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17 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 ...
Mark Joshi's user avatar
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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(...
Stéphane's user avatar
  • 2,486
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*}...
user53249's user avatar
  • 419
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 + \...
Gordon's user avatar
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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 ...
Adam N.'s user avatar
  • 233
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 ...
Quantuple's user avatar
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6 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 ...
Quantuple's user avatar
  • 14.7k
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 ...
phantagarow's user avatar
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 (...
alexbougias's user avatar
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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}...
Kevin's user avatar
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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, ...
StackG's user avatar
  • 3,046
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 $W_t$...
oliversm's user avatar
  • 1,399
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-...
Gordon's user avatar
  • 21.2k
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, ...
ChicagoCubs's user avatar
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 ...
Dave Harris's user avatar
  • 4,299
4 votes
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How accurate is the square root of time rule for VaR for a portfolio containing several different types of instruments

Effectively, I sense two questions here, 1) around the validity of the $\sqrt{T}$-assumption in the scaling of the risk horizon ; and 2) the quality of the $ \Delta$-$\Gamma$-approximation in ...
Kermittfrog's user avatar
  • 6,832
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 \...
StackG's user avatar
  • 3,046
3 votes

Central limit theorem and normality assumption of asset return distribution

No. I just published a paper on this. If return is defined as $$r_t=\frac{p_{t+1}q_{t+1}}{p_tq_t},$$ and since returns are not data while prices and volumes are, then it follows that the ...
Dave Harris's user avatar
  • 4,299
3 votes
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Bivariate Gaussian copula with exponential margins

$$C(u,v) = \mathbb{P}\left(X\leq N^{(-1)}(u),\quad \rho X + \sqrt{1-\rho^2}X^\perp \leq N^{(-1)}(v)\right)$$
M. Jeunesse's user avatar
  • 2,432
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 ...
Kiwiakos's user avatar
  • 4,337
3 votes

If equity returns are normally distributed, why are average equity returns not zero

Well there are two misconceptions in your assessment of how returns behave. 1) Returns can be normally distributed or not; 2) Even if they are normally distributed it does not mean that returns ...
phdstudent's user avatar
  • 8,431
3 votes

If equity returns are normally distributed, why are average equity returns not zero

It's news to me that in today's world anybody really believes that equity returns are normally distributed. For instance in US Senate testimony by a Goldman Sachs CFO, under assumptions of Gaussian ...
DJohnson's user avatar
  • 169
3 votes

What is the Probability Distribution of Max-Drawdown?

I think the answer you're looking for is very similar to this question Expectation of maximum draw down in the Brownian motion case. just like your assumption that return is normally distributed with ...
numerairX's user avatar
  • 609
3 votes
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Why can we assume that asset return rates are normally (or lognormally) distributed?

In the colloquial sense of the word "justified," it is not justified. I will describe why it is justified mathematically and under what circumstances and in what case it is not justified. ...
Dave Harris's user avatar
  • 4,299
3 votes

Transforming non-normally distributed interest rates for OLS regression

You do not need to transform the variables into normally distributed data in order to use them in a regression. That is not a requirement of ordinary least squares. If the error terms are normally ...
Dave Harris's user avatar
  • 4,299
3 votes

Asset return distribution

In the Black Scholes (1973) model, the stock price is assumed to follow a geometric Brownian motion $\mathrm{d}S_t=S_t\mu \mathrm{d}t + S_t \sigma \mathrm{d}W_t$. If you solve the SDE, $(S_t)$ is log-...
Kevin's user avatar
  • 16k
3 votes

Simulating covariance matrices with nonzero correlation

I will just clarify Point 2 in StackG excellent answer. (It's really a comment, but it's too long and has too much math symbols to fit in the comment field.) Suppose you're given a covariance matrix $...
Dimitri Vulis's user avatar
3 votes
Accepted

A simple question about VaR estimation

I think there is a mistake in your definition. It should be between "50th and 51st" sorted numbers. 95% VAR means 5% is in the tail. 5% * 1000 = 50. The 95% VAR will be the 50th worst ...
AlRacoon's user avatar
  • 6,632

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