Questions tagged [normal-distribution]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1
vote
3answers
151 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?
0
votes
0answers
16 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 ...
-4
votes
0answers
18 views

Wrong way risk Gaussian copula

I'm studying Wrong Way Risk. To calculate the conditional expected exposure on default I have this formula \begin{equation} EE_{conditional} (t) = E[Weight(t) P(0,t)max(V(t),0)] \end{equation} where ...
0
votes
0answers
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) ...
0
votes
0answers
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 ...
1
vote
1answer
67 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 ...
0
votes
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 ...
1
vote
1answer
59 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'...
4
votes
3answers
278 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 ...
-3
votes
1answer
82 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 ...
0
votes
1answer
118 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 ...
2
votes
1answer
182 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 $Z(t)=exp\{-\theta B(t)-\dfrac{1}{2}\theta^2t\}$...
0
votes
1answer
73 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 ...
3
votes
1answer
212 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 ...
1
vote
1answer
46 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 ...
3
votes
2answers
92 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 ...
1
vote
1answer
236 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 ...
1
vote
0answers
54 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 ...
0
votes
1answer
62 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 ...
1
vote
2answers
134 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 ...
1
vote
1answer
217 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 ...
7
votes
1answer
241 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 ...
0
votes
1answer
719 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 ...
2
votes
0answers
297 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 ...
0
votes
1answer
379 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 ...
1
vote
1answer
603 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. ...
1
vote
2answers
127 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 ...
0
votes
1answer
185 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 ...
2
votes
1answer
278 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 ...
1
vote
4answers
547 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
votes
1answer
632 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) ...
1
vote
2answers
475 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 ...
1
vote
2answers
142 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 ...
0
votes
2answers
301 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
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 ...
-2
votes
2answers
350 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 ...
2
votes
1answer
299 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 {...
8
votes
2answers
3k 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. ...
2
votes
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 ...
0
votes
1answer
117 views

How to compute conditional expectation of multivariate normal

$(x_1, x_2, x_3)$~$N(0, \Sigma(\sigma_{ij}))$ then how to calculate $$E[x_2| x_1\leq a, x_3\leq b]$$
8
votes
3answers
320 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 ...
-1
votes
1answer
156 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 ...
2
votes
2answers
331 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 ...
1
vote
2answers
3k 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 ...
3
votes
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 ...
0
votes
1answer
177 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 ...
1
vote
1answer
198 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-...
4
votes
1answer
910 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 ...
3
votes
2answers
289 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 ...
2
votes
1answer
313 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 $...