# Questions tagged [normal-distribution]

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### Monte carlo portfolio risk simulation

My objective is to show the distribution of a portfolio's expected utilities via random sampling. The utility function has two random components. The first component is an expected return vector ...
963 views

### Kurtosis in asset logarithmic returns

Assets such as stocks usually display kurtosis in their logarithmic returns. However, their logarithmic returns in a time interval $n$ are the sum of smaller logarithmic returns in $1/n$ time ...
4k views

### Why does the Markowitz mean-variance model require the assumption of normality?

Given $N$ assets, the Markowitz mean-variance model requires expected returns, expected variances and a $N \times N$ covariance matrix. The joint distribution is fully defined by these measures. ...
322 views

### Portfolio Theory: Why is so much effort put into the reduction of estimation errors?

In MPT, very much effort by researchers is put into developing methods and techniques to handle the rather poor performance of the estimated means, variances and covariances. There are shrinkage ...
616 views

### Normally Distributed Returns Become Leptokurtic Due to Compounding

I was running a bunch of simple simulations in excel the other day in excel. Using the NORM.INV(RAND(),0,1) to simulate daily stock returns I noticed that the more compounded the returns, ie, the more ...
797 views

I don't understand why in the formula $$\log{\frac{S_{t+\triangle t}}{S_t}} \sim \phi{\left((\mu - \frac{1}{2}\sigma^2)\triangle t, \sigma^2 \triangle t\right)}$$ the mean is $(\mu - \frac{1}{2}\sigma^... 1answer 242 views ### Show that$(W_t, \int_0^t W_s ds)$has a normal joint distribution I have to show that, if$W_t$is a 1-d Brownian motion then$\biggl(W_t, \int_0^t W_s ds\biggr)$has normal distribution. Hint: apply Ito formula to this bivariate process. Any idea or suggestion on ... 2answers 593 views ### Transformation to reduce standard deviation without changing median Consider some negative skew and high kurtosis return time-series$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 ... 1answer 662 views ### Portfolio choice problem of a CARA investor with n risky assets Ok, I am working on a problem that consists of the following: I am looking to solve the portfolio choice optimization problem (maximizing utility with a known utility function) in the case where all ... 2answers 4k views ### An alternative to the Gaussian distribution to describe/fit market stock returns After the financial crisis in 2008, many people (including me) don't really believe that stock returns can be described in terms of the normal distribution (Gaussian distribution). But besides the ... 3answers 281 views ### Why is it so rare for finance theory to depart from the normal distribution? I understand almost all of the theory that has been built upon in quantitative finance is based on the normal distribution, and obviously you wouldn't want to throw all of it out the window on a whim ... 1answer 941 views ### Correlation of a lognormal asset and a normal asset So if i want to calcualte the correlation between a pair of assets, my intuition is that i should calculate whatever correlation i plan on using; When we look at correlation, it's normally the ... 1answer 670 views ### Why does Bloomberg's HRH test the simple returns for normality? On a Bloomberg terminal, it is possible to use the HRH (Historical Return Histogram) function on individual assets. It basically generates a histogram of the (simple) returns and overlays them with a ... 1answer 693 views ### How to design back-testing (validation) for such modified Vasicek model? Consider a classical Black Scholes model , $$\frac{dS}{S} = \mu dt + \sigma dW$$ , where$dW$is a Brownian motion, that$W(t_1) - W(t_0) \sim N(0, t_1 - t_0)$. The back-testing strategy is straight-... 