13
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 ...
9
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 ...
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
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 ...
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
5
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 ...
5
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 $F$
...
5
One is exploring forward volatility of a price of a single asset (joint distributions from within a process), the other explores correlation of two prices at the same time for two different underlyings (glueing, otherwise unrelated, marginal distributions).
Forward starting options depend on the joint distribution of the (already chosen and used to price ...
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 $$F_{X,Y}(x,y) = ...
4
I would guess you are calculating the maximum likelihood estimator:
$ \hat{\theta} = \frac{1}{N} \sum (x_i - \bar{x}) (y_i - \bar{y}) $
instead of the unbiased estimator:
$ \hat{\theta} = \frac{1}{N-1} \sum (x_i - \bar{x}) (y_i - \bar{y}) $
The unbiased estimator has a bias of zero, i.e. :
$ E_{x|\theta}[\hat{\theta}] - \theta = 0 $
The unbiased ...
4
You did not mention it, but I think you also need to include the discount factor $D$ at the time $T$ of maturity of your option as a third variable. Denote the two interest rates as $r$ and $s$ and the pay-out function of your option as $f=f(r,s).$
The price of your option is then the expectation of the discounted cash flow:
$$ \text{price }=\mathbb{E}[f(r,s)...
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
$$\text{Pr}[\tau_1>t,\tau_2\leq t,\tau_3\leq t]=\text{Pr}[\tau_2\leq t,\tau_3\leq t] - \text{Pr}[\tau_1\leq t,\tau_2\leq t,\tau_3\leq t]$$
$$\text{Pr}[\tau_2\leq t,\tau_3\leq t]=C(1,q_2(t),q_3(t))$$
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
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 ...
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
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
It's very difficult to find accessible material on copulas. I'm still struggling to understand them myself. While I haven't come across any videos that explain copulas well, I have found the following resources very helpful.
A blog post: An intuitive, visual guide to copulas, by Thomas Wiecki, is truly introductory with absolute minimal math but it doesn't ...
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
Your confusion stems from you confusing several aspects of VaR and copulas. Note first that Portfolio Value at Risk measures the value at risk of a portfolio. This means the total loss of your portfolio is the sum of losses from single assets, instruments, entities, lines of business ... whatever
$$ S = \sum_{i=1}^n L_i. $$
Now what causes a large total loss?...
3
I think you're asking about the different tranches in a multi-tranch mortgage securitization such as a Collateralized Mortgage Obligation (CMO) was sized, and what the math behind it was. This was how they did it:
They used historical default statistics (probability of default and loss in the event of default) from prior episodes of high mortgage defaults, ...
3
I don't know if this will help solve your convergence issue, but a standard way of incorporating conditional heteroskedasticity in copula models is to build a copula-GARCH model. Each time series is first modelled with GARCH, and then the standardized innovations from the GARCH models of all the series are jointly modelled with a copula.
3
At the risk of arming you to create the next quant-apocalypse...
The statement that the expected loss does not depend on correlation is typically the result of modelling a portfolio as a sum of individual exposures: X+Y+..., and then using: E[X+Y+...]=E[X]+E[y]+.... This does NOT generalize to statements like "expectations don't depend on correlation&...
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
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: $$\mathbb{...
2
You need to estimate or assume a marginal distribution of the (u,v). Lets say you assume normality (don't do this), you would be able to perform a rosenblatt-transformation, to perform the task you describe.
https://en.wikipedia.org/wiki/Inverse_transform_sampling
This could be a useful resource.
2
Note that the survival copula $C_{\theta_A, \theta_B}(u, v)$ and the non-survival copula $C(u, v)$ are related by
\begin{align*}
C_{\theta_A, \theta_B}(\hat{u}, \hat{v}) = \hat{u}+\hat{v}-1 + C(1-\hat{u}, 1-\hat{v}),
\end{align*}
where $\hat{u}=\bar{F}_A(x)=1 - F_A(x)$ and $\hat{v}=\bar{F}_B(y)=1-F_B(y)$.
Then,
\begin{align*}
C(u, v) = C_{\theta_A, \theta_B}(...
2
You can have a look at Andrew Patton's "Copula toolbox for Matlab". It contains his code for the "Time-varying Symmetrised Joe-Clayton copula".
2
For non-normal asset price models you could look at the theory of Lévy-processes.
If we assume that you work in the physical probability measure $P$ and that the random numbers that you have generated are daily log-returns, then you can do the following:
Asset $i$ has starting price $S_0^i$ and for the future prices you can put
$$
S_t^i = S_0^i \exp(\sum_{k=...
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