# What different techniques exist for modeling exotics near payoff discontinuities in Finite Difference method?

If you are modeling an exotic, like a binary or a barrier, and hedging it with vanillas that have strikes quite close to the exotic's strike, then a large asset step size, for example, $\delta S = \frac {K_{max} -K_{min}}{\beta}$, with $\beta = 1 \space or \space 2$, where $K_{min}$ and $K_{max}$ are the min and max of strikes in the basket, does not allow the payoff shape to be correctly modeled. It introduces substantial error in the valuation. To resolve this sort of problem what approaches using FDM do exist?