Efficient swaptions price in Hull-White one factor model

The Hull-White one factor model is used to price interest rate options. The parameters of the model are often calibrated to simple liquid instruments, in particular European swaptions. It is therefore very important to have very efficient pricing formula for simple instruments. Such a formula is proposed here for European swaption. Based on a very efficient corrector type approximation the approximation is efficient both in term of precision and in term of spped. In our implementation the approximation is more than ten time faster than the direct pricing formula and more than twenty time faster than the Jamshidian trick.