Questions tagged [normal-distribution]

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expression on page 90 of shreve's stochastic calculus for finance II

Hi: In the middle of page 90, Shreve has an expression which implies that (I'm using $t$ where he uses $u$ only because I find it confusing to use $u$ and $\mu$ in the same expressions): $ E[\exp(\...
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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 ...
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Significance of annualized volatility over 100% on the normal distribution? [closed]

Assume stock is 50 dollars. From what I understand, an annualized vol of 20% means there is a ~68% chance the stock will be between 40 and 60 a year from now; a ~95% chance it will be between 30 and ...
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2 votes
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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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2 votes
1 answer
64 views

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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6 votes
1 answer
681 views

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 ...
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How to use CAPM model to calculate expected value of portfolio?

Let's assume that vector $(R_1, R_2, R_3)$ has multivariate normal distribution $N(\mu, \Sigma)$ where $\mu = (2, 6, 4)$ and $$\Sigma^{-1} = \begin{bmatrix} 2 & 2 & 2\\ 2 & 4 & 4 \\ 2 &...
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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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155 views

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 ...
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Entropy-implied volatility requires itself to be calculated?

\begin{align} H &= \frac{1}{2} \ln (2\pi\sigma^2) + \frac{1}{2}\\ &= \frac{1}{2} \ln (2\pi e \sigma^2) \end{align} is the analytical solution for the entropy of a Gaussian random variable, ...
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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 ...
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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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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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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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3 answers
400 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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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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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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4 votes
3 answers
381 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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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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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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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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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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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
70 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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2 answers
147 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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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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1 vote
0 answers
71 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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1 answer
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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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1 vote
3 answers
417 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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1 answer
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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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7 votes
1 answer
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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 ...
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1 answer
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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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2 votes
0 answers
337 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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1 answer
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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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1 answer
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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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1 vote
2 answers
227 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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1 answer
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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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3 votes
1 answer
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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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1 vote
4 answers
820 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 ...
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2 votes
1 answer
1k 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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4 votes
4 answers
868 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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1 vote
2 answers
208 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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2 answers
397 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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1 answer
69 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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-2 votes
2 answers
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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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2 votes
1 answer
382 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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8 votes
3 answers
5k 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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3 votes
1 answer
2k 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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1 answer
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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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8 votes
3 answers
350 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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