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

it doesn't require normality. What it requires is that the investor's decisions are determined by mean and variance. A normal distribution is determined by mean and variance, so if you assume joint ...
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8 votes
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Why is it so rare for finance theory to depart from the normal distribution?

The Geometric Random Walk: The Starting Point Let me begin by being a little more specific. The simplest, yet relatively sound model of asset prices that we have is this one: \begin{equation} ln S(...
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  • 2,356
8 votes

Is there a closed-form solution for the following integral?

Thanks to Gordon's help, we have that \begin{eqnarray*} F=exp\Big\{d + \frac{{c}^2}{2}\Big\}\Big[ \Phi\Big(\Phi^{-1}\Big(1+b\Big)-{c}\Big)- \Phi\Big(\Phi^{-1}\Big(a+b\Big)-{c}\Big)\Big] \end{eqnarray*}...
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  • 409
7 votes

An alternative to the Gaussian distribution to describe/fit market stock returns

My take on the whole issue is as follows: We cannot be sure to find the one and only true model, the only thing we can do is to identify the most prevalent so called stylized facts and try to model ...
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7 votes
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Show that $(W_t, \int_0^t W_s ds)$ has a normal joint distribution

Note that \begin{align*} \int_0^t W_s ds &= tW_t -\int_0^t sdW_s \tag{1}\\ &= \int_0^t (t-s)dW_s. \end{align*} Then, for $\lambda_1, \lambda_2 \in \mathbb{R}$, \begin{align*} \lambda_1 W_t + \...
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7 votes
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Reconciling Two Claims About Volatility Under Fat Tails

I don't think the claim that "Lévy alpha-stable distributions are better descriptors of returns" is universally accepted. While Mandelbrot (and others before him) has correctly identified ...
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  • 118
6 votes
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The Distribution of Future Stock Price

You ask 2 questions and I try to answer: 1) Why do we use geometric Brownian motion ($\ln S_t-\ln S_0$ is normally distributed)? In this case you have $$ S_t = S_0 \exp( (\mu-\sigma^2/2) t + \sigma ...
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  • 13.2k
6 votes
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Quantile normal and lognormal

Quantiles are preserved under monotonic transformations, hence the quantile for $Y$ is simply the exponential of the quantile of $X$, no need for corrections whatsoever (see here for instance). Put ...
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6 votes

Measure of a Brownian motion = normal distribution?

It is correct that $$ \mathbf{P}(t^{-1/2}W(t) \in[a,b])=Φ(b)−Φ(a), \forall t\in(0,\infty) $$ due to the stationary increments property of the Wiener process and the fact that you normalized the ...
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6 votes
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Why assume stock returns are normally distributed instead of just adjusting the kurtosis?

If we are talking about risk management (Hence, the risk neutral world), normality allows us to get closed form solutions. For instance, the Black and Scholes equation assumes Gaussian returns (...
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  • 1,346
6 votes
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Steven Shreve: Stochastic Calculus and Finance

Note merely that $B_t=B_s+(B_t-B_s)$ which is the sum of independent normally distributed random variables. In particular, $B_s$ is $\mathbb{F}_s$-measurable and $B_{t-s}$ is independent of $\mathbb{F}...
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6 votes
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Simulating covariance matrices with nonzero correlation

What does 'simulate a covariance matrix' mean? If the question means, generate an arbitrary correlation matrix for 1000 stocks, then we can choose any symmetric matrix with all 1s down the diagonal, ...
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  • 2,836
5 votes

Is it possible that under Black-Scholes: $\ln S_{T} \sim N \left ( \ln S_t - \frac{1}{2}\sigma^2(T-t), \sigma^2(T-t) \right )$

Starting from the Black-Scholes model that $$ \dfrac{dS}{S} = \mu \:dt + \sigma\:dW_t $$ where $W_t$ is a standard Brownian motion, and $\sigma$ and $\mu$ are constant where $\sigma > 0$. Here $W_t$...
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  • 1,359
5 votes
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Correlation of a lognormal asset and a normal asset

Let $(X_t)_{t\geq 0}$ denote a Geometric Brownian Motion $$ \frac{dX_t}{X_t} = \mu_X dt + \sigma_X dW^X_t,\ \ \ X(0) = X_0$$ such that $X_t$ is lognormally distributed $\forall t > 0$ $$ X_t = X_0 ...
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4 votes

Portfolio choice problem of a CARA investor with n risky assets

This problem is from the exercise for Chapter 2 of Kerry Back's Asset Pricing Book. The setup of the problem is rather simple. You want to \begin{equation*} \begin{aligned} & \underset{\phi}{\...
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  • 650
4 votes
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Why does Bloomberg's HRH test the simple returns for normality?

