I'm trying to calibrate Hull White model in VietNam market to value IRS, CSS products which are not publicly traded.
dr(t)=(θ(t)−αr(t))dt+σ(t)dW(t)
I only have historical VNIBOR curves data, we do not have trading interest derivatives yet.So basically we have R_i(0,1/250), R_i(0,1/4), R_i(0,1/2), ... for observation date i. I have done some research and got some confusions:
Some banks use IBOR 3M as a surrogate for instantaneous rate (fit with theoretical formula of r(t)). The reason was Overnight rate is too volatile. But is it too inaccurate to consider 3M as instantaneous? If cant use Overnight rate and has to use 3M rate, why don't we use a theoretical formula of R(t,t+3M)?
Some papers suggest using historical variance of r_i (if we use IBOR 3M as surrogate for instantaneous rate, r_i will be R_i(0,3M) data) as estimation for
Var(r(t)) = σ^2*(1-exp(-2at))/a.
However, the r(t) process in HW model doesn't have fixed mean like in Vasicek model.
So historical variance of r_i should contains both variance of r(t) and F(0,t) Should we use the historical variance of the new series x_i = r_(i+t) - F_i(0,t)?
I've read that we shouldn't use different maturities (e.i VNIBOR 3M and VNIBOR 6M) in 1 calibration because they has different underlying. Refer @castella08's comment here What instruments can be used to calibrate short-rate models? Does it mean for each s, R_i(0,s) should have different risk neutral parameters (a,σ)? However, in HW model, when we calculate the mean parameter θ(t) and F(0,t) in the E(r(t)) formula, we have to use different maturities R_i(0,s) with the same (a,σ). Does it conflict with the idea above?
My first approach is using the variance of x_i = r_(i+t) - F_i(0,t) as estimation of theoretical Var(r(t)) for different "window of view" t. F_i(0,t) observed at time i and r_(i+t) observed at time i+t. Then try to find (a,σ) that best fit those variances with theoretical formulas. Is it appropriate? If in practice Overnight rate is too unreliable, could I change to something like x_i = R_(i+t)(0,3M) - F_i(0,t,t+3M), I'm not sure the formula for 3M rate yet, as estimation of theoretical Var(R(t,t+3M)) formula?
