Calibration of Hull-White Two-Factor Interest Rate Model
The Hull-White one-factor (1F HW) interest rate model is a widely used model in finance for simulating the evolution of interest rates over time. It is based on a mean-reverting stochastic process where short-term interest rates revert to a long-term mean at a certain speed. The model incorporates two key parameters: the rate of mean reversion and the volatility of interest rate changes. One of the advantages of the Hull-White model is its flexibility, as it can fit the current term structure of interest rates, making it particularly useful for pricing interest rate derivatives like bonds, swaps, and options.
The Hull-White two-factor interest rate model (2F HW) is a generalization of the 1F HW model. In this model, the short rate incorporates two stochastic factors. Reference [1] presented methods for calibrating both the 1F and 2F HW models. Essentially, the calibration process for the 1F HW model proceeds as follows,
Interpolating the yield curve
Estimating the parameters using swaption pricing
As for the calibration of the 2F HW model, the author pointed out,
The 2F HW model aims at estimating short-term and long-term movements in interest rates. Similar to the 1F HW model, this second stage of the model, whereby it uses cap volatilities to price caps, is complex. Transforming these volatilities into accurate cap prices was difficult and was an important part of the model calibration. Furthermore, the calibrated parameters showed that not all parameters of the 2F HW model were statistically significant for this specific data; hence, the efficiency of the model depends on these parameters and the data used. The model might help researchers and practitioners gain higher accuracy by applying it to different datasets with the given calibration techniques. Nevertheless, the 2F HW model provides a dynamic picture of interest rates for different maturities, and thus it is useful.
In short, the calibration of the 2F HW model operates similarly to the 1F HW model, except that in the second step, cap prices obtained from the Black model are used instead of swaption prices.
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References
[1] Vithanalage, W. C. Calibration of two factor Vasicek and Hull-White models with contemporary data, University of Calgary, 2024
I recently read an intersting paper about forecasting the term structure:
https://www.sas.upenn.edu/~fdiebold/papers/paper49/Diebold-Li.pdf
They basicly assumed the yield curve can be built from 3 mathematical parts and that they estimate the historic betas, after which they tried to forecast the betas with an AR 1, the results showed significance over a random walk.