Questions tagged [distribution]

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3
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2answers
133 views

Compare two distributions for forecasting returns

Let's imagine that we have two separate models, both used to forecast the return for the next period. Both models are estimated everyday, and both models outputs a probability distribution. How can we ...
2
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0answers
31 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 ...
4
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0answers
103 views

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
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0answers
25 views

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
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1answer
121 views

How to price a barrier using monte carlo when return distribution is not iid?

this question is actually related to set the stop loss and stop return. Say after a liquidity shock, I want to place two stops, one being stop loss and another being stop return. If I use, say 10 ...
1
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1answer
94 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 ...
7
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0answers
814 views

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 ...
0
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1answer
428 views

Theoretical distribution of (geometric) Brownian motion (with drift)

I am working on a simulation study which focuses on both the Brownian motion with drift (1) and the geometric Brownian motion (2). I denote them by $X_t$. What are the theoretical distributions of ...
2
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2answers
105 views

Verifying that the extreme value copula is indeed a copula

Given the extreme value copula as defined in Schölzel/Friederichs (2008), how does one verify that $\frac{\partial C(u_1, u_2)}{\partial u_1} \geq 0?$ For the LHS, I have $$\exp\left[\log(u_1u_2)A\...
2
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1answer
218 views

Showing the Gaussian shift theorem for bivariate case

I was reading about the Gaussian shift theorem in "An Introduction to Exotic Option Pricing" by Peter Buchen and came across a question that I can't seem to figure. In the book, he uses F(Z) (a ...
1
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0answers
31 views

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) ...
1
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0answers
82 views

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 ...
1
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0answers
40 views

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 ...
2
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0answers
64 views

$\int_{0}^1W_x(t)dW_y(t)/(\int_{0}^1W_x^2(t)dt)^{1/2}$ 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 ...
4
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2answers
95 views

Approach to add scenarios to OpRisk loss distribution

There is quite a lot of literature on OpRisk modelling. My question focuses on a loss distribution approach (LDA). Let's look at a basic model. A Poisson-distributed $N$ and loss sizes $X_i$ and from ...
2
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0answers
119 views

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 ...
1
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0answers
17 views

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....
1
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1answer
120 views

Why should we care if the “squares of returns are independently distributed over time” to choose an adequate model of the distribution of returns?

In a Time Series Book by Hashem Pesaran, he mentions that there are a number of issues that need to be addressed in order to choose an adequate model for predicting asset returns. I understand the ...
0
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2answers
308 views

Log normal price simulation

I'm trying to figure out a spreadsheet I have which simulates 50000 returns in excel using the following function: LOGNORM.INV(RAND(),0,0.35)-1 Question: How ...
0
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1answer
61 views

Quantile with periodic investing

Short Version Can I get a quantile of such an expression? \begin{equation} \sum_{k=1}^{n} A_k\exp(\mathcal{N}(t_k\mu-\sigma\sqrt{t_k}/2,\sigma))) \end{equation} I know I can do it for one part of ...
2
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1answer
319 views

Quantile normal and lognormal

Let's assume we have a normal distribution $X\sim \mathcal{N}(\mu,\sigma^2)$. In a normal distribution the quantile can be calculated as follows: \begin{equation} \Phi_X ^{-1}(p)=\mu +\sigma {\sqrt {...
-1
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1answer
348 views

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^...
0
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1answer
165 views

Why/When local volatility is preferred over implied distribution sampling?

Let's say we have an option whose payoff is path dependent (let's say it's asian option with observations every month). Then why these are usually priced with local vol instead of sampling from ...
1
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1answer
296 views

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 ...
-2
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1answer
81 views

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 ...
1
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1answer
235 views

Question on implied vol (surface) and strikes

there have been loads of papers on skews ATM / OTM, volatility premium and such. Lots of explanations for why iv is different on same stock with different strikes focused on preference of informed ...
-1
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1answer
45 views

How to simulate the exponential law over an interval of the form [0,T]?