2answers 292 views ### Is it possible that under Black-Scholes:$\ln S_{T} \sim N \left ( \ln S_t - \frac{1}{2}\sigma^2(T-t), \sigma^2(T-t) \right )$I have a slide on which there is written that under Black-Scholes model: $$\ln S_{T} \sim N \left ( \ln S_t - \frac{1}{2}\sigma^2(T-t), \sigma^2(T-t) \right )$$ Now, here there is a good explanation ... 1answer 218 views ### Determining if a time series is random I originally posted this in the Data Science Stack Exchange. Another poster suggested I post it here. The idea would be to identify "orderly" segments within a market time series and use them to ... 1answer 2k views ### How to use the Feymann-Kac formula to solve the Black-Scholes equation I have the Black-Scholes equation for European option with maturity$T$and strike$K$$\begin{cases}\frac{\partial u}{\partial t} = ru - \frac{1}{2} \sigma^2 x^2 \frac{\partial^2 u}{\partial x^2}-r ... 2answers 93 views ### Do we need to assume underlying returns are normal in BSM model, given Central Limit Theorem? I am trying to get a better understanding of Central Limit Theorem and how it can be used in life and in finance. From what I have read, the BSM model assumes the underlying asset's simple returns ... 1answer 1k views ### Gaussian vs Student Copula applied to finance I would like to get your opinion on the following topic: I am comparing the behaviour of Gaussian and Student-t Copulas. I employ the follwing procedure: Simulate N=100,000 samples from a Student ... 1answer 777 views ### What is the correct Stutzer index and Sharpe ratio relation, assuming a normal returns distribution? Assuming the returns distribution is normal, then there is a relation between Stutzer index and Sharpe ratio. However, I found in the following paper 2 different equation: Paper I (page 10-11)‎ ... 0answers 88 views ### How to trade risk-adjusted returns? Why does dividing daily returns by daily range eliminates fat tails and results in an (almost) gaussian distribution? And how could that distribution be exploited to enter trades? 1answer 986 views ### a simpler test for normality given skewness, kurtosis and autocorrelation and size of time series I typically do a JB (Jarque Bera) test and DW (Durbin Watson) tests for check for normality given skewness, kurtosis and autocorrelation of the data. However this requires a CHI distribution table ... 2answers 333 views ### Expectation and Cholesky Decomposition Assume that the random vector (X,Y) is (bivariate) normally distributed. Show that$$ \Bbb E[X|Y=y]= \Bbb E[X]+ \frac {Cov[X,Y]}{Var[Y]}(y-\Bbb E[Y])$$Also,$$ Var[X|Y=y]= (1-\rho^2) Var[X]$$I ... 2answers 816 views ### The Distribution of Future Stock Price In Hull, we are presented that$$\frac{\Delta S}{S_{0}}=\mu \Delta t+\sigma\sqrt{\Delta t}\cdot \varepsilon.$$Following some algebra,$$ \begin{align*} \frac{\Delta S}{S_{0}} &=\mu \Delta t+\... 1answer 188 views ### Steven Shreve: Stochastic Calculus and Finance The lecture notes have the following theorem: Let\theta\in \mathbb{R}$be given and$B(t)$stands for the Brownian motion which is a martingale, then$Z(t)=exp\{-\theta B(t)-\dfrac{1}{2}\theta^2t\}$... 1answer 302 views ### Quantile normal and lognormal Let's assume we have a normal distribution$X\sim \mathcal{N}(\mu,\sigma^2)$. In a normal distribution the quantile can be calculated as follows: \begin{equation} \Phi_X ^{-1}(p)=\mu +\sigma {\sqrt {... 2answers 161 views ### Problem with obtaining densities For my research I need to obtain a series of densities, however, I am encountering some problems. The first problem is perhaps very simple, but the answer eludes me. Let's say I have an observation ... 1answer 317 views ### Bivariate Gaussian copula with exponential margins I got little bit lost in the formulas. Assume to have two random variables distributed exponentially$X_i \sim Exp(\lambda_i)$and$X_j \sim Exp(\lambda_j)$. Thus, the distribution functions are$...
Introduction My goal is to retrieve a single Value-at-Risk (VaR) of a N(0, H) random variable $X$ at the $\alpha \in (0,1)$ confidence level where H is a known d-dimensional positive definite matrix ...