What do you mean by estimation errors?

Yes you can

Implied Volatility as it is only the market's prediction of it from a Black- Scholes model, with quoted option prices , need not necessarily follow the statistical properties of a variance- co variance matrix .'Volatility smile' is an example for it

These are two entirely different statistics. Never substitute one for the other. Implied volatility is a notoriously poor estimate to begin with. It only works within the context a certain set of assumptions.

When trading options on a weighted basket of stock market indices, say a call option on a basket consisting of 40% S&P500, 20% FTSE100, 20% EUROSTOXX50 and 20% TOPIX one would usually use implied volatilities from the option markets combined with some "implied correlation" (there is a weekly "implied correlation" fixing of all the exotic option desks of the major banks exchanging their views on these parameters). This proceedure is a conseqnence of the need to hedge the position using single index options on the S&P500, FTSE100, EUROSTOXX50 and TOPIX in each of the single markets (our example US, UK, Europe, Japan).

However, if you want to do a portfolio optimization of an international diversified stock index portfolio you may either use historical variances and covariances (just like Markowitz did in the 1950s) or you have a model which forecasts the variances (volatilities) and correlations (e.g. GARCH for variances and maybe a cointegration model for the correlations) and use those as input. According to my practical experience it does not make such a difference as in times of crisis all correlations suddenly jump to +100% and all volatilities go equally up as well.

Thanks Markus for the insight. my research is in understanding the reasons why classical M_V framework fails when the inputs to the optimizer suffer from estimation errors and how we can try to fix the problem using robust estimators and robust optimization techniques (which takes into account the estimation errors).Also in the process i am trying to model volatility with the purpose of using it as an input to the optimizer. Same with the Variance -covariance matrix. As a practitioner you suggest using Garch type forecasting models for the inputs like volatility and correlation which i absolutely agree but i was also wondering that if a market implied covariance matrix can be estimated then we can use both the historical and the implied matrix as an input to the optimizer

Now I understand what you hav ein mind. Your idea has some similarity with HEAVY models (Nourelidin-Shephard-Sheppard, J. of Applied Econometrics, 2011) where a GARCH-type model is combined with a realized volatility model (the latter uses high frequency data to extract daily covariances). To my understanding, you might combine a GARCH-type model with a market implied covariance so that your input to the MV framework takes into account both the historical data as well as the market implied covariances. You migth try to model the sequence of market implied covariances, being thus closer to the HEAVY model class, or simply take the implied market covariance as an additional information source to improve a, say, standard GARCH model

It depend on what you want to do with it.

Assume that the variance -covariance itself is going to be used as input to portfolio optimization.

In this case, a forward looking volatility measure such as the implied volatility can be utilized since it gives an idea of how stock traders view the market as whole (measuring fear in the market). This would represent the diagonal element of the variance covariance matrix, the off diagonal in this context would be estimated using bi-variate GARCH processes.

Dear Anirban, you can look at these two papers for further insights:

http://www.scientificbeta.com/download/file/estimating-covariance-matrices-portfolio-optimisat (not forward looking)

and

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1474212

(forward looking)

Dear Guillaum.

Thanks for the info. I ll have a look. I think not much work has been done with respect to emerging markets' data set like Indian Stock market

Benefits of using implied volatility(IV) over historical volatility(HV) could be huge. However we have to be careful regarding the context. e.g. if you are using Fishers information matrix to estimate the standard errors(SE) of option pricing model then using IV as a proxy to any HV parameter will certainly make sense as all the parameters are estimated in implied sense i.e. from forward looking expectation. However if it is a joint estimation and/or estimation based on historical return and then using the physical-risk neutral relationships to switch to implied sense estimates, then giving proxy to HS by IV creates mathematical inconsistencies!!

Hope this helps.

Yes I agree with Dr; Sharif answer.

A standard way of incorporating implied volatilities is using historical correlation structure. That us, denote C the historical var-covar matrix, and HS, IV - diagonal matrices with historical and implied volatilities in the diagonals. Then correlation structure is K = HS-1  * C * HS-1, and the combination looks like CC = IV * K * IV. In all, one can first calculate D = HS-1 * IV, and then immediately get the result CC = D * C * D.

This approach does not create any math inconsistencies.

While implied vol is "the wrong number to plug into the wrong formula to get the right Price" (as Rebonato, I think, states) it carries a lot of current Information or at least market consensus which is forward-looking: Nonetheless, I also agree with Sharif Mozumder when mixing up different sources and, hence, measures of risk!

Thanks Marcus. But if taken as a forward looking measure, in using the implied volatility as an input to the var-covar matrix my aim is to see:

Whether the portfolio which uses implied volatility performs better in comparison to other methods of estimating var-covar matrix since its well known that historical var-covar matrix has estimation issues.

Its an exploratory research based question

Lets no get into BS pricing and the problems associated with it. At some point I believe that unless we have a better OPM BS serves some purpose. 

Beware of the sensitivity of mean-variance models to input data mis-specification! Better use more robust risk measures and the associated models, such as CVaR optimization and scenario optimization.

In principle of course you can use forward looking estimates, iv they are more reliable.

Book Practical Financial Optimization. Decision making for financ...

I have new empirical evidence that would make most implied volatility estimates useless. I will write up a draft soon. It will make all papers depend on implied volatility estimates pretty much useless.

Both implied and historical volatility can be used, but neither is especially useful or accurate in modelling time-varying volatility.