13
votes
Accepted
Copula Correlations
This may not be a consequence of biased estimators or sampling error. I don't think it is a coincidence that
$$\frac{6}{\pi} \arcsin\left(\frac{0.9}{2} \right) = 0.891457\ldots \approx 0.891$$
...
- 3,365
9
votes
Accepted
Copulas simply explained
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 ...
- 206
8
votes
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How to price this basket option?
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 ...
- 14k
7
votes
Copulas simply explained
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 $...
- 13.3k
7
votes
Copulas simply explained
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 ...
- 27k
5
votes
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Portfolio VaR with Copula?
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 ...
- 1,484
5
votes
Accepted
Simulating from a multivariate clayton copula
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^...
- 363
5
votes
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Why do we not use copula for forward starting options?
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 ...
- 5,038
4
votes
Copula Correlations
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}{...
- 8,089
4
votes
Accepted
Gaussian copula calibration to option price
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 ...
- 1,933
4
votes
Accepted
Optimal Hedging Ratio using Copula Models
Using a static copula model implies $\rho_{s,f,t}\equiv\rho_{s,f}$. In such case fitting a copula model to obtain $\rho_{s,f}$ is an overkill, since it can be estimated very simply by the empirical ...
- 1,788
3
votes
Accepted
Any video lecture on copula function, a statistics concept for measuring dependence?
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 ...
- 46
3
votes
Copula Correlations
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 ...
- 13.3k
3
votes
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Verifying that the extreme value copula is indeed a copula
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, ...
- 20.5k
3
votes
Accepted
How to sample from a copula in matlab
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 ...
- 1,487
3
votes
Accepted
Copulas and default probability
$$\text{Pr}[\tau_1>t,\tau_2\leq t,\tau_3\leq t]=\text{Pr}[\tau_2\leq t,\tau_3\leq t] - \text{Pr}[\tau_1\leq t,\tau_2\leq t,\tau_3\leq t]$$
$$\text{Pr}[\tau_2\leq t,\tau_3\leq t]=C(1,q_2(t),q_3(t))$...
- 2,362
3
votes
Accepted
Bivariate Gaussian copula with exponential margins
$$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)$$
- 2,362
3
votes
How to combine Gaussian marginals with Gaussian copula to obtain multivariate normals?
You can express the Normal distribution by Sklar's Theorem in terms of Gaussian Marginals and Gaussian Copula as follows:
$$F(x_1,...,x_n)=C(F(x_1),...,F(x_n))=C^{Gau}(N(x_1),...,N(x_n))$$
So the ...
- 5,649
3
votes
Accepted
Copula- AR simulation
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 ...
- 5,649
3
votes
Does Value-at-Risk have any mathematical equivalence to copulas?
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 ...
- 1,933
3
votes
What was the probability distribution used for Mortgage Backed Securities (MBSs) during the subprime crisis?
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 ...
- 421
3
votes
How to include heteroscedasticity in copula modelling
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 ...
- 1,788
3
votes
Beginner's resources on copulas and impact of correlation on loan defaults?
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 ...
- 31
2
votes
Ito integrals and copulas
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 ...
- 106
2
votes
Fitting Copula and Simulation
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 ...
- 86
2
votes
Accepted
Relation of survival and non-survival Marshall-Olkin copula
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{...
- 20.5k
2
votes
Accepted
Clayton-Gumbel (BB1) and Joe-Clayton (BB7) time-varying copulas
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".
- 36
2
votes
Simulate (imaginary) asset prices using random numbers that follow a Frank Copula
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 ...
- 13.3k
2
votes
Copulas simply explained
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 ...
- 184
2
votes
Gaussian vs Student Copula applied to finance
The first graph with $\rho=0.1$ is straightforward. The t-copula presents more tail dependence than the gaussian copula. Hence, when you look at the tail, there is more probability mass in the case of ...
- 247
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