# Copula-GARCH模型下的两资产期权定价

Garchmodels unlocks univariate and multivariate GARCH models in one framework.

## Copula-Garch Model

Ciprian Necula Senior Lecturer, DOFIN, Faculty of Finance and Banking, Copula-GARCH模型下的两资产期权定价 Bucharest University of Economics, Bucharest, Romania Romania; Researcher, Center for Advanced Research in Finance and Banking (CARFIB),Bucharest University of Economics, Bucharest; Researcher,Centrul de Analiză şi Prognoză Economico-Financiară (CAPEF) [email protected]

• https://doi.org/10.1080/1331677X.2010.11517408

## A Copula-Garch Modelcopula-Garch Model

In the present study we develop a new two-dimensional Copula-GARCH model. This type of two-dimensional process is characterized by a dependency structure modeled using a copula function. For the marginal densities we employ Copula-GARCH模型下的两资产期权定价 a GARCH(1,1) model with innovations drawn from a t-Student distribution. The model can be easily extended by using more sophisticated processes Copula-GARCH模型下的两资产期权定价 for the marginal densities. The static specification of the model assumes that the dependency structure of the two data series does not vary in time implying that the parameters of the copula function are constant. On the other hand, the dynamic specification models explicitly the dynamics of these parameters. We econometrically estimate the parameters of the two specifications using various copula functions, focusing on the mixture between the Gumbel and Clayton copulas. We employ daily index returns from two emerging and two developed financial markets. The main finding is that including a varying dependency structure improves the goodness-of-fit of the Copula-GARCH model. 1

U ovom smo istraživanju razvili novi dvodimenzionalni Copula-GARCH model. Ovu vrstu dvodimenzionalnih procesa karakterizira zavisna struktura stvorena koristeći spojnu funkciju (kopulu). Za marginalne gustoće koristili smo GARCH(1,1) model s inovacijama preuzetim iz t-Student distribucije. Model se može lako proširiti koristeći sofisticiranije procese za marginalne gustoće. Statička specifikacija modela pretpostavlja da zavisna struktura dva niza Copula-GARCH模型下的两资产期权定价 podataka ne varira u vremenu te tako podrazumijeva da su parametri spojne funkcije konstantni. S druge strane, dinamička specifikacija eksplicitno određuje dinamiku ovih parametara. Ekonometrijski procjenjujemoparametre dvije specifikacije koristeći razne spojne funkcije, uz naglasak na mješavinu između Gumbelove i Claytonove kopule. Koristili smo dnevne indekse zarade s dva razvijena i dva financijska tržišta u razvoju. Glavni nalaz upućuje na to da uključivanje promjenjive zavisne strukture poboljšava sukladnost distribucije Copula-GARCH模型下的两资产期权定价 Copula-GARCH modela.

## HankXiong/Arma-Garch-Copula

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Built Copula-GARCH模型下的两资产期权定价 ARMA-GARCH-Copula model to model the dependence between SP500 and TSX log return from 2006 - 2018

By comparing the different copula's average distance with empirical copula, I find that t copula fits the dependence best which means that the extremes are more likely to happen Copula-GARCH模型下的两资产期权定价 and contrary to most results, in relatively short time, the dependece is not asymmetric.

Simulate returns based on the model, estimate VaR and compare their performance within different models and with traditional VaR method through back-test. The comparison confirms the superiority of the Arma-Garch-Clayton copula in estimating VaR.

Please see more details about copula by reading attached Copula-GARCH模型下的两资产期权定价 my math thesis 'Copula and Its Application in Estimating Portfolio Value-at-risk'

## How to forecast from GARCH-copula model?

I am reading to understand Copula-GARCH模型下的两资产期权定价 how to forecasting time-series data from the GARCH-copula model. I am looking Copula-GARCH模型下的两资产期权定价 forward to understanding the steps. From my understanding, we should follow the following steps:

fit a GARCH model to the data.

transform the residuals to the copula data (uniform margins. Then, estimate the model parameters).

Generate 100 1-day a hed forecast from copula

Simulate from copula

transform the simulated data to the original margins.

I really understand the Copula-GARCH模型下的两资产期权定价 first 3 steps. However, I have no idea about the fourth step. What is the process for this step? what is the process Copula-GARCH模型下的两资产期权定价 of forecasting using copula? a help with an example will be much appreciated.

How to fit a copula GARCH model?

1. For each series (margins):
(a) fit a univariate GARCH model (e.g. using ugarchspec followed by ugarchfit from the rugarch package in R),
(b) obtain standardized residuals,
(c) apply probability integral transform (PIT) to obtain Uniform[0,1] pseudo observations.
The latter two Copula-GARCH模型下的两资产期权定价 steps can be acomplished by a single function pit applied on the fitted uGARCHfit object.
2. Fit a copula on the pseudo observations from all series.

How to simulate (generate) from a copula GARCH model?
Suppose you have a fitted copula-GARCH model and so Copula-GARCH模型下的两资产期权定价 for each margin you can obtain the predicted conditional variance $$\hat\sigma_^2=\hat\omega+\hat\alpha_1\hat\varepsilon_t^2+\hat\beta_1\hat\sigma_t^2$$ and the predicted conditional mean $$\hat\mu_=.$$ e.g. $\hat\mu_=\hat$ (Copula-GARCH模型下的两资产期权定价 assuming constant conditional mean) or $\hat\mu_=\hat+\hat\varphi_1x_t+\hat\theta_1\hat\varepsilon_t$ (assuming ARMA(1,1) conditional mean). This can be done as follows:

1. Simulate (generate) "pseudo observations" from the fitted copula. The result is a number of i.i.d. random vectors with each coordinate coming from a Uniform[0,1] distribution.
2. For each margin:
(a) transform the Copula-GARCH模型下的两资产期权定价 margin's generated values by "inverse PIT" that is specific to the margin (using a quantile function such as qnorm with zero mean Copula-GARCH模型下的两资产期权定价 and unit variance),
(b) multiply all the generated values by the predicted conditional standard deviation $\hat\sigma_$ due to the GARCH model specific to that margin,
(c) add the predicted conditional mean $\hat\mu_$ due to the conditional mean model specific to that margin.

The idea basically is to have a qualitative idea what the joint distribution is (that is the copula-GARCH model), then disassemble the data generating mechanism by learning its parameters on the way (estimate the model, obtain pseudo observations), simulate the most basic inputs (the pseudo observations) Copula-GARCH模型下的两资产期权定价 and assemble the joint distribution again (by transforming the simulated basic inputs into simulated values of the actual joint distribution according to the Copula-GARCH模型下的两资产期权定价 copula-GARCH model).

If you simulate a large enough number of realizations from the copula and then transform them as indicated above, you can inspect the empirical distribution of the resulting data. It can serve as an estimate of the predicted distribution.

## garchmodels

A parsnip backend for GARCH models in the tidymodels framework.

## Tutorials

Getting Started with Garchmodels: A walkthrough of the tidy modeling approach with the package.

Tuning Univariate Copula-GARCH模型下的两资产期权定价 Garch Models: Learn how to tune parameters of univariate garch models.

## Why Garchmodels?

Garchmodels unlocks univariate and multivariate GARCH models in one framework.

In a single framework you will be able to find what you need:

Univariate Methods: garchmodels connects to the rugarch package.

Multivariate Methods: garchmodels connects to the rugarch and rmgarch packages. Available methods include DCC-Garch (Dynamic Conditional Correlation Garch), Copula Garch and GO-Garch models.