monte Carlo approach and methods for generating random returns
Recently I found myself reading more about Monte Carlo approach in m.v. portfolio optimization framework.
I already discuss the topic on this forum (if interested please consider the following links - ...
- 359
3
votes
1
answer
423
views
Minimizing variance vs. expected shortfall: distributions where the difference is salient
In portfolio theory in finance, given a set of $n$ assets to choose from, one often selects portfolio weights so as to maximize expected return and minimize some measure of risk, e.g. variance or ...
- 2,632
0
votes
3
answers
234
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 ...
- 2,970
2
votes
1
answer
212
views
Is it always better to use the entire distribution of a financial returns series, not just $\mu$ and $\sigma$?
In finance models that use historical returns for inputs, including option pricing models, forecasting and portfolio optimization, only the statistical moments of the returns distribution, $\mu$ and $\...
- 2,970
0
votes
1
answer
208
views
Which financial time series have a PDF and/or CDF?
Consider the following types of financial time series for a single publicly-listed stock:
Price data
Log returns
Cumulative returns
Each is computed from the item listed before it: log returns are ...
- 2,970
0
votes
0
answers
63
views
Density of a portfolio's returns is the weighted average of asset distributions?
The expected return of a portfolio can be formulated as a weighted average of the constituent assets' returns:
$$r_p = w_1 r_1 + w_2 r_2 + \dots + w_N r_N + \epsilon$$
Does it also follow that the ...
- 2,970
1
vote
1
answer
333
views
Why do cumulative returns have a bimodal distribution?
Regular returns (log-differenced prices) have statistical distributions that are bell-shaped and unimodal (one mode/peak) despite being non-normal and fat-tailed.
Cumulative returns, on the other hand,...
- 2,970
0
votes
1
answer
147
views
Option implied distributions
I am having a bit of trouble understanding how to obtain the option implied distributions.
I have strike levels, deltas and implied vols for a call option that expires in 6 months. Roughly 40 data ...
user38671
1
vote
1
answer
111
views
Kurtosis of a straddle
I want to determine the kurtosis of a straddle. My question is closely related with the following topic here.
According to the following paper of Ben-Meir and Schiff (2012) the expected value of a ...
- 73
4
votes
1
answer
225
views
Change of measure
I am looking at the derivation of the Hill estimator. It is $ \bar{F}(x) = 1 - F(x)$ the right tail of the distribution. In the derivation they use the equation
$$ \frac{1}{\bar{F}(u)}\int\limits_u^\...
- 103
0
votes
5
answers
367
views
Why worry about fat tails, if you can use stoploss?
Sorry this might sound a silly question, but -humbly- I don't understand why models assume that returns range from [-∞,+∞] instead of [-stoplimit, +takeprofit].
A common objection to most models is "...
- 450
1
vote
0
answers
91
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
0
votes
1
answer
50
views
What does 'near term order flow to be distributed across short term options' mean?
Please see the red phrase below. Guide to Option Pinning at Options Expiration | Investing With Options
What Have Weekly Options Done To Pinning?
That's a great question for a graduate student ...
user31928
0
votes
0
answers
44
views
Degree of freedom input for Monte Carlo simulation of asset returns with multivariate t distribution
How do I calculate or estimate the degrees of freedom in order to perform a Monte Carlo simulation of asset returns with multivariate t distribution using R functions? I am able to calculate the mean ...
- 13
1
vote
1
answer
450
views
Value at Risk (VaR): Normal distribution with gamma distributed volatility
If I was to do a 99% VaR calculation on a portfolio with normally distributed returns $\mathcal{N} (\mu,\sigma)$, the 99% VaR would be $\mu - 2.33\sigma$.
Instead of having a constant volatility, let'...
2
votes
1
answer
93
views
Sampling from an empirical distribution
I want to sample from the empirical distribution of returns. To do so, I do not want to make the preliminary assumption of which distribution the returns follow, rather I would like to sample from the ...
- 791