# Calibration of 1F Hull White short-rate model to market data

I want to calibrate the Hull White 1 factor short rate model to market data. The main purpose is to simulate interest rate paths, which I will use to calculate the net pv of banking liabilities.

Some sources suggest the use of market volatilities (of caps or swaptions), while I also encounter the use of market prices. Can someone explain to me the difference (if any) between the use of volatilities and prices for the calibration of the Hull White model. Which method is preferred with regards to my situation?

Suppose we have a set of $N_T$ maturities $\tau_t$ and a set of $N_k$ strikes $K_k$ .For each maturity-strike combination $(\tau_t,K_k)$ we have a market price (for example) $Caplet(\tau_t,K_k)=C_{tk}$ and a corresponding model price $Caplet(\tau_t,K_k,\Lambda)=C^\Lambda_{tk}$ in which $\Lambda$ is Hull-Whit's Parameters. The first category minimize the error between quoted and model prices The error is usually defined as the squared difference between the quoted and model prices, or the absolute value of the difference; relative errors can also be used. For example, parameter estimates obtained using the mean error sum of squares (MSE) loss function are obtained by minimizing $$\frac{1}{N}\sum\limits_{t=1}^{{{N}_{T}}}{\sum\limits_{k=1}^{{{N}_{K}}}{{{w}_{t,k}}}}{{({{C}_{t,k}}-C_{t,k}^{\Lambda })}^{2}}\ ,\ \ N={{N}_{T}}\times {{N}_{K}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$$ $\omega_{tk}$ is weight parameter. The second category are those that minimize the error between quoted and model implied volatilities. Again, the error is usually defined as the squared difference, absolute difference, or relative difference, between quoted and model implied volatilities: $$\frac{1}{N}\sum\limits_{t=1}^{{{N}_{T}}}{\sum\limits_{k=1}^{{{N}_{K}}}{{{w}_{t,k}}}}{{(I{{V}_{t,k}}-IV_{t,k}^{\Lambda })}^{2}}\ \ ,\ \ N={{N}_{T}}\times {{N}_{K}}\,\,\,\,\,\,\,\,\,\,\,\,\,(2)$$ The main disadvantage of Equation (2) is that it is numerically intensive.Indeed, at each iteration of the optimization, we must first obtain every Caplet Price and then apply a root-finding algorithm such as the bisection algorithm to extract the implied volatility $IV_{t,k}^{\Lambda }$ from $C_{t,k}^{\Lambda }$.