[R-SIG-Finance] [R] Fitting distributions to financial data using volatility model to estimate VaR

[Alexios Ghalanos]

Sent from my iPhone

On Apr 10, 2013, at 0:28, "R. Michael Weylandt" <michael.weylandt at gmail.com> wrote:

> Moving to R-SIG-Finance which seems to be the correct mailing list.
> 
> On Sun, Apr 7, 2013 at 8:41 AM, Stat Tistician
> <statisticiangermany at gmail.com> wrote:
>> Ok,
>> I try it again with plain text, with a simple R code example and just
>> sending it to the r list and you move it to sig finance if it is
>> necessary.
>> 
>> I try to be as detailed as possible.
>> 
>> I want to fit a distribution to my financial data using a volatility
>> model to estimate the VaR. So in case of a normal distribution, this
>> would be very easy, I assume the returns to follow a normal
>> distribution and calculate a volatility forecast for each day, so I
>> have sigma_1,sigma_2,...,sigma_n,. I can calculate the VaR via (mu
>> constant, z_alpha quantile of standard normal):
>> VaR_(alpha,t)=mu+sigma_t * z_alpha. This is in case, I have losses, so
>> I look at the right tail. So for each day I have a normal density with
>> a constant mu but a different sigma corrensponding to the volatility
>> model. Let's assume a very simple volatility model, e.g. (empirical)
>> standard deviation of the last 10 days and the mu is set to zero. The
>> R code could look like (data):
>> 
>> volatility<-0
>> quantile<-0
>> for(i in 11:length(dat)){
>> volatility[i]<-sd(dat[(i-10):(i-1)])
>> }
> 
> Let's clean this up:
> 
> library(xts)
> # Load data into an xts object called "dat" here.
> 
> library(TTR)
> vola <- runSD(dat, 10) # volatility is a ttr function name, so not using that
> 
>> for(i in 1:length(dat)){
>> quantile[i]<-qnorm(0.975,mean=0,sd=volatility[i])
>> }
> 
> # Similarly, quantile is a function name and not a good idea to override
> 
> quan <- qnorm(0.975, mean = 0, sd = vola)
> 
>> # the first quantile value is the VaR for the 11th date
>> 
>> #plot the volatility
>> plot(c(1:length(volatility)),volatility,type="l")
> 
> Can just do
> plot(vola) # if vola is an xts object. Also, the c() is superfluous
> (and arguably dangerous) there
> 
>> 
>> #add VaR
>> lines(quantile,type="l",col="red")
> 
> library(PerformanceAnalytics)
> chart.BarVaR(dat) # is probably easier.
> 
> 
>> 
>> Now, I want to change the volatility model to a more advanced model
>> (EWMA, ARCH, GARCH) and the distribution to a more sophisticated
>> distribution (student, generalized hyperbolic distribution). My main
>> question is now, how can I combine the volatility model and the
>> distribution, since in case e.g. of a Student's-t distribution with
>> parameters mu (location), v (df), beta (scale) I cannot just plug the
>> sigma in, because the distribution has no sigma?
> 
> Be careful here -- it certainly has a standard deviation, just not a
> parameter called sigma. Keeping those two ideas distinct is important
> here. They only coincide for the normal distribution. (Among the major
> ones)
> 
> You're also using something other than the classical t-distribution
> (as defined by the all-hallowed Wikipedia) if it's a three parameter
> distribution.
> 
>> 
>> One solution I already know is, that I take the variance formula of
>> the corresponding distribution - in case of a Student's-t distribution
>> this would be sigma=beta v/(v-2). I have an estimate for sigma. So for
>> each day I do the ML estimation with a modified log-likelihood where I
>> insert for the scale parameter: beta=sigma * (v-2)/v and do the
>> estimation.
>> 
>> First of all, is this correct?
> 
> No -- I believe that the t-distribution has sigma^2 = v / (v-2).
> 
> But even with that correction, it's not what I would do.
> 
> I would first fit beta, nu, and mu by ML and then estimate sigma from there.
> 
> Without checking, I'm not positive that sample standard deviation
> gives a particularly good estimator for the df of a t-distribution. It
> wouldn't surprise me if it did though.
> 
> In that same vein, I'm not sure this process gives an estimate that's
