# 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?

Thanks in advance

## 1 Answer

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 $C_{t,k}^{\Lambda }$.

I hope it will be useful for you.

• Thanx for your explanation! One more question; why do people in practice use IV then? If it is more intensive than just optimize with respect to quoted and calculated prices, it seems to me that it just requires an extra step, there should be some kind of advantage to the use of IV in the calibration right? May 24, 2016 at 7:49
• Did a quick check at the bloomberg terminal. Swaptions and caplets are naturally quoted in vols, therefore I suspect this is the 'why'. May 24, 2016 at 8:43
• This category is sensible, since options are often quoted in terms of implied volatility, and since the fit of model is often assessed by comparing quoted and model implied volatilities.
– user16651
May 24, 2016 at 10:13
• Also, implied volatilities are easier to interpolate. As explained, the goal is to calibrate over a (discrete) set of options. Often, we need to interpolate these data, without introducing (ideally..) arbitrages. This is more easily done on IV than on prices, on which the non-arbitrage constraints are not as clear. Aug 11, 2021 at 16:34