Improving Binomial Lattice Option Valuation with Extrapolation

Summary

The note asks how to compare the convergence speed of binomial lattice option values as the number of steps increases. It observes that the model with the fastest convergence can differ by option moneyness, using at-the-money and deep in-the-money puts as examples. This motivates seeking a quantitative approach instead of selecting a model by visually inspecting value-versus-step charts.

For vanilla puts, the cited response points to research recommending smoothed, truncated Tian-parameter lattices combined with Richardson extrapolation. The approach adds a relatively low-cost tree with a larger time step and uses it to extrapolate lattice values toward the continuous-time limit. The cited paper is described as reviewing established techniques for improving lattice performance. The recommendation is explicitly framed for vanilla options; the note does not provide convergence metrics, numerical comparisons, or evidence that the same choices are best for exotic contracts.

Key ideas

• Binomial lattice convergence speed can vary with both the model and option moneyness.
• Visual comparison of lattice values against step count may not provide a consistent way to rank models.
• Richardson extrapolation uses an additional coarse tree to estimate the value at a zero time step.
• The cited recommendation combines smoothing, truncation, Tian parameters, and extrapolation for vanilla options.
• The note does not establish that this recommendation applies to exotic options.

Tags

# Binomial lattice convergence


# Binomial lattice convergence












How do I measure how quickly a binomial lattice converges to an option value as the number of steps is increased?

I'm charting option value versus number of steps for various binomial lattice models and while the values generally converge as the number of steps increases, the rate at which they converge differs. For example, Model A converges fastest for at-the-money puts but Model B converges fastest for an otherwise identical option that is deep in-the-money.

Eyeballing the charts it seems obvious (sometimes) which model pick for a particular scenario, but how do you quantify this?

## Answer by Brian B (score 5, accepted)

https://quant.stackexchange.com/a/9023

You don't mention if the puts in question are exotic or vanilla, but assuming they are vanilla, you should read this paper by Chen and Joshi. In it, they find optimal performance by using smoothed, truncated Tian-parameter binomial lattices with Richardson extrapolation -- where the idea is to run one extra low-cost (long $\Delta T$) tree in order to extrapolate the lattice values to $\Delta T=0$.

Aside from a carefully-derived recommendation, the paper is an excellent review of most of the known tricks for extracting performance from binomial lattices.

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.