How do you simulate an exponential random variable over an interval $[0, T]$ with $T > 0$?
0
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2answers
2k views

What the implied distribution really is?

From volatility surfaces we have a implied distribution of $S_T$. This distribution is the real world distribution or this is a risk neutral distribution?
1
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0answers
309 views

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?...
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0answers
194 views

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 ...
1
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1answer
725 views

How to simulate asset returns using student t?

I am currently trying to simulate an asset return using the student-t distribution, but I can't find how I should do this. I began with the Geometric Brownian motion and just changed in order that ...
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0answers
118 views

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 ...
1
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0answers
153 views

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 ...
3
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1answer
162 views

What is the limiting distribution of loss portfolio?

I am working through this paper on Vasicek's portfolio loss distribution. On page 3 he mentions that by the law of large numbers, $$\lim_{n\to\infty}\sum_{k=0}^{\lfloor nx \rfloor} \binom{n}{k}s^k(1-...
1
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1answer
90 views

Calculating VaR of an Incomplete Distribution

I am currently completing a multiple choice question that has stumped me. An asset has its price and its corresponding probability described as: 100, 0, -50, -70 and -90 with probabilities 50%, 12%,...
3
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1answer
462 views

Use NIG distribution to model stock path

I would like to use Monte Carlo simulation to price some options. First I use standard approach where stock price is discribed by the following process: $$S_T = S_0\exp \left[(r - 0.5\sigma^2)T + \...
2
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0answers
109 views

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, $\nu$...
2
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3answers
230 views

Distribution of pay-off of an exotic option

Can any assumptions be made about the pay-off of an exotic option? For example, might we say the distribution of the pay-off a vanilla option would be Normal? I have built a valuation tool that ...
2
votes
1answer
364 views

How do I get Value-at-Risk for a GED distribution in R?

I need to calculate parametric Value-at-Risk using a GARCH model assuming a GED distribution. How can calculate it in R? thank you
1
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3answers
77 views

find the qth lower tail quantile

I have daily currency returns. For each month, I have to find the return associated to the 5% lower tail quantile for each currency (the lowest return or the second lowest return). Could you please ...
3
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1answer
2k views

Density plot of the skew-t distribution

I am using the sgt package in R to recreate the plot from Hansen's paper ( available here http://www.ssc.wisc.edu/~bhansen/papers/ier_94.pdf on page 8) using random ...
2
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0answers
125 views

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=...
4
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2answers
816 views

Is it possible to deal with non-normal distribution in Black-Litterman model?

Suppose that I know that the normality assumption about my data is unrealistic (as it is very frequently): is it possible to apply any distribution that I judge the right one to the Black-Litterman ...
1
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1answer
381 views

Creating the histogram for the distribution of the portfolio returns

Given log returns for some stocks $A$ and $B$, which are the constituents of our hypothetical portfolio in equal weights, how does one actually come up with a distribution of the log returns of the ...
1
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3answers
607 views

Normal Inverse Gaussian distribution - any consensus on an accurate quantile function?

I am making use of the Normal Inverse Gaussian distribution in my work to model underlying interest rate implied volatility risk drivers. What is particularly nice about this distribution for my ...
1
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0answers
417 views

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 ...
1
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1answer
2k views

Calculate VaR for a liabilty taking a exponential distribution?

An insurance company faces the liability loss off $L = \begin{cases} 0, & \mbox{with probability } 0.75 \\ Z, & \mbox{with probability } 0.25\end{cases}$ where $Z\sim Exp(\mu)$. I want to ...
2
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2answers
170 views

Problem with obtaining densities

For my research I need to obtain a series of densities, however, I am encountering some problems. The first problem is perhaps very simple, but the answer eludes me. Let's say I have an observation ...
1
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0answers
2k views

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 ...
3
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1answer
990 views

Log-likelihood of skew-t distribution

I am trying to estimate GARCH models with the use of Hansen's (1994) skew-t distribution. I am using matlab's ARMAX-GARCH-K toolbox, where the log-likelihood is calculated as: ...