For small changes, the log-return $\ln \frac{S_{t_i}}{S_{t_{i-1}}}$ is close to the simple return $\frac{S_{t_i}-S_{t_{i-1}}}{S_{t_{i-1}}}$: \begin{align*} \ln \frac{S_{t_i}}{S_{t_{i-1}}} &= \ln \...
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4 votes
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Kurtosis in asset logarithmic returns

Generally Kurtosis measures the degree to which a distribution is more or less peaked than a normal distribution. Positive kurtosis indicates a relatively peaked distribution. Negative kurtosis ...
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4 votes

Kurtosis in asset logarithmic returns

Perhaps an answer coming from a different angle and giving you some perspective: The typical approach taken by statistics is top-down: Just looking at the data and finding patterns and stylized facts (...
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  • 26.7k
4 votes
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Is it possible that under Black-Scholes: $\ln S_{T} \sim N \left ( \ln S_t - \frac{1}{2}\sigma^2(T-t), \sigma^2(T-t) \right )$

Thanks to @Phun and @oliversm I solved the problem. So I'm posting here the solution in case someone will need it. Under Black-Scholes assets dynamics are determined by a Geometric Brownian Motion, ...
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4 votes
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Expectation and Cholesky Decomposition

By Cholesky decomposition, you can express the normal random variables $X$ and $Y$ in the form \begin{align*} Y &= E(Y) + \sqrt{Var(Y)}\, \xi,\\ X &= E(X) + \sqrt{Var(X)}\left(\rho \xi+\sqrt{1-...
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  • 20.4k
4 votes

Determining if a time series is random

So there are several issues with your posting that you will need to resolve. The first one is your concept of randomness and distinguishing between a random event and a non-random event. To ...
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  • 4,064
4 votes
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Proving Scaled Random Walk Approaches Normal Distribution

$X_j$ can be either 1 or -1 with 50% probability each. So this step is just applying the expectation to both possible cases. See definition of the Expectation... \begin{align} {\mathbb E}\bigl[ X \...
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  • 2,836
3 votes
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Bivariate Gaussian copula with exponential margins

$$C(u,v) = \mathbb{P}\left(X\leq N^{(-1)}(u),\quad \rho X + \sqrt{1-\rho^2}X^\perp \leq N^{(-1)}(v)\right)$$
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  • 2,362
3 votes
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Problem with obtaining densities

You know that : $X \sim N(\mu,\sigma^2)$. $Z = \large\frac{X-\mu}{\sigma}$. $\text{Var}(Z) = \large\frac{1}{\sigma^2}\text{Var}(X) = \large\frac{1}{\sigma^2}\sigma^2 = 1$. So that $Z \sim N(0,1)$. ...
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  • 2,504
3 votes

Kurtosis in asset logarithmic returns

At what scale do you see kurtosis? Daily data? Single stocks or indices? Let us not look at single stock data, because you always find crazy stocks whose price process breaks all rules. Talking ...
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  • 13.2k
3 votes

Kurtosis in asset logarithmic returns

I think there are a few conflating ideas here. With respect to the sum of logs idea, I think you're thinking about infinitely divisible distributions (https://en.wikipedia.org/wiki/...
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  • 515
3 votes
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Normally Distributed Returns Become Leptokurtic Due to Compounding

Basically, what you are asking is: What is the distribution of $$ Y = \prod_{i=1}^n X_i $$ where the $X_i$ are i.i.d. and $X_i \sim N(\mu, \sigma^2)$. In general, $Y$ has a very complicated ...
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  • 2,310
3 votes

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

In MPT investors maximize ex ante expected return for a given level of ex ante variance. Gaussian-ity or iid-ness of returns are not requirements. The problem is estimating these ex-ante quantities ...
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