> much superior to the rolling empirical standard deviation. Though it
> might be useful for forward looking predictions...
> 
> Bigger question -- have you looked at TTR::volatility or
> PerformanceAnalytics::VaR. There's a lot of useful stuff in there.
> 
> I'll try to read through the rest of this later -- about to have
> dinner with the family,
> Michael
> 
>> 
>> I looked at several papers, but I did not understand, how they did
>> this? No matter what volatility model they use, I cannot understand
>> the connection of distribution and volatility model. For example,
>> consider this paper:
>> http://www.math.chalmers.se/~palbin/mattiasviktor.pdf
>> 
>> On page 50 they are showing the hyperbolical distribution with
>> different volatility models, how did they do this?
>> 
>> Also, I do not understand table 6.2 on page 49: If they have estimated
>> several distributions over the time, they have lots of estimates, but
>> they just show one distribution? I mean, where does it come from? The
>> 3d picture clearly differnt distributions over time, so they have
>> estimated the distribution after 5 days (page 48), but in the table is
>> just one specific distribution with specific parameters? And they give
>> the volatility models in the rows?
>> 
>> A second famous paper is the Technial Document by JPMorgan:
>> RiskMetrics Technical Document - Fourth Edition 1996, December:
>> 
>> http://www.google.de/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&ved=0CDUQFjAA&url=http%3A%2F%2Fzhiqiang.org%2Fblog%2Fdownload%2FRiskMetrics%2520-%2520Technical%2520Document.pdf&ei=RSJhUd7YJIbktQaQ-YCAAw&usg=AFQjCNGpCXUdLSVHQtYJMl7MccLGQtdkDw&sig2=HBxWDrRTMN7rVqWu-Yp1zQ&bvm=bv.44770516,d.Yms
>> 
>> 
>> Especially page 238 is interesting: "According to this model, returns
>> are generated as follows"
>> 
>> r_t=sigma_t xi_t
>> 
>> sigma^2_t is calculated by EWMA
>> 
>> 
>> xi is distributed according to the generalized error distribution. So
>> they do not assume the returns to follow a certain distribution, but
>> they assume the returns condition on the volatility to follow a
>> certain distribution, right?
>> 
>> Now my question is, how can one calculate the VaR in this case? On
>> page 242 they give a short summary, describing the steps. The third
>> step in the most intersting: "Third, we use [...] volatility estimated
>> and the [...] probability distributions ([...] generalized error
>> distribution) evaluated at the parameter estimates to construct VaR
>> forecasts at the 1st and the 99th percentile."
>> 
>> My question is now, how do they do it?
>> 
>> They describe their fitting steps in the steps before, but I am not
>> getting the following point:
>> 
>> Do they fit the distribution to the original return series, calculate
>> the volatility (σt) and then just calculate the VaR with
>> VaR_t=sigma_t*q_alpha where q_alpha is the quantile of the fitted
>> distribution
>> 
>> or
>> 
>> do they fit the distribution to the standardized returns
>> (xi_t=r_t/sigma_t), calculate the volatility and then just calucate
>> the VaR with VaR_t=sigma_t*q_alpha where q_alpha is now the quantile
>> of the fitted distribution which was fitted using the standardized
>> residuals?
>> 
>> Another question is: Did they set the mean of the return to zero?
>> 
>> My main point is, how to fit a sophisticated distribution to financial
>> data using a volatility forecast for each day, which was generated by
>> EWMA, ARCH or GARCH and how to calculate the VaR for each day over a
>> time horizon and how to implement this. The paper is doing this, but I
>> am simply not getting it. I worked on this for days now and I am
>> really stuck here.
>> 
>> And one other question: If I am doing the way with the modified
>> log-likelihood: This is really challenging in case of a hyperbolic
>> distribution, since the variance is not calculated easily anymore? See
>> wikipedia.
>> 
>> Thanks a lot for your help.
>> 
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> 
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