Questions tagged [distribution]
The distribution tag has no usage guidance.
48
questions with no upvoted or accepted answers
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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]},\...
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votes
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Arbitrage free smoothing of volatility smile - cubic spline - implementation procedure
I am studying the paper Arbitrage-Free Smoothing of the Implied Volatility Surface, from Matthias R. Fengler (https://core.ac.uk/download/pdf/6978470.pdf).
The problem I want to solve is much simpler ...
- 81
5
votes
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1k
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Fitting Student t-distributions to log-returns
It seems that some tail-risk centric groups are bent on using Paretian and t-distributions to account for tail risk when fitting log-returns. It has been observed, however, that with and without ...
user6430
4
votes
0
answers
99
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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?
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4
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0
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143
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Alternative Method for Determining Option-Implied pdf
As I am refining a pricing model to incorporate skew, and not just ATM volatilities, I need to create random realizations of the underlying consistent with the skew-implied pdf. When searching, one ...
- 1,651
3
votes
0
answers
294
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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 ...
- 2,980
3
votes
0
answers
326
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Example how to model stock price with Pareto distribution according to Mandelbrot and Taleb
There's a paper by B. Mandelbrot and N. Taleb Mild vs Wild Randomness that says that Pareto distributions is a better fit for modelling price changes.
...
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3
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144
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What is the relation between return volatility and return rank volatility, and how can I control the latter?
I have no experience in finance, but I've been playing around with a virtual portfolio.
I'm trying to control the "rank volatility" distribution - that is, the volatility of a stock's daily rank in ...
- 221
2
votes
0
answers
76
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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
2
votes
0
answers
117
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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 $...
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2
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0
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743
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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
2
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0
answers
52
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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 ...
- 1,436
2
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0
answers
69
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$\int_{0}^1W_x(t)dW_y(t)/(\int_{0}^1W_x^2(t)dt)^{1/2}$ normally-distributed?
I have came across the following stochastic integrals:
$$\frac{\int_{0}^1W_x(t)dW_y(t)}{(\int_{0}^1W_x^2(t)dt)^{1/2}}$$
which was claimed to be standard normally distributed ($W_x$ and $W_y$ are ...
- 21
2
votes
0
answers
150
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Bivariate risk neutral distribution through copula
I want to build a bivariate risk-neutral distribution from two liquid assets (A and B) through the use of a copula. As A and B are liquid, I have the marginal distributions from the market. All I have ...
- 143
2
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0
answers
201
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Simulating t-distributed returns by calibrating degrees of freedom $\nu$ from variance or kurtosis
A slight twist (I hope) on the familiar problem of simulating log returns from a t-distribution. My two questions concern calibration to sample data.
First, one can infer the degrees of freedom, $\nu$...
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2
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0
answers
153
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Gaussian Copula with t margins
I am trying to fit a Gaussian Copula with t margins to my data (log returns of two stocks). It has already worked for a Gaussian Copula with normal margins with:
normcopula_dist = mvdc(copula=...
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2
votes
0
answers
103
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How to choose a window for curve fitting and prediction?
I am using Pareto distribution to fit a serie of survival rates (with least square).
My ultimate goal is to use this fitting curve for prediction. Thus I would mainly focus on the tail of the ...
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1
vote
0
answers
44
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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 ...
- 5,662
1
vote
0
answers
40
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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 $r_u$ ...
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1
vote
0
answers
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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 ...
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1
vote
0
answers
49
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Skewness and kurtosis measures when full distribution is not available
I have asked this question here, but did not get any answer. I was wondering if anybody knows a method of deriving skewness and kurtosis measures from different quantiles, mean, and/or variance. I do ...
- 1,830
1
vote
0
answers
53
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relationship between option vol and option payoff
Has anyone thought of the relationship between the option vol and distribution of option payoff? for example, I have 1000 paths of simulated underlying prices, keeping all inputs the same but only ...
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1
vote
0
answers
131
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Pricing call option on bond under CIR model by simulating noncentral chi square distribution
In the original paper of CIR model, there is a pricing formula about call option on bond
$$
\begin{array}{l}{C(r, t, T ; s, K)} \\ {=P(r, t, s) \chi^{2}\left(2 r^{*}[\phi+\psi+B(T, s)] ; \frac{4 \...
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1
vote
0
answers
32
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Sample distribution of cross-sectional statistics of returns
Currently doing an application of VaR on sample of industry portfolios in the US. I have a matrix of $n$ industry portfolios with $m$ time-series observations. I calculate cross-sectionally (for each ...
- 1,396
1
vote
0
answers
37
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Finding the distribution and moments of returns with GARCH models (in R if possible)
I understand the GARCH type models and I know how to fit a model to a time series. But, there is a paper which calculates the moments of the distribution of returns (Variance, Skewness, and Kurtosis) ...
