11
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 ...
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
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
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 $\...
6
You ask 2 questions and I try to answer:
1) Why do we use geometric Brownian motion ($\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
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 ...
5
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 ...
5
Since both $ER$ and $S$ are gaussian random, why not just assume their dependence is captured by their covariance, and make your draws from the bivariate normal distribution? It is hard to construct any other way of making two marginal gaussians cointegrated.
Even if the variables were not gaussian, you would probably find yourself relating them using a ...
5
This is simply the integral of the pdf from -0.5 to 0.5 (scaled to the SD of the distribution), also known as the cumulative distribution function or cdf. The cdf(x) function is indicated on the following wikipedia link: Normal Distribution.
The normal cdf(x) function computes the integral on [-Infinity, x], so to compute on your interval [x1,x2], is ...
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
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).
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
$$ dY_t = \mu_Y dt + \sigma_Y dW_t^Y,\ \ \ ...
4
The term 1/2 * sigma-squared arises through the application of Ito's Lemma. Keep in mind that the assumption is of a stock price that follows geometric BM with a constant drift and volatility. If you set up a delta-hedge portfolio and apply Ito calculus you will end up with an adjustment in the distribution by exactly above term. Another way of interpreting ...
4
For small changes, the log-return $\ln \frac{S_{t_i}}{S_{t_{i-1}}}$ is close to the simple return $\frac{S_{t_i}-S_{t_{i-1}}}{S_{t_{i-1}}}$:
\begin{align*}
\ln \frac{S_{t_i}}{S_{t_{i-1}}} &= \ln \Big(1+ \frac{S_{t_i}-S_{t_{i-1}}}
{S_{t_{i-1}}} \Big)\\
&\approx \frac{S_{t_i}-S_{t_{i-1}}}{S_{t_{i-1}}}.
\end{align*}
Note also that, assuming the SDE
\...
4
Thanks to @Phun and @oliversm I solved the problem. So I'm posting here the solution in case someone will need it.
Under Black-Scholes assets dynamics are determined by a Geometric Brownian Motion, and we can define the price of a security at time $t+\Delta t$ as:
$$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 ...
3
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 ...
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
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 ...
3
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 ...
3
You can refer to Shreve's book, Volume II, Section 4.4.3 .
Assume that we have a generalized geometric Brownian motion
$$dX_t = \sigma_t dW_t + (\alpha_t - \frac{1}{2} \sigma_t^2) dt ,$$
where the drift coefficient and the volatility are functions of $t$ also. $(dX_t)^2 = \sigma_t^2 dt + \mathcal{O}(dt^{3/2})$ .
Assume that the asset price is
$$ S_t = S_0 ...
3
Basically, what you are asking is: What is the distribution of
$$ Y = \prod_{i=1}^n X_i $$
where the $X_i$ are i.i.d. and $X_i \sim N(\mu, \sigma^2)$.
In general, $Y$ has a very complicated distribution. Check out the discussion in
https://math.stackexchange.com/questions/161757/what-is-the-distribution-of-a-random-variable-that-is-the-product-of-the-two-...
3
You know that :
$X \sim N(\mu,\sigma^2)$.
$Z = \large\frac{X-\mu}{\sigma}$.
$\text{Var}(Z) = \large\frac{1}{\sigma^2}\text{Var}(X) = \large\frac{1}{\sigma^2}\sigma^2 = 1$.
So that $Z \sim N(0,1)$.
However note that the pdf evaluated for X and Z have different
domains.
The following figure illustrate it :
$X$ is plotted in a) and $Z$ in b)
Their ...
3
$$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)$$
3
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 using ex-post transformations of the recent history. Many of the sophisticated techniques that you mention try to offer robustness of ex post estimators in the ...
3
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 have a mean of zero. In fact the mean as you say is slightly positive.
So what can we do? Well we can test the data. I took the SPX returns between 1980 and 2012 ...
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