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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

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 ...


4

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 ...


4

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 ...


3

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 ...


2

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).


2

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 ...


2

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 ...


2

As @Joshua Ulrich points out your distribution gets wider. Approximately, what you do is, you simulate $$ Y = X_1 + \dots + X_n $$ and $X_i$ is standard normal. Of yourse the variance increases with $n$ (and standard deviation with $\sqrt{n}$). But: greater variance does not mean heavier tails as suprises. If you want to put this in an easy number (besides ...


2

I think some some terminology got mixed up here. Let $r_t$, $t=1,\ldots,T$ be a series of iid excess returns with the estimated mean excess return $\bar{r}= \sum_{t=1}^Tr_t$. Then the Stutzer Index $S$ is defined as $ S=\frac{|\bar{r}|}{\bar{r}}\sqrt{2I_p}$ with $I_p$ being the "Stutzer Information Statistic", $I_p=\max_\theta -\log(\frac{1}{T}\sum_{t=1}^T ...


2

Your SDE has no closed-form solution, so you'll have to apply the Euler method to obtain an approximate terminal distribution. Once you have the terminal distributions, any time series you want to validate has a highly multivariate probability density (due to the fact that each day's data comes from a slightly different distribution). You can transform ...


1

As Quartz says it is possible to make non-linear transformations taking into account skew and kurtosis, but this is mostly is limited to univariate processes (one approach for a t distribution is to match moments). For multivariate processes, it is considerably more difficult. A more general solution is to rely on Entropy Pooling. You could take views on ...


1

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 ...


1

If I understand well, one part of your analytic is already normalized ((closeVsLow - closeVsHigh) / myrange), but not the other (volume). If you just aim to compare the values of AD from any stock with the other, why not normalizing volumeby the usual daily volume (median of the daily volume) of the stock during the last 60 days? Moreover, if you really ...



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