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Questions tagged [distribution]

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0
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1answer
21 views

A quick and dirty loss distribution and Credit VaR

I need to create a loss distribution for a credit portfolio as the first steps to estimate the portfolio Credit VaR. I have historical monthly account snapshots (payment history) of all accounts ...
0
votes
1answer
24 views

Calculate the implied loss rate on a loan, given the interest charged

My bank has a retail credit portfolio of 100 million in loans. I know the payment history,balance history of all these loans since inception. Are there any tools to calculate an expected loss, a loss ...
0
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1answer
67 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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0answers
27 views

Why Jarque - Bera values are so high? Is this normal? [closed]

Please advise whether the following is a normal occurrence: In the above table I have Autocorrelation at lag1, LB, Skew, Kurt and JB test. I have noticed that whenever the value of Kurt increases, ...
1
vote
1answer
66 views

Distribution of simple returns vs logreturns

I understand that stock prices are conditionally modeled using a log normal distribution by the relationship $ y_t/y_{t−1}∼logN(μ_{daily},σ^2_{daily})$ $y_t∼logN(log(y_{t-1})+μ_{daily},σ^2_{...
29
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4answers
19k views

How to derive the implied probability distribution from B-S volatilities?

The general problem I have is visualization of the implied distribution of returns of a currency pair. I usually use QQplots for historical returns, so for example versus the normal distribution: ...
3
votes
2answers
89 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
votes
0answers
22 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
74 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 ...
0
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0answers
30 views

How does CVaR change when the mean and variance of the loss distribution change?

I have a CVaR constraint in my optimization problem and I want to change the mean and standard deviation of loss distribution during each iteration. How can I get the new CVaR based on the old CVaR ...
6
votes
3answers
6k views

How can I estimate the degrees of freedom for a Student's T distribution?

I am doing research estimating the value at risk for non-normally distributed assets. I need help in the process of estimating the parameters of Student's t distribution and which method to use. I ...
1
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0answers
21 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 ...
6
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0answers
147 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 ...
1
vote
1answer
64 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 ...
1
vote
3answers
511 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 ...
0
votes
1answer
139 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
votes
2answers
101 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\...
5
votes
0answers
912 views

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 ...
2
votes
1answer
86 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
vote
0answers
30 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
vote
0answers
54 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
vote
0answers
35 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 ...
0
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0answers
38 views

History to use to fit a gaussian mixture distribution?

I have to model the distribution of past returns for several time horizon using a gaussian mixture distribution. I first build my time series of past returns thanks to a rolling sum of my past log ...
2
votes
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 ...
5
votes
2answers
79 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 ...
3
votes
0answers
87 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
vote
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
vote
1answer
97 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 ...
-1
votes
1answer
241 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
votes
2answers
226 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
votes
1answer
57 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 ...
24
votes
3answers
3k views

Tools in R for estimating time-varying copulas?

Are there libraries in R for estimating time-varying joint distributions via copulas? Hedibert Lopes has an excellent paper on the topic here. I know there is an existing packaged called copula but ...
2
votes
1answer
248 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 {...
0
votes
1answer
130 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
vote
1answer
226 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
votes
1answer
69 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
votes
1answer
102 views

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. ...
6
votes
6answers
3k views

Consensus on Cauchy distribution for stock prices

What is the general consensus for using a Cauchy distribution to model stock prices? I can't find much after researching online and wonder if it has been tried and discarded. My motivation is to find ...
1
vote
1answer
167 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 ...
2
votes
3answers
172 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 ...
-1
votes
1answer
43 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
votes
2answers
1k 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?
4
votes
2answers
960 views

Brownian Bridge's first passage time distribution

Let's say we have a Brownian Bridge $Y_{b,T}(t)$ such that $Y_{b,T}(0)=0$, $Y_{b,T}(T)=b$. Let's say we are interested in the first passage time of $Y_{b,T}(t)$ at level $b$: $\tau_b = \{\min \tau; ...
3
votes
1answer
748 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: ...
1
vote
0answers
264 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?...
1
vote
2answers
906 views

How to combine Gaussian marginals with Gaussian copula to obtain multivariate normals?

in the book "Numerical Methods and Optimization in Finance" I red the following: "Combining the Gaussian copula with Gaussian marginal gives a fancy way of expressing multivariate normals. However, ...
1
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0answers
146 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
vote
1answer
453 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 ...
1
vote
0answers
83 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
vote
0answers
66 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 ...