I am trying to price a digital option and the final price under different number of time steps are as follows: enter image description here

Is it possible to have a graph like this?


It is quite common to see non-smooth convergence in tree models and this is not specific to digital options.

The problem usually that the tree is constructed independent of the contract to be priced. Thus, the location of the strike relative to the two surrounding nodes might vary widely between two successive step sizes. For European plain vanilla options, a common approach is to "tilt" the tree such that one of the terminal nodes coincides with the strike. The same approach can be applied to European digital options as well.

A good reference is Tian (1999) "A Flexible Binomial Option Pricing Model", Journal of Futures Markets, Vol. 19, No. 7, pp. 817-843. The below plot is taken from this paper and compares the convergence of a standard Cox-Ross-Rubinstein tree to the tilted one.

Comparison of binomial tree convergence.


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