Questions tagged [normal-distribution]

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Proving Scaled Random Walk Approaches Normal Distribution

I'm reading Stochastic Calculus for Finance II: Continuous-Time Models by Steven Shreve and I don't understand how he went from the equation on the left to the middle one. If it helps, this section is ...
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1answer
46 views

A simple question about VaR estimation

"A 99% VaR using 1,000 (simulation) replications should be expected to have only 10 observations in the left tail, which is not a large number. The VaR estimate is derived from the 10th and 11th ...
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63 views

question about significance level

A case study in a exam material goes like this: "Assume that the bank reports a daily VAR of \$100 million at the 99% level of confidence. Under the null hypothesis that the VAR model is ...
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3answers
178 views

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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24 views

Price of Call & Put Spreads as Volatility Tends to Infinity in Bachelier Model

In the standard Black Scholes model, as we take volatility to infinity, the price of call spreads goes to zero and the price of put spreads goes to the difference in strikes. I ran a simulation using ...
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24 views

Does annualizing daily returns make them more normally distributed? [duplicate]

Asset returns are usually non-normal because they are skewed and have fat tails. Does annualizing or downsampling daily returns (transforming a sample to a lower frequency, such as daily to monthly) ...
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22 views

Minimal bounds to enclose most sample paths of a GBM (Geometric Brownian Motion)

For a (generalized) Brownian motion $Y = F(t,W)$, starting at $InitialValue$ and running for a total of $T$ time, if I want to "enclose" (in a visual way) "most" of the possible sample paths, I could ...
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1answer
100 views

How accurate is the square root of time rule for VaR for a portfolio containing several different types of instruments

Assuming that your value at risk model is based on normality assumptions, e.g. using a Delta-Gamma normal model does the approximation hold perfectly for a portfolio of stocks and options? What about ...
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0answers
13 views

Finding fifth and sixth polynomials for Headrick (2002) method for non-normal multivariate distribution

I am trying to perform a 3-asset class return Monte Carlo simulation. As the asset class returns are non-normal, I found the following function rHeadrick from the ...
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1answer
70 views

Value at Risk (VaR): Normal distribution with gamma distributed volatility

If I was to do a 99% VaR calculation on a portfolio with normally distributed returns $\mathcal{N} (\mu,\sigma)$, the 99% VaR would be $\mu - 2.33\sigma$. Instead of having a constant volatility, let'...
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3answers
284 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 ...
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1answer
83 views

Can someone prove (or disprove) this assertion about the normal distribution? [closed]

Let $X$ be distributed as a $Normal (\mu, \sigma^2)$. Then for a fixed $\mu$ it is always the case that: \begin{equation} \frac{90th quantile-10th quantile}{\sigma}=constant \quad \forall \sigma>0 ...
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1answer
123 views

How to extract standard deviation from normal distribution in R

If I have some point forecast and an 80% confidence interval, with the forecast assumed to be normally distributed with a constant variance, how do I extract the actual variance? Let us work with the ...
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1answer
189 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 $Z(t)=exp\{-\theta B(t)-\dfrac{1}{2}\theta^2t\}$...
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1answer
75 views

Two commodities which are normal distributed and perfectly correlated

The daily price change in commodity 1 is distributed $N(0,0.15^2)$ and the daily price change in commodity 2 is distributed $N(0,0.3^2)$. The two commodities are 100% correlated. 1) Does the relative ...
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1answer
220 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 ...
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1answer
49 views

Asset return distribution

What is the basis for assumption that asset prices follow a log normal distribution? Then how is it transformed to say that asset return follows a normal distribution? How this relationship between ...
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2answers
94 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 ...
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1answer
310 views

Why assume stock returns are normally distributed instead of just adjusting the kurtosis?

Most standard models assume stock returns are normally distributed even though everyone agrees that real-world returns have fat tails. We've all heard stories of hedge funds that went bankrupt cause ...
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0answers
58 views

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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1answer
66 views

Transforming non-normally distributed interest rates for OLS regression

I am studying the effects of short- and long-term interest rates on bank risk-taking in the Euro zone countries. To analyse the effects, I will use, amongst other, an OLS regression. However I have ...
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2answers
159 views

Why can we assume that asset return rates are normally (or lognormally) distributed?

In many theories of financial mathematics it is assumed that asset return rates are normally distributed (e.g. VaR models) or lognormally distributed (e.g. Black-Scholes model). In practice, asset ...
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1answer
234 views

Measure of a Brownian motion = normal distribution?

Consider some model where the process increments are normally distributed, e.g. Vasicek: $$dr(t) = \left(\theta - ar(t)\right)dt + \sigma dW(t).$$ We usually say that $W(t)$ is a Brownian motion ...
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1answer
244 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 ...
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1answer
755 views

Why should we use log returns? Log normality

According to this link, there are some reasons we have to use log returns. But I can not understand the first reason provided in the link: First, log-normality: if we assume that prices are ...
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0answers
298 views

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 ...
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1answer
396 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 ...
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1answer
644 views

What is the Probability Distribution of Max-Drawdown?

How to obtain the probability distribution of Maximum Drawdown, starting from the probability distribution of Daily Returns? Here the details: Suppose I have a time serie of N=1000 daily returns. ...
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2answers
131 views

Distribution of data for GBM

I am running some Monte Carlo simulations with GBM on time series of commodity prices. First of all, the price data is annual between 1900-1950. I would firstly like to know if it is bad practice to ...
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1answer
215 views

Determining the probability of arriving at a price by a time T

A useful calculation for ascertaining the risk of something might be determining the probability of a realization of a set of stock prices $X$ being greater than or equal to some future price $x$. I ...
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1answer
287 views

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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4answers
597 views

If equity returns are normally distributed, why are average equity returns not zero [closed]

So I am getting confused between assumption of equity returns normality and why then equity markets in the long term on average go up i.e equity risk premium. Does this not already poke wholes in the ...
2
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1answer
678 views

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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2answers
517 views

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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2answers
146 views

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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2answers
303 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 ...
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1answer
59 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 ...
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2answers
354 views

Modeling stock performance in excel

I am trying to model the ending value of a stock after a certain number of years, I need it for a bigger project but I made this sample sheet to get help. This sheet is assuming that annual returns ...
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1answer
305 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 {...
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2answers
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. ...
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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 ...
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1answer
121 views

How to compute conditional expectation of multivariate normal

$(x_1, x_2, x_3)$~$N(0, \Sigma(\sigma_{ij}))$ then how to calculate $$E[x_2| x_1\leq a, x_3\leq b]$$
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3answers
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 ...
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1answer
158 views

Finance beta: normally distributed?

If we assume normally distributed return (or normally distributed log Returns) for an asset and the market, can be then also say that the betas derived by this are also normally distributed? How ...
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2answers
334 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 ...
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2answers
4k views

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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1answer
3k 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 ...
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1answer
180 views

EM for conditional Gaussian model

Let $$X_1\sim N(\mu_{X_1},\sigma_{X_2}^2)$$ $$X_2\sim N(\mu_{X_2}, \sigma_{X_2}^2)$$ where $\mu_{X_2}=c+aX_1$. Also, I have data $D$ (with missing values on $X_1,X_2$). How can I update/estimate the ...
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1answer
204 views

$\mathbb{P}$ and $\mathbb{Q}$ probability measure/distribution interpretations

I'm trying to understand probability distributions implied from market prices and was reading through this reference explaining the interpretation of $N(d_1)$ and $N(d_2)$ in the log-normal vol Black-...
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1answer
966 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 ...