Questions tagged [normal-distribution]
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99 questions
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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 ...
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1
answer
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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 ...
- 13.5k
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3
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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.
...
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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 ...
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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 ...
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In Black-Scholes, why is $\log{\frac{S_{t+\triangle t}}{S_t}} \sim \phi{((\mu - \frac{1}{2}\sigma^2)\triangle t, \sigma^2 \triangle t)}$?
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^...
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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 ...
- 73
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1
answer
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Is there a closed-form solution for the following integral?
The integral under consideration is as follows:
$$
F=\int_{a}^{1} \exp\Big\{c\Phi^{-1}(x+b) + d\Big\}\; \mathrm dx,
$$
where $0<a, b<1$, and $c>0, d\in\mathbb{R}$ are constants, and the ...
- 419
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2
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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 ...
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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 ...
- 1,261
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1
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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 ...
- 2,611
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1
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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 ...
- 215
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sub-Gaussian random variables in financial economics
Unlike financial time series that typically possess fat tails, sub-Gaussian random variables have strong decay in the tails of their distribution. do sub-Gaussian random variables or processes appear ...
- 3,090
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3
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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 ...
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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 ...
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1
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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 $K$
$$\begin{cases}\frac{\partial u}{\partial t} = ru - \frac{1}{2} \sigma^2 x^2 \frac{\partial^2 u}{\partial x^2}-r ...
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4
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Central limit theorem and normality assumption of asset return distribution
Can central theorem justify normality assumption of assets return distribution? And if it can why the empirical evidence show this assumption, which many finance models are based on, is a far cry from ...
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Why does Bloomberg's HRH test the simple returns for normality?
On a Bloomberg terminal, it is possible to use the HRH (Historical Return Histogram) function on individual assets. It basically generates a histogram of the (simple) returns and overlays them with a ...
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Statistical distribution of Max Drawdown
Are there any good papers/ references on the statistical distribution of Max Drawdown over a specified amount of time given a specified Sharpe? Assuming returns are iid normally distributed
I’ve been ...
- 500
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How to design back-testing (validation) for such modified Vasicek model?
Consider a classical Black Scholes model ,
$$\frac{dS}{S} = \mu dt + \sigma dW$$
, where $dW$ is a Brownian motion, that $W(t_1) - W(t_0) \sim N(0, t_1 - t_0)$.
The back-testing strategy is straight-...
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2
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483
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Is it possible that under Black-Scholes: $\ln S_{T} \sim N \left ( \ln S_t - \frac{1}{2}\sigma^2(T-t), \sigma^2(T-t) \right )$
I have a slide on which there is written that under Black-Scholes model:
$$\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 ...
- 253
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Why is the price of an ATM straddle not the same as the "dollar move" from implied volatility?
Knowing that implied volatility represents an annualized +/-1 Standard Deviation range of the stock price, why does the price of an ATM straddle differ from this? Also for simplicity, no rates, no ...
user46424
3
votes
1
answer
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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 ...
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1
answer
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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 $...
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2
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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 ...
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How to compute a single Value-at-Risk (a single quantile) of portfolio returns taking into account correlation between individual returns?
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 ...
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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 ...
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answer
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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) ...
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0
answers
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Spread vol for interest rate spread options in normal environment
Suppose I am long spread option with underlying : rate A - rate B. The vega on the option would be positive. But if I want to compute the option vega with respect to individual rates, can I use the ...
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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?
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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 ...
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3
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Simulating covariance matrices with nonzero correlation
How would you simulate a covariance matrix of 1,000 stocks where each pair has nonzero correlation?
I have literally no idea how to start with this.
Any suggestions?
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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]$$
I ...
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The Distribution of Future Stock Price
In Hull, we are presented that
$$\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+\...
2
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1
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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 $Z(t)=exp\{-\theta B(t)-\dfrac{1}{2}\theta^2t\}$...
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1
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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 {...
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2
answers
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Box-Muller Method Proof
Here we want to show that the Box-Muller method generates a pair of independent standard Gaussian random variables. But I don't understand why we use the determinant? For me when you have two ...
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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 ...
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Price Option B Knowing The Price of a Similar Option A
How do we find the implied volatility from the price in a call option and apply it to another option without a calculator? Or is there actually a better way?
For example, given a 25-strike 1.0-expiry ...
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1
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Showing that the shortfall-to-quantile ratio of a normal distribution goes to one
I dont get why $$\lim_{x \to \infty}
\frac{\mu \{1 - \Phi(x)\} + \sigma \phi(x)}{(\mu + \sigma x) \{1 - \Phi(x)\} }
= \lim_{x \to \infty}
\frac{1}{1 - \sigma \frac{1 - \Phi(x)}{(\mu + \...
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2
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Price is Log-normal distributed, yet the return is non-normal
I have a price series. The natural logarithm of the price shows good normality. As shown in the standardized normal probability plot below:
However, by viewing the standardized normal probability ...
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1
answer
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Distribution of discrete Geometric average and Stock Price
If we have $$S_t = S_0 e^{(r-\frac{1}{2} \sigma ^2) +\sigma W_t}$$ and a discrete geometric average of stock prices $$G_n = (\prod_{i=1}^{n} S_{t_i})^{\frac{1}{n}} $$ where the monitoring points are ...
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Show that the Ito integral is Gaussian
Let $f(t), 0 \leq t \leq T$ be a deterministic function with $f(t) = \sum_{i=1}^na_{i-1}1_[t_{i=1}, t_i)(t)$ with $0 \leq t_0<t_1<...<t_{n-1} = T$. Show that the stochastic integral $I_t(f) ...
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Variability in the Expected Shortfall estimator
Are there any results for calculating the variability in the Expected Shortfall measure. I am looking for Large sample confidence intervals under Normality for Expected Shortfall or calculation of ...
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1
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normalized accumulation distribution
I am looking for a way to take an accumulation/distribution indicator and normalize it
so I can compare a bunch of stocks with stock prices that have no relationship with each other.
EDIT: This ...
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0
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Game Theory Brainteaser
Seeking help / thought process guidance on the following interview problem, which seems centred on game theory
Setup: there’s a number X which we can measure once with error following N(0, 1). We can ...
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0
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A question in information strucutres and probability measures - How are they connected?
Suppose that $\mathcal{I}=(X,\sigma^{\mathcal{X}},\mu)$ is an information strucutre, which is a probability space, where
$X=X^1\times X^2$ is the cartesian product of the individual finite sets of ...
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0
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RiskMetrics VAR calculations and conditional distribution of sum of log returns
According to Tsay's book in Chapter 7, for the Risk Metrics model:
A nice property of such a special random-walk IGARCH model is that the
conditional distribution of a multiperiod return is ...
- 666
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votes
1
answer
509
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Interpretation of cross-correlation matrix when one sample distribution is not normal
I am looking at the variance of (log) price changes in securities vs. the amount of social media discussion about them. I'm not interested in building a model. I'm just looking to see if there is a ...
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1
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Creditworthiness indicator for copula one-factor model
In this paper in equation 15 on page 261 dealing with one factor copula model, one is using creditworthiness indicator as one of a variables. It is defined as
\begin{equation}
Y_c = \sqrt{\rho_c} Z +...
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