Improving Binomial Tree Convergence by Aligning Nodes with the Strike

Summary

The discussion explains why option values computed with a binomial tree may converge unevenly as the number of time steps changes. This behavior is common in tree models and is not limited to digital options. When the tree is built without reference to the contract, the strike can fall at different positions relative to nearby terminal nodes as the step count changes, causing successive estimates to move irregularly.

For European vanilla options, one common remedy is to tilt the tree so a terminal node lands on the strike. The response says the same approach can be applied to European digital options. It cites a published study comparing standard Cox-Ross-Rubinstein and tilted trees, but the excerpt supplies no numerical results or implementation details. The technique addresses strike placement in the tree; the discussion does not claim it removes every source of pricing error or establish how it performs for other payoff types or exercise styles.

Key ideas

• Binomial tree prices can converge non-smoothly as the number of steps changes.
• The strike's position relative to terminal nodes can shift between successive tree sizes.
• Tilting the tree to place a terminal node at the strike is a common approach for European vanilla options.
• The response says strike alignment can also be used for European digital options.
• The cited comparison does not provide results or implementation guidance in the excerpt.

Tags

# Number of Time Steps in Binomial Option Pricing - Problem?


# Number of Time Steps in Binomial Option Pricing - Problem?












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

Is it possible to have a graph like this?

## Answer by LocalVolatility (score 3, accepted)

https://quant.stackexchange.com/a/36975

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.