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Choosing Finite-Difference Steps for Barrier-Option Gamma

Article Quant Q&A · Author: vanilla_skies
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Summary

The document addresses scattered gamma plots for single-barrier options, including up-and-out and down-and-in contracts, despite smoother plots for vanilla options. The suggested cause is numerical instability when gamma is estimated with a finite-difference method using a price increment that is too small. At very small steps, the calculation can be affected by finite machine precision and produce visibly noisy estimates.

The proposed remedy is to increase the underlying-price step used in the finite difference. The questioner reports that changing the step from 0.0001 to 0.01 eliminated the scattered appearance. This is a brief empirical example rather than a general calibration rule: an appropriate step depends on the option, pricing model, scale, and numerical method, and the document offers no broader validation or error analysis.

Key ideas

  • Scattered barrier-option gamma estimates can result from an excessively small finite-difference step.
  • Finite machine precision can make numerically differentiated values noisy.
  • Increasing the underlying-price increment may smooth the estimated gamma.
  • The reported successful step change is specific to one calculation and is not a universal setting.

Tags

OptionsDerivatives pricingStatistics
Full text
# For single barrier options, why is a plot of gamma so scattered compared to other greeks?


# For single barrier options, why is a plot of gamma so scattered compared to other greeks?












Is this to be expected or is there something wrong with the model?

I am getting scattered gamma plots for all types of barriers like U&O, D&I, etc

However a basic vanilla options has a smooth gamma plot.

## Answer by user59852 (score 4)

https://quant.stackexchange.com/a/68884

Your gamma seems to be "quantized" like if your calculation happens to be at the machine limit in term of precision. Maybe you aren't using a "dS" large enough if you compute derivatives using a finite difference approach.

Try increase the step for your numerical calculation.

## Answer by vanilla_skies (score 0)

https://quant.stackexchange.com/a/69092

it had to do with dS being too small. After changing dS from .0001 to .01 the scattered plots went away

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.