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1
vote
0
answers
102
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Probability of outlier events for laplace distribution
I've read that the laplace distribution is better for forecasting purposes than the normal distribution due to it better accounting for fat tails. However, when I run the numbers in matlab, laplace ...
- 115
1
vote
0
answers
50
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Bootstrapping to Judge the Fit of a Sampled Return Distribution
Consider the following: I have sampled yearly stock returns from a specified distribution.
What I want to do is compare how well my sampled distribution fits the empirical distribution of yearly ...
- 137
1
vote
0
answers
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Question about Paul Kupiec's "concentrated Bond loss rate distribution"
I wonder if anyone here has read the following paper by Paul Kupiec in which he approximates a loss rate distribution for a portfolio composed of (possibly) concentrated bond positions.
https://www....
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1
vote
0
answers
335
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Probability Integral Transform: Standardisation
I've been applying the probability integral transform as shown here to standardise date for input into a neural network:
https://math.stackexchange.com/questions/592076/mapping-cdfs-to-each-other?...
- 879
1
vote
0
answers
308
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Expected shortfall of stable distribution by Stoyanov
I've been working on calculating parametric ES assuming the returns follow Paretian stable law. Given the four parameters - $\alpha, \beta,\sigma,\mu$- Stoyanov introduces closed form solution of the ...
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1
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0
answers
174
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Interpretation of Skew and Kurtoisis - strategy backtesting
I am working on my dissertation and i would like to provide a nice interpretation of two tables which i will present below.
I have 10 portfolio buckets which i sort on 6 different attributes. One of ...
- 210
1
vote
0
answers
295
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student-t asset path
I am trying to simulate an asset path based on a t-distribution. I found a lot of ressources and the fact that it will be difficult to do a path. But now I changed my Geometric Brownian Motion ...
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1
vote
0
answers
742
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Large deviations theory and extreme value theory
I'll enter into details of both, sooner or later, but for the moment I'm concerned about the differences (and relationships, if any) between these two theories. Can someone give me a brief, but still ...
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1
vote
0
answers
2k
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Skewed Generalized Error Distribution's (SGED) pdf
I want to use the SGED distribution of Theodossiou for GARCH estimation, however, I am struggling to understand which is the correct pdf function of the distribution. Let me just say that the ...
- 491
1
vote
0
answers
85
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Cross-sectional moments
I got a seminar topic named Forecasting risk from cross sectional moments? Could at least someone tell me what should I write about and if there is any paper that I could read. Thank you very much in ...
- 11
1
vote
1
answer
366
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Distribution of realized volatility for stock prices from a GBM
If you generate random stock price paths according to a GBM with daily increments, what will be the distribution of the realized volatility? Assume that the realized volatility is measured over daily ...
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0
answers
49
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Taking a set of normally distributed random variables as the sample space to fitting an exponential distribution
Disclaimer, this is my first question/interaction in this forum.
Let's assume I have random variables that are normally distributed. Then, say I take the observations that are greater than the mean, i....
- 1
0
votes
0
answers
37
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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 ...
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46
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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
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0
answers
102
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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 ...
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0
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0
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572
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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 ...
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0
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443
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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 ...
- 2,980
0
votes
0
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65
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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$$
Does it also follow that the ...
- 2,980
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0
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47
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Degree of freedom input for Monte Carlo simulation of asset returns with multivariate t distribution
How do I calculate or estimate the degrees of freedom in order to perform a Monte Carlo simulation of asset returns with multivariate t distribution using R functions? I am able to calculate the mean ...
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2
answers
255
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Estimating distribution of rate of return
Let $f[t]$ be the price of a stock at time $t$. We can calculate the rolling rate of return of the stock in a window of length $n$ by computing:
$$r[t] = \frac{f[t] - f[t-n]}{f[t-n]}$$
$r[t]$ is ...
-1
votes
1
answer
532
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Intensity of Exponential Distribution
How do I show the following:
Suppose $\lambda=-\frac{S'(x)}{S(x)}$, where $S(x)=1-F(x)$ is survival probability.
Show that $\lambda$ is the intensity of the exponential distribution with cdf $F(x)=1-e^...
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-1
votes
1
answer
177
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Why do I get this error using ghyp-distribution function?
I want to fit multivariate GH distribution on my data, and then generate simulations for that distribution. Using the instructions given in ghyp package, I wrote following lines of code in R.
...
-2
votes
1
answer
118
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How to compute Pr(S>100) when S follows Geometric Brownian Motion?
I have been trying to resolve this problem, under (b), but I cannot find the correct answer. For i=1, my ultimate answer (P=1) deviates from the correct answer (P=0.7580). Please let me know whether ...
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