there are many many different types and flavorsthe simplest one being GARCH(1,1)havingS[n+1] = S[n] + S[n] mu + sqrt{v[n]} B[n]the stock price discrete sequence in the real worldwith mu the real world drift and possibly time-dependentand B[n] sequence of independent Brownian Motionsv[n] the varianceGARCH(1,1) is v[n] = (1-a-b) V + b v[n-1] + a v[n-1] B^2[n-1]

GARCH(1,1)v[n] = (1-a-b) V + b v[n-1] + a v[n-1] B^2[n-1]was invented by Engle and Bollerslev in mid 80'swe have a>0, b>0, a+b<1V corresponds to the Long Term Variancethe weight "a" accounts for the last innovationthe weight "b" accounts for the last variance hence the "clustering"

... moving on, in GARCH(1,1) the stock and vol process are uncorrelated however it is possible to a dd a leverage paramete "c" to the vol process to create a correlationthis is called NGARCH (Engle & Ng)v[n] = (1-a-b) V + b v[n-1] + a v[n-1] (B[n-1] -c)^2both GARCH(1,1) and NGRACH converge weakly to a continuous processdv = (omega - theta v) dt + xi v dZThis was proven by D. Nelson in early 90's

I recommend, MatLab Financial Toolbox, which implemented the followings: ugarch - Univariate ARCH/GARCH parameter estimation. ugarchllf - Log-likelihood function of univariate GARCH parameters. ugarchpred - Forecast volatility based on a univariate GARCH process. ugarchsim - Simulate a univariate GARCH process.MatLab is the best simulation tool out there for scientific as well as Computational Finance.I am using it with together with the Wavelet Toolbox plus the Digital Signal Processing Toolbox for some Computational Finance modelling. It is excellent to just drag & drop , change the block Transfer function in less time then run simulations.Sione.

A few more to add to Reza's succint replies.EGARCH The EGARCH model was proposed by Nelson and Cao (1992) , here the nonnegativity constraints as in the linear GARCH model are taken out and so there are no restriction on the parameters in this model.GARCH-M - The GARCH in the Mean model has the added regressor that is the conditional standard deviation.IGARCH -- Garch processes are weakly stationary , as the mean / variance and autocovariance are finite and constant over time and so they came up with these stationary garch models. Reza shall help me I have to locate my Hamilton and MillsAnd yes you can do all this and more with Finmetrics - splus toolbox.

Heston & Nandi also propose a Square Root GARCH model, which should converge to the famous Heston Stchastic Volatility modelI said "should" because I think there are still some issues on this

How do GARCH processes work?In developing volatility estimators of HF data, GARCH dynamics are used to simulate the price process at the tick time scale. It's not at all clear, however, that GARCH process assumptions correspond to all the necessary properties of empirical data.

Recent work by Andersen and Bollerslev argues that the daily squared return is an unbiased estimator of true volatility is an extremely noisy estimator. On the other hand, by taking the sum of intraday squared returns is not only the best measure of realize volatility, but also improve the forecasting of volatility. In the 24-hours foreign exchangse and the S&P500 futures (intraday + overnight) the best measure of daily stock volatility is given by re-scaled sum of squared intraday and intranight returns. By regressing the SP500 daily squared returns on the daily GARCH forecast provides R^2 of 0.035, while the regression of the sum of squared intraday and intranight returns on daily forecast the R^2 IS 0.150. (See M. Martens). An intraday GARCH model, which clearly models the volatility of overnight and intraday returns, provides the best daily volatility forecasts. Thus, the regression rises from 0.150 for the daily GARCH(1,1) to 0.420 for the intraday GARCH model.

From Bollerslev, Engle, Nelson papers about arch models, in particular on GARCH(p,q) model Garch(1,1) is the simplest.given that implementing this model could be a good approximation of volatility of single stocks or stock indexes, I’d like to point meaning of all parameters used in variance V[n] v[n]=(1-a-b)V + bv[n-1] + a v[n-1]B^2[n-1]in the Garch (1,1) [discrete case] conditional variance observed at time t is linear function of a constant, of the past variance at time t-1 and of a square factor related to time t-1.The term ‘conditional’ implies certain information about v[n] at time t-1 and that’s not always true; 1.I read with extreme interest on the tech forum an excellent thread of Reza about passage from discrete Garch to SDE and need some clarifications about calculation of risk premiums being volatility not constant in the SDE case 2.Could be interesting comparing Black-Scholes versus GARCH option pricing models given following process for conditional variance in Garch(1,1): h(t+1)=w+alfa*epsilon(t)*h(t)+beta*h(t) any ideas?Thanks.

