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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?

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Some techniques I can think of include

  • Use a brownian bridge to get a crossing probability for points near the boundary
  • Use implicit stepping in your PDE solver (which increases smoothness) as opposed to explicit stepping (which "rings" near discontinuities)
  • Employ control variates, by using the same grid to price related instruments having easy analytic pricing solutions
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can you expand on the brownian bridge idea? –  phubaba Jun 16 at 20:38

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