# Questions tagged [distribution]

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### 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: ...
5k 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 ...
6k 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 ...
23k views

### Copulas simply explained

I try to understand the basic idea of copulas, however I am still struggling and hope that someone can help me. I understood that in general a copula is a function which links several marginal ...
8k views

### What distribution to assume for interest rates?

I am writing a paper with a case study in financial maths. I need to model an interest rate $(I_n)_{n\geq 0}$ as a sequence of non-negative i.i.d. random variables. Which distribution would you advise ...
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 ...
1k views

### Is volatility for the next day forecastable? To any extent?

In a more general way: is there 1) a methodological approach to quantify the correctness of a model that produces a probability distribution for the, say, S&P 500 index return for the next ...
459 views

### What distribution should I apply to estimate the likelihood of extreme returns?

Say I have a limited sample, a month of daily returns, and I want to estimate the 99.5th percentile of the distribution of absolute daily returns. Because the estimate will require extrapolation, I ...
1k views

### Kolmogorov-Smirnov test for Generalized Pareto Distribution

I've fitted my data to a generalized pareto distribution as to model the returns in the tails more accurately. The interior is fitted with kernel distributions. I would like to now test whether the ...
337 views

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 $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t\in[0,T]},\... 8 votes 3 answers 380 views ### Can Gaussianity of returns depend on the time frame? I would be interested in knowing if the fact that returns are Gaussian is disproved on all time frames, or if, for example, the 5 minute intra-day time frame could exhibits Gaussian returns assuming ... 8 votes 3 answers 8k views ### How can I estimate the degrees of freedom for a Student's T distribution? I am doing research estimating the value at risk for non-normally distributed assets. I need help in the process of estimating the parameters of Student's t distribution and which method to use. I ... 7 votes 1 answer 789 views ### Benfords law and quantitative finance Benford's law has been applied in various ways for detecting fraud (e.g. elections or accounting). But what are the most useful applications of Benford in quantitative finance? Are there any? I have ... 7 votes 1 answer 2k views ### Fitting distributions to financial data using volatility model to estimate VaR I want to fit a distribution to my financial data using a volatility model to estimate the VaR. So in case of a normal distribution, this would be very easy, I assume the returns to follow a normal ... 7 votes 0 answers 2k views ### Arbitrage free smoothing of volatility smile - cubic spline - implementation procedure I am studying the paper Arbitrage-Free Smoothing of the Implied Volatility Surface, from Matthias R. Fengler (https://core.ac.uk/download/pdf/6978470.pdf). The problem I want to solve is much simpler ... 6 votes 2 answers 676 views ### Transformation to reduce standard deviation without changing median Consider some negative skew and high kurtosis return time-series$X_t$. I do not know the functional form of the pdf of$X_t$and have about 150,000 data points. Suppose that I was to create an ... 6 votes 2 answers 637 views ### Do futures follow physical or risk-neutral distributions I've spent a while looking for an answer to this question and while I feel it is a simple question I have not found an answer. I know prices of option contracts follow an implied, risk-neutral ... 5 votes 3 answers 492 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 ... 5 votes 1 answer 317 views ### Density of Geometric BM via Fokker-Planck Attempting to derive density of a GBM (which we know is log-normal) the long way, using the Fokker Planck-equation. Can't figure out where I went wrong - would appreciate a few sets of extra eyes! ... 5 votes 2 answers 115 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 ... 5 votes 0 answers 1k 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 ... 4 votes 1 answer 182 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 ... 4 votes 2 answers 994 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 ... 4 votes 1 answer 230 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^\... 4 votes 2 answers 1k 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; ... 4 votes 1 answer 222 views ### What are some common models for one-sided returns? One typically models the log returns of a portfolio of equities by some unimodal, symmetric (or nearly symmetric) distribution with parameters like the mean and standard deviation estimated by ... 4 votes 1 answer 2k views ### Value at Risk Monte-Carlo using Generalized Pareto Distribution(GPD) I have created a VBA program to calculate VaR by using Monte Carlo, I have simulated Brownian Motion. This method might be ok for 100% equity portfolio, but let's say this portfolio may have fixed ... 4 votes 1 answer 189 views ### Ito integrals and copulas Let X_{t} and Y_{t} be two brownian motions and let their joint distribution be given by F. So in regularly correlated BM's where dX_{t}dY_{t}=\rho dt, we have a bivariate normal distribution ... 4 votes 0 answers 99 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? 4 votes 0 answers 143 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 ... 3 votes 2 answers 295 views ### Produce the random variable for an asset from a uniformly distributed random varible I'm working on a quant interview question from the book called Quant Job Interview Questions And Answers (by Mark Joshi and other authors). I cannot understand the following question(not the answer, ... 3 votes 2 answers 204 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 ... 3 votes 2 answers 257 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 ... 3 votes 2 answers 2k views ### Distribution for High Kurtosis Can you please advise which distribution to follow when your skewness is 0.28 and Kurtosis value is 51. Since it's leptokurtic and positively skewed I would like to fit distribution and also wanted to ... 3 votes 1 answer 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 ... 3 votes 2 answers 937 views ### Stock Returns Distribution in Heston Model There is a paper by Dragulescu and Yakovenko (DY) in 2002 proposing a pdf for the stock returns in the Heston model. However, in a paper by Daniel, Bree and Joseph, they actually perform statistical ... 3 votes 2 answers 571 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_{daily})) ... 3 votes 1 answer 589 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 + \... 3 votes 1 answer 448 views ### Fitting stochastic variance distributions to index return data I want to calculate option prices based on a realistic distribution of the underlying. The underlying is a liquid index such as Eurostoxx50. I think of two aproaches, both of them incorporate ... 3 votes 1 answer 662 views ### Is there an easily implementable alternative to lognormal growth (something with fatter tails)? I have a toy model in Excel for the growth of a investment portfolio. I assume iid lognormal annual growth factors: =EXP(mu+sigma*NORM.S.INV(RAND())) where mu and ... 3 votes 1 answer 360 views ### What are$d_1$and$d_2$for Laplace? What are the formulae for d1 & d2 using a Laplace distribution? 3 votes 1 answer 432 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 ... 3 votes 1 answer 1k 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: ... 3 votes 0 answers 294 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 ... 3 votes 0 answers 326 views ### 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. ... 3 votes 1 answer 172 views ### VaR calculation accuracy/comparison/effectiveness through different R packages My question is what would be the better( in terms of estimation accuracy) method of VaR calculation among below two:, also any small code snippet will be great as a starting point for me. 1st method: ... 3 votes 0 answers 144 views ### 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 ... 2 votes 1 answer 216 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$\...
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 {...