10
This may not be a consequence of biased estimators or sampling error. I don't think it is a coincidence that
$$\frac{6}{\pi} \arcsin\left(\frac{0.9}{2} \right) = 0.891457\ldots \approx 0.891$$
Copula construction involves applying nonlinear transformations to random variables which need not preserve correlation.
If random variables $X$ and $Y$ are ...
8
No offense but it will be much more complicated than what you think... I'm not even sure that you are familiar with risk-neutral pricing in the first place? I'll try to give you some clues.
This security is called a basket option. On top of the multi-asset feature, there are non-trivial mechanisms embedded in the contract you mention:
an auto-callable ...
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
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
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
The algorithm is certainly useful in that it is non-parametric, fast, and versatile. Meucci summarizes the advantages nicely:
Unlike traditional copula techniques, CMA a) is not restricted to few
parametric copulas such as elliptical or Archimedean; b) never
requires the explicit computation of marginal cdf’s or quantile
functions; c) does not ...
5
Here is a working paper that you may be interested in.
5
In general setting this is quite a tough problem and it looks like just switching from regular multivariate probability to copulas doesn't make it easier. In general case you need to rely on numerical methods for integration.
There is a nice overview of the problem in Copula Theory and Its Applications: Proceedings of the Workshop Held in Warsaw, 25-26 ...
5
If the density of $(X,Y)$ is known, then you may obtain the density of the sum $X+Y$ simply by applying the Jacobi's transformation formula, which describes the density of the transformed random variable $g(X,Y)$ for $g(x,y) = (x+y, x)$. Integrating out the $x$-component yields the density of $X+Y$. See Jacod/Protter Probability Essentials ch. 12 for details....
5
In general you don't need copulas to calculate VaR on portfolio. You can use historical method if you have time series of returns for the assets in your portfolio. If you have sufficiently enough data this will allow you to take into account correlation risk, non-normality of returns.
Example of code in R for equally weighted portfolio without assuming any ...
4
Look here
for multivariate distribution on the positive quadrant ... quite difficult.
http://xianblog.wordpress.com/tag/multivariate-analysis/
I have been thinking about this for weeks and months in the context of credit risk (modelling default intensities jointly) and I think this does not work.
4
Is'nt it true that your first line can be written as
$$
F_{X,Y}(x,y_2) - F_{X,Y}(x,y_1),
$$
where $F_{X,Y}$ is the joint cdf of $(X,Y)$. If we assume that the distributions of $X$ and $Y$ are continuous without atoms (I have to check the exact formulation), then it is clear from Sklar's theorem that there is exactly one copula $C$ such that $$F_{X,Y}(x,y) = ...
4
You don't really have a multivariate case: we can only define VaR (in its usual sense) for a one-dimensional output. Recall that
$$
\operatorname{VaR}_\alpha(X) = \inf\{v:F_X(v)\geq \alpha\}
$$
and since in your case $X = X_1+X_2$ you just need to compute $F_X$ in terms of $X_1$ and $X_2$. For the notation of partial derivatives, I denote the generic ...
4
I would guess you are calculating the maximum likelihood estimator:
$ \hat{\theta} = \frac{1}{N} \sum (x_i - \bar{x}) (y_i - \bar{y}) $
instead of the unbiased estimator:
$ \hat{\theta} = \frac{1}{N-1} \sum (x_i - \bar{x}) (y_i - \bar{y}) $
The unbiased estimator has a bias of zero, i.e. :
$ E_{x|\theta}[\hat{\theta}] - \theta = 0 $
The unbiased ...
3
$$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)$$
3
As you know, simulating AR(1) is to simulate the distributed error path.
Assume the bivariate errors distributed $\sim F(x),\sim F(y)$ with copula $C(u,v)$ to model their dependence.
Then the bivariate joint error distribution is given by Sklar's theorem:
$$F(x,y)=C(F(x),F(y))$$
You can simulate from this distribution using Conditional Sampling:
To ...
3
Do you refer with 'negative tail dependence' to the case that one variable has a extremely low value and the other random variable has an extremely large value, i.e.,
$$\tau=\lim_{p \rightarrow 0} \frac{Pr[x>Q_x(1-p),y<Q_y(p)]}{p},$$
where $Q_x(1-p)$ and $Q_y(p)$ refer to the $(1-p)$-th quantile of the random variable $x$ and the $p$-th quantile of ...