Hi,I am using Matlab’s Garch toolbox for volatility forecast. I am still new in this area,,, I am having trouble interpreting the output of garchpred,, for example,sigmaForecast,meanForecast,sigmaTotal] = garchpred(coeff,... nasdaq,10); [sigmaForecast,meanForecast,sigmaTotal]ans = 0.0120 -0.0005 0.0120 0.0120 -0.0005 0.0170 0.0121 -0.0005 0.0208 0.0121 -0.0005 0.0241 0.0122 -0.0005 0.0270 0.0122 -0.0005 0.0297 0.0123 -0.0005 0.0321 0.0123 -0.0005 0.0344 0.0124 -0.0005 0.0366 0.0124 -0.0005 0.0386How do I now determine the volatility for t+1 ,..t+N ,, How does this help me to forecast returns for t+1 ?Regards

Why GARCH is a generalisation of ARCH?Lets take ARCH(1) process:1) h(t) = w + a*r^2(t-1)where h(t) is a conditional volatility at day t, r^2(t-1) is squared return at day t-1 and w and a are coefficients in the conditional variance equation.In words, we believe that variance h(t) depends of the magnitude of the return in previous day r(t-1), which is represented by the squared return r^2(t-1).Now suppose that we believe that volatility h(t) depends not only on the return over the previous day r(t-1), but also on the return two days ago r(t-2). Thus we arrive to the ARCH(2) process:2) h(t) = w + a1*r^2(t-1) + a2*r^2(t-2)There is much empirical evidence that financial volatility has a long memory and therefore many previous returns should be included in the conditional variance equation. We arrive to the ARCH(p) process:3) h(t) = w + a1*r^2(t-1) + a2*r^2(t-2) +…+ ap*r^2(t-p)There are a number of problems associated with the inclusion too many parameters in a model, such as the reduction of degrees of freedom and necessity for larger datasets. To avoid these problems, we generalise ARCH(p) as follows:4) h(t) = w + a*r^2(t-1) + b*h(t-1)This is GARCH(1,1). We have included h(t-1). What for? Here is the answer. What is h(t-1)?5) h(t-1) = w + a*r^2(t-2) + b*h(t-2)What is h(t-2)?6) h(t-2) = w + a*r^2(t-3) + b*h(t-3)What is h(t-3)?7) h(t-3) = w + a*r^2(t-4) + b*h(t-4)and so on.Remember that for including all these historical conditional variances in our model we had to include only h(t-1) with one parameter. Without this generalisation we would have to include p squared returns with p parameters.

in a way it is not surprising that a GARCH(1,1) could fit te data well, indeed it has an ARMA(1,1) equivalent representation which itself could be written as an AR(+\infty)ARCH(p) cannot do this !

Last edited by reza on November 8th, 2003, 11:00 pm, edited 1 time in total.

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QuoteOriginally posted by: mrbadguy2.Could be interesting comparing Black-Scholes versus GARCH option pricing models given following process for conditional variance in Garch(1,1): h(t+1)=w+alfa*epsilon(t)*h(t)+beta*h(t) any ideas?Thanks.There are two published papers on GARCH option pricing. One by Lehnert (2003), "Explaining Smiles: GARCH options pricing ..."; the other by Heston and Nandi (2000), "A Closed-Form GARCH Options Valuation Model", RFS. Both papers have empirical comparisons with BS-like pricing models.I have a question for all the practitioners out there. What kind of GARCH model do you use? What do you use it for?

I am using Splus to fit my GARCH models. After doing forecasts I compare with observed data. As observed data, I use the squared returns (working with log returns).To my surprise the values forecasted with GARCH models, are mostly to high. 2/3 of my forecasts are to high.Compared to a simple model, such as weighted moving average of last 8 observations, the GARCH models (also the modified) perform very bad.Why does this occur? I thought the developed GARCH models should be better in this case.Am I using some strange values as observed data, or what values do the Splus use when calculating forecasts?The mean in the model I just model as a constant, and this is small, so it should not make this much difference.Using MatLab give me the same results.Anyone knows what I am talking about?//Muzzex

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