16
votes
Accepted
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 ...
- 6,743
8
votes
Accepted
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(...
- 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*}...
- 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 ...
- 26.7k
7
votes
Accepted
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 + \...
- 20.4k
7
votes
Accepted
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 ...
- 118
6
votes
Accepted
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
$$
S_t = S_0 \exp( (\mu-\sigma^2/2) t + \sigma ...
- 13.2k
6
votes
Accepted
Quantile normal and lognormal
Quantiles are preserved under monotonic transformations, hence the quantile for $Y$ is simply the exponential of the quantile of $X$, no need for corrections whatsoever (see here for instance).
Put ...
- 13.9k
6
votes
Measure of a Brownian motion = normal distribution?
It is correct that
$$
\mathbf{P}(t^{-1/2}W(t) \in[a,b])=Φ(b)−Φ(a), \forall t\in(0,\infty)
$$
due to the stationary increments property of the Wiener process and the fact that you normalized the ...
- 71
6
votes
Accepted
Why assume stock returns are normally distributed instead of just adjusting the kurtosis?
If we are talking about risk management (Hence, the risk neutral world), normality allows us to get closed form solutions. For instance, the Black and Scholes equation assumes Gaussian returns (...
- 1,346
6
votes
Accepted
Steven Shreve: Stochastic Calculus and Finance
Note merely that $B_t=B_s+(B_t-B_s)$ which is the sum of independent normally distributed random variables. In particular, $B_s$ is $\mathbb{F}_s$-measurable and $B_{t-s}$ is independent of $\mathbb{F}...
- 13.8k
6
votes
Accepted
Simulating covariance matrices with nonzero correlation
What does 'simulate a covariance matrix' mean?
If the question means, generate an arbitrary correlation matrix for 1000 stocks, then we can choose any symmetric matrix with all 1s down the diagonal, ...
- 2,836
5
votes
Is it possible that under Black-Scholes: $\ln S_{T} \sim N \left ( \ln S_t - \frac{1}{2}\sigma^2(T-t), \sigma^2(T-t) \right )$
Starting from the Black-Scholes model that
$$
\dfrac{dS}{S} = \mu \:dt + \sigma\:dW_t
$$
where $W_t$ is a standard Brownian motion, and $\sigma$ and $\mu$ are constant where $\sigma > 0$. Here $W_t$...
- 1,359
5
votes
Accepted
Correlation of a lognormal asset and a normal asset
Let $(X_t)_{t\geq 0}$ denote a Geometric Brownian Motion
$$ \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 ...
- 13.9k
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}{\...
- 650
4
votes
Accepted
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 \...
- 20.4k
4
votes
Accepted
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 ...
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 (...
- 26.7k
4
votes
Accepted
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, ...
- 253
4
votes
Accepted
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-...
- 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 ...
- 4,064
4
votes
Accepted
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 \...
- 2,836
3
votes
Accepted
Bivariate Gaussian copula with exponential margins
$$C(u,v) = \mathbb{P}\left(X\leq N^{(-1)}(u),\quad \rho X + \sqrt{1-\rho^2}X^\perp \leq N^{(-1)}(v)\right)$$
- 2,362
3
votes
Accepted
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 $Z \sim N(0,1)$. ...
- 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 ...
- 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/...
- 515
3
votes
Accepted
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 ...
- 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 ...
- 4,217
Only top scored, non community-wiki answers of a minimum length are eligible