Thanks a lot, Calculator!As for theta(t), I think the tree building technique can fit the model to current yield curve, so no need to calibrate it. If using simulation, one can choose to calculate theta(t) with analytical formula or calibrate with your simulation engine. No large discrapencies. Please correct me if I am wrong.

One more question about the spreadsheet.Is it a tree-based calibration? How about using analytical formula of cap price to calibrate? Anyone can share experience with me?

Please forgive me if I sound too ignorant, I know there is a formula to calibrate theta(t), that involves instantaneous forward rate and first derivative of instantaneous forward rate, how do you code these in, what variables represent these? I am using the simulation approach.Thankscr

theta(t) is "calibrated" by enforcing that you can price zero coupon bonds exactly. In other words, Arrow-Debreu prices at each node is foundQ(i+1,j) using theta(i) and then one can use this to price zero coupon bonds at time step i+1, ad infinitum (see Hull chapter on trinomial models).As an alternative, one can use the approximate formula for theta which involves forward rates and derivatives of it (assuming you are using a smooth function to interpolate the yield curve, i.e. cubic spline).Also for unequal time steps, there is a different implementation proposed by Hull-White in their 2000 paper and is discussed in the Interest Modelbook by Brigo and Mercurio. Basically let's say you are on lattic j at time step i (as before the lattice spacings are sqrt(3*dt_i) where dt_iis the time stepsize at step i). Then at time step i+1, you link this node j with the node k such that the short rate value there is the closest to the expected value from j with the correct probabilities that takes you from j to k. This ensure positivity of probabilities while tree is recombining. This also gives you implicitly what k_max and k_min shouldbe.

jrquant1, may I ask some naive questions?"theta(t) is "calibrated" by enforcing that you can price zero coupon bonds exactly." 1. When you calibrate theta(t), have you finished calibration of alpha and sigma (assuming constant alpha and sigma)?2. How to calibrate alpha and sigma? By enforcing your tree can price caps exactly? or by analytical formula?My understanding is to calibrate alpha and sigma by analytical formula, then calibrate theta(t) by enforcing your tree to price zero coupon bonds exactly. However, whenever I use the Excel Solver Tool or my own solver to minimise the errors between the equivalent black cap prices and the prices under the relevant models, I always get negative alpha. I've asked this question in This threadcomputedrisk, I haven't solved above problem, so I am afraid my understanding may misguide you.

domilar04,I agree with your steps of calibraiton. After all, the theta(t) calibration is way to correct for the error introduced by the tree construction.As noted in Brigo's book, getting negative alpha is common when calibrating the HW model to the cap volatilities due to the humped nature of it. Other models such as the "shifted" Cox-Ingersoll-Ross give much better fit. From I've observed, usually calibration to the swaption surface typically gives alpha on the order of 0.05-0.10 and sigma is about 0.015 - 0.02. We mainly use interest-rate models to price hybrids so we don't delve into more complicated models but maybe other experienced interest rate quant can better address this calibration problem.

Thanks a lot!I agree that it's reasonable to get negative alpha in certain situation. What confusing me is murex system got positive alpha with same set of caplets, same yield curve and same model. Also the spreadsheet pasted by doublebarrier2000 seems using tree-based calibration.

Hi,I've calibrated my Hull-White model to swaptions at end 2004 data and i get a sigma of around 0,7% and an alpha of 0,01.This is lower than the figures jrquant1 gives below.Are these reasonable outcomes or are these wrong for sure?Thanks,Richard

Hi,I've calibrated my Hull-White model to swaptions at end 2004 data and i get a sigma of around 0,7% and an alpha of 0,01.This is lower than the figures jrquant1 gives below.Are these reasonable outcomes or are these wrong for sure?Thanks,Richard

Dear Colleague,I read at Wilmott Forum that your task of calibrating a HW model with time dependent parameters was successfully reached. Could you kindly send your spreadsheet to me?Thanks in advance

I would recommend Marc Henrard's method for calibration. Using 2004 swaption vol. surface I got alpha of 1.83% and sigma 0.68% which are close to rplat's results.