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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. ...


7

What you refer to as the 99.5th percentile is known as the "Value-at-Risk." You are correct that you will need to make a distributional assumption, and there is a popular and well-researched approach to this problem, though I'm not certain it could be called "standard." I would recommend you use the "truncated Levy flight" distribution. James Xiong at ...


7

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


7

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 ...


7

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 ...


6

I think an extremely interesting strand of research on this topic is represented by extensions of vine copulas with time-varying parameters. For vine copulas in general have a look at this site from the Technische Universit√§t M√ľnchen: Vine Copula Models One of their research projects, which is the most relevant in this context, is:Time varying vine copula ...


6

You could try using the Gaussian Affine Term Structure Models (GATSM), with the right boundary conditions to stop rates being negative (in the style of their Black implementation). See, for example, Monika Piazzesi, the "Affine Term Structure Models" if you want to enter/modify the basis or the work of Krippner, for example "Measuring the stance of monetary ...


6

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 ...


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

The question requires you to provide a method which uses uniform random variables and transforms them to generate realizations of the described asset values. To give a bit more general answer: this is solved by the inverse transform sampling method. The main idea is to obtain realizations of a random variable $x$ with any given distribution function $F(x)$, ...


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

Because we are modelling the underlying price process, not the value process of your stop-loss portfolio...


5

You can't have a precise argument without a precise definition. In general, the appropriate notion of integral here is the Lebesgue-Stieltjes integral. In a fairly general setup, let $F: \mathbb R \to \mathbb R$ be a right-continuous function that is of locally bounded variation, that is $$V_F([a,b]) := \sup\lbrace \sum_{i=1}^n \vert F(x_{i+1}) - F(x_i ) \...


4

The consensus nowadays is that stable distributions are not a well fit, although they do possess heavy tails. In particular Cauchy has too fat tails. The reasons for this are disparate, however the first that comes to mind is that empirically longer horizons show a decrease in tail thickness, approaching normality for 1-year returns (although this has been ...


4

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 ...


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.


4

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 $75\%...


4

In this related question How to derive the implied probability distribution from B-S volatilities?, it is shown how to infer the implied probability density of the future prices of a risky asset from a continuum of call prices written on that asset (Breeden-Litzenberger identity). The developments, which I invite you to read, basically rely on the fact ...


4

You should post on mathematics.stackexchange I answer but I should not. Let $X $ be an exponential r.v. of parameter $\lambda $ $$P (X<u|X <T)=\frac {P (X<min (u,T))}{P (X <T )} $$ So for $0\leq u\leq T$ $$P (X<u|X <T)=\frac {1-\exp (-\lambda u)}{1-\exp (-\lambda T)} $$ So if $U $ is an uniform on $ [0,1]$ then $$Y= -\frac {1}{\lambda }\...


4

Answer If you assume your returns are independent (yes your models might loosen this assumption) then the two models, $Q_1$ and $Q_2$ assign probability distributions to the returns on any given day, $i$: $q_1^i(r^i)$ and $q_2^i(r^i)$. Presumably you are interested in the model that can more accurately predict the state of the market over subsequent ...


4

Assume we knew the density function $f$ of the FX price that we observe in the market. Then the market price of a call option $C(K)$ with strike $K$ would be \begin{align*} C(K)&=e^{-rT} \int_0^{\infty}(s-K)^+ f(s)ds \\ &=e^{-rT} \left( \int_K^{\infty} s f(s)ds - K \int_K^{\infty}f(s)ds \right) \tag*{(1)}. \end{align*} $C(K)$ is a market price and ...


3

I wrote a proof deriving the distribution of returns for all asset and liability classes. If there were no budget constraint, limitation on liability and liquidity had no cost, then you can prove the distribution of returns follows a Cauchy law. The budget constraint triggers skew that becomes larger and larger the higher the return. The reason is that ...


3

What about this sketch of an answer: Let's put $T=1$ in your formula to simplify the notation. Then $Y_b(t)$ is a Brownian bridge where $Y_b(0)=0$ and $Y_b(1)=b$. This can be written as $Y_b(t) = b\ t + Y_0(t)$, that is to say the standard Brownian bridge (from zero to zero) with an added drift $b\ t$. The standard Brownian bridge can be written in terms ...


3

By accident i stumbled upon a new work submitted yesterday (http://arxiv.org/abs/1208.5896). It seems most papers are talking about a specific dataset following Benfords law. But that seems not to be what you are looking for. You should look into this bibliography here ( about Benford's law in general): http://arxiv.org/abs/math/0607168 There are also ...


3

the risk neutral drift is needed for pricing of derivatives. For a $100\%$ equity portfolio you can take the real world drift - sometimes a good guess is a drift of zero. For fixed-income you could do the same and might need more sophistication for the variance term. If you have short-dated bonds then you will need a special model for the pull-to-par. For ...


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

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) Their ...


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

The rsgt is a skewed generalized t distribution, whereas your picture is a skewed student-t distribution. Try using fGarch package. Plot reproduced: library(fGarch) x<-seq(-2.5, +2.5, by=0.001) plot(x, fGarch::dsstd(x, mean = 0, sd = 1, nu = 30, xi = 1 + 0.5), type = "l", ylim=c(0, 2.4), lty = 1, xlab="z", ylab=expression(paste("...


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