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there are has been a lot of papers on the analysis of the convergence of binomial trees for European options. You can regard a tree as an explicit finite difference method. The conclusions are that the location of the nodes near the strike determine the error. So if $\kappa$ is the fraction of the distance between the strike and the lower node as opposed to ...


Rather than thinking about the steps, think about the piecewise regions where your value is constant. When using the explicit scheme, time zero option value at any stock price for your simple digital option is basically just a function of which antecedent nodes (accounting for backwards timestepping) were above or below the strike. Slight modifications of ...

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