15
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 normality then there is no point in the investor being interested in anything else.
(we try to discuss assumptions thoroughly in our book, Introduction to ...
8
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(t+1) = \mu - \Psi_{t+1}(-1) + ln S(t) + \epsilon(t+1), \; \epsilon(t+1) | F_t \sim N(0,\sigma^2).
\end{equation}
where $\Psi_{t+1}(u) := ln E_t \left( \exp( -u ...
7
So we have the BS-Model
$$dS_t=S_t(\mu dt +\sigma dW_t)$$
W.l.o.g we assume $S_0=1$. Itô's lemma implies that
$$S_t=\exp{(\sigma W_t+(\mu-\frac{1}{2}\sigma^2)t)}$$
We know that $W_t$ is normally distributed with mean $0$ and variance $t$. Now have a look at the r.v.
$$X_t=\sigma W_t+(\mu-\frac{1}{2}\sigma^2)t$$
$\sigma W_t$ is the random part and $\...
7
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 + \lambda_2 \int_0^t W_s ds &= \lambda_1\int_0^t dW_s + \lambda_2 \int_0^t (t-s)dW_s\\
&=\int_0^t \big(\lambda_1 + \lambda_2(t-s)\big)dW_s,
\end{align*}
...
6
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
$$
S_t = S_0 \exp( (\mu-\sigma^2/2) t + \sigma B_t),
$$
which means that you model positive prices. Furthermore the log-return
$$
\ln(S_t/S_0) = (\mu-\sigma^2/2) t + \sigma B_t,
$$
is normally distributed. ...
6
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 them parsimoniously. The following paper was already mentioned in the comments:
Empirical properties of asset returns: stylized facts and statistical issues by ...
6
I think one has to distinguish between pricing and fitting/reproducing empirical stock returns. A model might fit the empirical stock returns extremly well but fail to reproduce derivative prices. In my answer I will assume that you are interested in reproducing the empirical stock returns.
Mandelbrot and the Stable Paretian Hypothesis
The most salient ...
6
Another way of seeing it is that the $-\frac12\sigma^2$ is just a correction term that comes from Jensen's inequality.
You need this when switching from supposedly symmetric returns (normal distribution) to the skewed price process (log-normal distribution).
6
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 otherwise, let $q$ denote the quantile $\alpha$ of $X$ i.e.
$$\Bbb{P}(X \leq q) = \alpha$$
then
\begin{align}
\Bbb{P}( X \leq q ) &= \Bbb{P}( \underbrace{\...
6
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 random variable by dividing by its standard deviation.
$\mathbf{P}$ is a probability measure on an abstract space, not a random variable. Hence, you probably mean ...
6
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 (Equivalently, the stock follows a geometric Brownian motion). Your thought is correct, although you can not simply adjust for kurtosis. You need to define properly ...
6
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 $\mathbb{F}_s$. Thus,
\begin{align*}
\mathbb{E}_s[Z_t] &= \mathbb{E}_s\left[\exp\left(-\frac{1}{2}\theta^2t+\theta B_t\right)\right] \\
&= \mathbb{E}_s\left[\exp\...
6
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, so long as every element is between -1 and 1 and the matrix is positive semi-definite. The large size of the matrix means that putting random values in every ...
5
It's not possible with a simple linear transformation like the one you mentioned: since scale and thus the distance between mean and median are required to change, either the mean or the median will not be preserved.
Therefore you must use nonlinear transformations, which will complicate quite a bit mantaining skew and kurtosis and imho will not be ...
5
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 $W_t$ is a Brownian motion under the physical measure $\mathbb{P}$. We can then use Girsanov's theorem to change the measure to risk neutral measure $\mathbb{Q}$ ...
5
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 $\forall t > 0$
$$ X_t = X_0 e^{(\mu_X - \frac{1}{2}\sigma_X ^2)t + \sigma_X W_t^X}$$
Let $(Y_t)_{t\geq 0}$ denote an Arithmetic Brownian Motion
$$ dY_t = \mu_Y dt + \sigma_Y dW_t^Y,\ \ \ ...
4
The term 1/2 * sigma-squared arises through the application of Ito's Lemma. Keep in mind that the assumption is of a stock price that follows geometric BM with a constant drift and volatility. If you set up a delta-hedge portfolio and apply Ito calculus you will end up with an adjustment in the distribution by exactly above term. Another way of interpreting ...
4
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}{\text{maximize}}
& & \phi'\mu + \frac{1}{2} \alpha \phi' \Sigma \phi\\
& \text{subject to}
& & 1'\phi = w_0
\end{aligned}
\end{equation*}
The ...
4
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 \Big(1+ \frac{S_{t_i}-S_{t_{i-1}}}
{S_{t_{i-1}}} \Big)\\
&\approx \frac{S_{t_i}-S_{t_{i-1}}}{S_{t_{i-1}}}.
\end{align*}
Note also that, assuming the SDE
\...
4
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 indicates a relatively flat distribution.
In time series we can encounter high kurtosis which is caused by "fat tails" (higher frequencies of outcomes) at the ...
4
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 (like excess volatility, volatility clustering, fat tails, no autocorrelation in returns but significant autocorrelation in absolute returns etc.) The problem ...
4
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, and we can define the price of a security at time $t+\Delta t$ as:
$$S_{t+\Delta t}=S_{t}\exp\left(\left(r-\frac{1}{2}\sigma^{2}\right)\Delta t+\sigma\sqrt{\...
4
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-\rho^2} \eta\right),
\end{align*}
where $\rho = \frac{Cov(X, Y)}{\sqrt{Var(X)Var(Y)}}$ is the correlation, $\xi$ and $\eta$ are two independent standard normal ...
4
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 understand the problem, I think I should tell you a story.
You go home for a family reunion and see a tree you used to climb as a small child. You see the branch you ...
4
$X_j$ 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 \bigr] = \sum_i i \cdot p(x = i)
\end{align}
It's the sum over all possibilities of the probability of getting that value (both ${\frac 1 2}$ in your case) ...
3
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 about daily data of indices: they could be thought of the sum of hourly returns or other returns of high frequency (minute returns, milliseconds ...).
What are the ...
3
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/Infinite_divisibility_(probability)). These ideas are indeed used to build more complicated models (i.e. Levy processes) for asset returns.
With regards to the Efficient Market ...
3
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 $Z \sim N(0,1)$.
However note that the pdf evaluated for X and Z have different
domains.
The following figure illustrate it :
$X$ is plotted in a) and $Z$ in b)
Their ...
3
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 distribution. Check out the discussion in
https://math.stackexchange.com/questions/161757/what-is-the-distribution-of-a-random-variable-that-is-the-product-of-the-two-...
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