In the current low rates environment, the classic stochastic alpha beta rho (SABR) formula used to compute option-implied volatilities leads to arbitrages. In "Arbitrage free SABR", Hagan et al ...

The trapezoidal rule with second-order backward difference formula (TR-BDF2) finite-difference scheme is applied to the Black-Scholes-Merton partial differential equation on a nonuniform grid.