[R-SIG-Finance] Simulating returns using copula and SPD distribution

[alexios ghalanos]

You've missed a step. Think about it:
1. You take z.
2. Fit an spd.
3. Transform it to U using pspd.
4. Fit a copula on U.
5. Sample from copula to generate u.
6. Need to translate u back to z (step 1) using the quantile function
(qspd) and the estimated parameters from step 2.

i.e. you've missed step 6.


-Alexios

On 28/05/2014 16:53, Guillaume PEALAT wrote:
> Hi,
> 
> I am fitting some SPD distributions to past returns and some Copula to
> simulate the dependance.
> I am struggling with the simulation part, as the returns I simulate a
> clearly out of line after some time.
> 
> I have attached some sample code in order to clarify the point.
> 
> I first calibrate the SPD distribution on the past returns:
> 
> #Calibrating the SPD distributioon the residuals of an ARMA+GARCH
> 
> *spec.u1 = ugarchspec(mean.model=list(armaOrder=c(1,1)),
> distribution="std")*
> 
> *fit.u1 = ugarchfit(spec.u1, return.u1)*
> 
> *z.u1= residuals(fit.u1, standardize=TRUE)*
> 
> *gpd.u1 = spdfit(z.u1, upper=1. - tailFraction, lower=tailFraction)*
> 
> 
> #Calibrating the SPD distributioon the residuals of an ARMA+GARCH
> 
> *spec.u2 = ugarchspec(mean.model=list(armaOrder=c(1,1)),
> distribution="std")*
> 
> *fit.u2 = ugarchfit(spec.u2, return.u1)*
> 
> *z.u2= residuals(fit.u2, standardize=TRUE)*
> 
> *gpd.u2 = spdfit(z.u2, upper=1. - tailFraction, lower=tailFraction)*
> 
> 
> Then, once we have all this set, we need to calibrate the Copula of our
> choice
> 
> 
> #Calibrate the Copula
> 
> #We extract the residuals from the xts into a vector
> 
> *res.u1 = as.vector(z.u1[,1])*
> 
> *res.u2 = as.vector(z.u2[,1])*
> 
> #We transform the margin to uniform
> 
> *U.u1 = pspd(res.u1, gpd.u1)*
> 
> *U.u2 = pspd(res.u2, gpd.u2)*
> 
> 
> 
> #We create the matrix to help us calibrate the Copula
> 
> *test <-list()*
> 
> *test[[1]] = U.u1*
> 
> *test[[2]] = U.u2*
> 
> *test_matrix = do.call(rbind, test)*
> 
> 
> 
> *set.seed(123)*
> 
> 
> 
> #We define the Copula we want to use
> 
> *tcopula<-tCopula(param=0.5, dim=2, dispstr = "ex", df =6)*
> 
> #We fit the Copula with maximum Likelihood
> 
> *fit.mpl <- fitCopula(tcopula, t(test_matrix), method="ml")*
> 
> 
> 
> #We get back the parameters
> 
> #Rho
> 
> *rho.fit = fit.mpl at copula@parameters[1]*
> 
> #Degrees of Freedom
> 
> *df.fit = fit.mpl at copula@parameters[2]*
> 
> 
> 
> #Create fitted t-copula object
> 
> *t.cop.fit = tCopula(param=rho.fit, dim=2, df=df.fit)*
> 
> 
> 
> *margins = c("t", "t")*
> 
> *paramMargins = list(list(df=df.fit),list(df=df.fit))*
> 
> 
> 
> #Create fitted custom distribution object
> 
> *myBvd.tcop.fit = mvdc(copula=t.cop.fit, margins = margins, paramMargins=
> paramMargins)*
> 
> 
> 
> If all is well, I should be able to run some simulations and simulate my
> returns from this set up:
> 
> 
> #Generate random numbers from the Copula
> 
> *u = rMvdc(nsim, myBvd.tcop.fit)*
> 
> *u.1 = u[,1]*
> 
> #Generate the simulated returns
> 
> *sim.u1 = ugarchsim(fit.u1, n.sim=nsim, startMethod="sample", m.sim=1,*
> 
> *                    custom.dist=list(name="sample",distfit=matrix(u.1,
> ncol=1)))*
> 
> #Get back the simulated returns
> 
> *sim.u1.returns = sim.u1 at simulation$seriesSim*
> 
> 
> 
> The problem lies in the order of magnitude the simulated returns. I can get
> some occurrence of +26% for example, which is clearly wrong.
> 
> 
> Could someone point me on the wrong part?
> 
> I have also a remaining question:
> 
> In the univariate case, we can simulate the random number with *rspd(n =
> days, gpd.u1)*, and they will be in accordance with the gpd distribtuion
> calibrated.
> 
> In the multivariate case, can I consider this step to be taken care of by
> the random number of the Copula since it has been calibrated on the margins
> transformed to Uniform by the gpd distribution function?
> 
> 
> Thanks
> 
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> 
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