3
Suppose you have the copula $C(u_1,u_2)$, then you could compute the conditional copula
$$c_{u_1}(u_2)=\frac{\partial C(u_1,u_2)}{\partial u_1} \; .$$
Now, you can generate a pair of independent uniformly distributed random values $(U,V)$. Let's say a particular realistation is $(u,v)$. Then the pair
$$(u,c_u^{-1}(v))$$
will be distributed according to ...
3
Note that, you only need to show that
\begin{align*}
A\left(\frac{\log(u_2)}{\log(u_1u_2)}\right)-\frac{\log(u_2)}{\log(u_1u_2)}A'\left(\frac{\log(u_2)}{\log(u_1u_2)}\right) \ge 0,
\end{align*}
or, for any $t \in (0, 1)$,
\begin{align*}
A(t) - t A'(t) \ge 0.
\end{align*}
Recall that $A$ is a convex function from $[0,\, 1]$ to $[1/2,\, 1]$, $A(0)=A(1)=1$, and ...
3
This is an interesting observation that you have. The interesting part is "consistently smaller".
The normal copula is based on a multivariate normal distribution. The correlation you get out is the correlation parameter you put in.
Everything else is most probably due to an issue in your approach. If you did not say "consistently smaller", I would say it ...
3
Since I think this is of interest for other people, I will post the approach I found:
First, let $C_n(u_1,\ldots,u_n)$ be a $n$ - dimensional Clayton copula with generator function $F$ and inverse $F^{-1}$. Then,
Generate $n$ independent r.v. from $U (0,1)$
Calculate $n-1$ derivatives of $F$, where $F_{n-1}$ denotes the $n-1$-th - order derivative of $F$
...
2
It depends on the assets which copula is best and other methods may still be better and comparable in complexity.
If you want to use copula's for equities you can take a look at Clayton copula. While the Gaussian copula is symmetric the Clayton copula has asymmetric tail dependency. This makes modeling the increase in correlation during a crisis possible.
2
if you agree that the marginal probability $P(u\le Y\le v)=F_Y(v)-F_Y(u)$, then your formula follows immediately, because next you simply plug the marginals into the copula.
your 3rd equation for the joint probabilities is incorrect for $P(Z\le z,u\le Y\le v)$, I'm not sure where you got it from
2
Implementations of the BBx families are available from the VineCopula R-package from CRAN. Spatially and spatio-temporally varying bivariate copulas are provided through the R-package spcopula from r-forge. Temporal support will need some additional work as it was not part of the initial design. The tuning of the copulas' parameter can be done via a ...
2
You can express the Normal distribution by Sklar's Theorem in terms of Gaussian Marginals and Gaussian Copula as follows:
$$F(x_1,...,x_n)=C(F(x_1),...,F(x_n))=C^{Gau}(N(x_1),...,N(x_n))$$
So the distribution equals the copula function with the respective inverse marginals as arguments.
You can aswell combine any types of Copula and (continuous) different ...
2
0/ Let's me use more common notations to avoid misunderstanding. We will consider $B_t^x$ and $B_t^y$ - two correlated Brownian motions, e.g. $<dB_t^x,dB_t^y>=\rho dt$.
Just to recall, Ito's process:
$$X_t = X_0 + \int_0^t \mu(s,\omega) ds + \int_0^t \sigma(s,\omega) dB_s^x\\
dX_t=\mu(t,\omega) dt + \sigma(t,\omega) dB_t^x$$
1/ Single BMs: $$\mathbb{...
2
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!
2
$$\text{Pr}[\tau_1>t,\tau_2\leq t,\tau_3\leq t]=\text{Pr}[\tau_2\leq t,\tau_3\leq t] - \text{Pr}[\tau_1\leq t,\tau_2\leq t,\tau_3\leq t]$$
$$\text{Pr}[\tau_2\leq t,\tau_3\leq t]=C(1,q_2(t),q_3(t))$$
2
For non-normal asset price models you could look at the theory of Lévy-processes.
If we assume that you work in the physical probability measure $P$ and that the random numbers that you have generated are daily log-returns, then you can do the following:
Asset $i$ has starting price $S_0^i$ and for the future prices you can put
$$
S_t^i = S_0^i \exp(\sum_{k=...
2
"convoluted expression" in American usage just means a complicated, big mathematical expression, sometimes also called "hairy" or "messy". It is ugly to work with and to look at, so you prefer not to deal with it if possible. Nothing more than that.
There is also a mathematical operation called "convolution ("Faltung" in German) but it has nothing to do ...
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