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Hi all,I would like to know how you calibrate a shifted two-factor gaussian G2++ model :[$]r(t) = x(t) +z(t) + \varphi (t, \alpha) \\r(0) = r_0, \\with : \\dx(t)=-ax(t)dt+\sigma dZ_{1}(t) \\x(0)=0, \\dz(t)=-bz(t)dt + \eta dZ_{2}(t) \\z(0)=0, \\[$]where :[$](Z_{1}, Z_{2})[$] is a two-dimensional Brownian motion with instantaneous correlation [$]\rho_{1,2}[$]and :[$]r_{0}, a, b, \sigma, \eta [$] are positive constants and [$]\alpha = [r_{0}, a, b, \sigma, \eta , \rho_{1,2}][$][$]\varphi (., \alpha)[$]Knowing that the solution of the above is :[$]r(t)=x(s)e^{-a(t-s)}+z(s)e^{-b(t-s)}+\sigma \int_s^t e^{-a(t-u)}dZ_{1}(u) + \eta \rho \int_{s}^{t} e^{-b(t-u)}dZ_{1}(u) + \eta\sqrt{1-\rho^2}\int_{s}^{t}e^{-b(t-u)}dZ_{2}(u) + \varphi(t, \alpha)[$]I have to calibrate the model parameters [$]a, b, \sigma, \eta , \rho_{1,2}[$] on the LIBOR zero-coupon and swaptions volatilities. Also, what is the market practice? Can I arbitrarily fix a and b for example?Thanks.

I think a and b is fixed by practise. Basically ppl would do is to calibrate swaps to the volatility parameters.Correct me if i am wrong.

Yes, but since that I will still have 4 remaining parameters, if I am not wrong I will need to use 4 instruments. Which are the best instruments for this purpose?

I can't speak to the best practices in your particular market, but the general technique for calibrating a model is: 1) Work out pricing formulas for observable products. That will give you a function of whatever parameters your model has.2) Choose a measure to determine what you want your fit to be optimal with respect to. L^2 is typical, corresponding to least squared difference of observed prices to predicted prices. (observed price 1 - predicted price 1)^2 + (observed price 2 - predicted price 2)^2 + ...3) Minimize You want to use as many observed prices as you can get your hands on, not just a number equal to the parameters in your model.

QuoteOriginally posted by: muaddibI can't speak to the best practices in your particular market, but the general technique for calibrating a model is: 1) Work out pricing formulas for observable products. That will give you a function of whatever parameters your model has.2) Choose a measure to determine what you want your fit to be optimal with respect to. L^2 is typical, corresponding to least squared difference of observed prices to predicted prices. (observed price 1 - predicted price 1)^2 + (observed price 2 - predicted price 2)^2 + ...3) Minimize You want to use as many observed prices as you can get your hands on, not just a number equal to the parameters in your model.Presently, since I do not have access to Bloomgerg or Reuters, I do not have a lot of datas especially CDS volatilities quoted on the market and swaptions volatilities.

There is a good and detailed description of how to do this in the most recent edition of Tuckman and Serrat. We usually reserve the term "calibration" for fitting parameters to a cross-sectional dataset at a given point in time, but if you can't get your hands on enough interest rate options data, you may consider fitting the parameters to the historical yield curve behavior. In this context, principal component analysis is usually an intermediate step, so that you would fit your volatility, correlation and mean reversion parameters to the shape and level of the first 2-3 PCs. In terms of trying to pin down a and b, you probably want one of them to be <5% to preserve a decent amount of vol for long-dated forward rates and the other one to be materially higher (maybe in the 30%-50% range).