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8

Upon close reading, this appears to be 3 (interesting) questions, not one. I'm not sure if the mods have the tools needed to split it up, so I'm just going to write down the three questions as I see them and then deal with them one by one. Note, it is simpler for me to talk about variance instead of volatility. This has no material impact on the answer. ...


6

Well there are two main things to consider here. Many implementation of Black-Litterman use the market portfolio and the ex post volatility and correlation structure to back out implied returns to use as prior. As far as I know, there is no standard way to reverse-engineer the optimization problem in the presence of nonnormal markets. (the first guess is ...


5

The VaR of level $\alpha$ a loss random variable (the bigger the worse) is the quantity $q$ such that the loss is bigger with probability $1-\alpha$. Thus we need a $q$ such that $$ P[L>q] = 1-\alpha, $$ where we can imagine $\alpha=99\%$ and thus we need the starting point of the $1\%$ tail. Because we have a probability of a loss of size $0$ of ...


4

I found Coping With Copulas by Thorsten Schmidt really helped me to get a more basic understanding of copulas. As well as looking at some simple examples in R and thinking about different directions the transformations can happen. To answer your actual question I'll attempt to describe the steps involved as simply as I can. Let's say you use the copula ...


4

Surely, there is; search for aggregational gaussianity in Google Scholar or ScienceDirect. In fact, 5 minutes returns are leptokurtic and fat-tailed; then as you increase timeframe, returns become more and more normal. Yearly data is almost normal, if you have enough points.


3

Your question is not clear. What you might want to say is what distribution should the futures price follow, under the risk-neutral or physical probability measure. In this sense, it will depend on your intention. For potential future exposure, you may want to use the physical measure for the price evolution, while the distribution will depend on your model ...


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) ...


3

When possible, I look at implementations in IMSL and the GSL for really good accuracy. Neither one appears to implement the Wald (inverse gaussian) or its quantile function. Matlab does have the distribution (as inversegaussian) so you could roll your own with fzero() or another root-finder based on that if you are unhappy with the accuracy, or for testing ...


3

My main reference will be "Dan Xu, Christian Beck - Transition from lognormal to chi-square superstatistics for financial time series" Non-equilibrium statistical mechanics (more specifically, superstatistics) gives some ideas of explaining the relation between time frame and its distribution: "...to regard the time series as a superposition of local ...


3

Hi bcf: This is a good question. As you pointed out below, \begin{align*} p_0 &= \delta(y-y_0)\\ &=\delta(e^w-y_0). \end{align*} Then, \begin{align*} p_0 * g &= \int_{-\infty}^{\infty}\delta(e^z-y_0) g(w-z) dz\\ &=\int_{0}^{\infty}\delta(u-y_0) g(w-\ln u) \frac{1}{u}du\\ &= \frac{1}{y_0}g(w-\ln y_0). \end{align*} Consequently, your last ...


2

You might also look at the boost package which should (I'm no expert for this) be usable within C#. It comes with an implementation of the inverse normal distribution which is explained in the online documentation http://www.boost.org/doc/libs/master/libs/math/doc/html/math_toolkit/dist_ref/dists/inverse_gaussian_dist.html Here they claim quite a high ...


2

The best introduction to copulas I know, i.e. with rigour and intuition, is the following. THE QUANT CLASSROOM BY ATTILIO MEUCCI A Short, Comprehensive, Practical Guide to Copulas Visually introducing a powerful risk management tool to generalize and stress-test correlations


2

Normal distribution makes most sense these days for ratesthat are very low, or even negative, like euribor, chf libor Normal distribution is what is assumed by option brokers impliedvolatility quotes for these currencies


2

In the theory of copulas you want to model a multivariate (often bivariate) distribution and keep the marginals fixed. Thus you have random variables $X$ and $Y$ with cdf $F_X(x) = P[X \le x]$ and $F_Y(y) = P[Y\le y]$ and you want to find some $F_{X,Y}(x,y) = P[X \le x, Y\le y]$ such that when you look at marginals you get $F_{X,Y}(x,\infty) = F_X(x)$ and ...


1

Have a look at ?dnorm, and rather use the standardized value as argument in your function, in addition to mean and sd: a_<-dnorm((0.001-0.0001)/0.4, mean=0, sd=1) Hope it helps [EDIT] Likewise from ?dsgt st<-(0.001-0.0001)/0.4 skewt<-dsgt(st, mu=0, sigma=1, lambda=0.1, p = 2, q=5, mean.cent=TRUE, var.adj=TRUE) results in skewt=0.4302996 (close ...


1

1- It seems to me there is a problem in the original code the variable b should be defined as b= sqrt(1 + 3*lamda^2 - a^2) 2- The likelihood is defined just after equation 8. in the paper. You have to take into account the $ \frac{1}{\sigma}$ term (in $ \frac{1}{\sigma} \times g(..) $ , ie to scale the densitie) . So the - 0.5*log(h(t)) refers to this ...


1

If high frequency returns are iid and the mean and variance are finite and vthe variance is greater than zero then the Central Limit theorem holds Then, regardless of the distribution of the high returns, when aggregated over time the aggregated returns will tend in distribution to a Normal distribution. The Lindeberg-Lévy-Feller version of the Central Limit ...


1

You are concerned about non-normality, heteroskedasticity, and autocorrelation in your data. The normality of errors is not an assumption of OLS (it is for MLE regression). That is, you can conclude that OLS is the best linear unbiased estimator (BLUE) without assuming normality. Nevertheless, there are a number of techniques within the context of robust ...


1

The issue I have with these approaches is that they use the unconditional distribution to eliminate the latent volatility. However, when the volatility process has very weak mean reversion one would need a very long and clean sample to make robust parameter identification from the unconditional density. They just throw away all the information from the ...


1

Gatheral (Amazon) has a quite extensive discussion on that, and dives into calibration issues. In summary, what you describe appears to be less of a modeling issue, and more of a calibration problem. This is primarily because the model functions (such as the Heston model) are not by nature convex in their input parameters. This is simply result of the fact ...


1

There is a brief and not overly technical introduction here: http://prescientmuse.blogspot.co.uk/2015/01/a-brief-introduction-to-copula.html And an application of use in a trading system with full R code here: http://prescientmuse.blogspot.co.uk/2015/02/vanilla-trading-algorithm.html Hope that helps!



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