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Bjerksund–Stensland Boundary Behavior When Carry Equals the Risk-Free Rate

Article Quant Q&A · Author: bruno
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Summary

The document addresses a possible numerical or formula issue in implementations of the Bjerksund–Stensland 2002 American option approximation, focusing on its early-exercise boundaries. The response identifies the lower boundary, B0, as potentially problematic when the cost of carry equals the risk-free rate. This condition can arise for an option on a non-dividend-paying equity underlier.

The suggested workaround is to use a small nonzero difference between the risk-free rate and cost of carry when calculating B0. The note does not provide a full derivation, implementation details, or validation across option parameters, and it does not resolve issues with the upper boundary. Treat the adjustment as a numerical workaround to investigate, rather than as a general correction for every far out-of-the-money calculation or every implementation.

Key ideas

  • The discussion concerns early-exercise boundaries in the Bjerksund–Stensland 2002 approximation.
  • The B0 boundary may be questionable when cost of carry equals the risk-free rate.
  • A small assumed difference between those rates is suggested for computing B0.
  • The response does not establish a general fix or analyze the upper boundary.

Tags

OptionsDerivatives pricingVolatility
Full text
# Far OTM calculation issue on Bjerksund-Stensland


# Far OTM calculation issue on Bjerksund-Stensland












Has anyone come across and fixed calculation issues on boundaries using Bjerksund-Stensland 2002 (Hull, Haug or Rouah implementations) ? Thanks in advance

## Answer by Yanyi Yuan (score 0)

https://quant.stackexchange.com/a/55705

Is your question about the early exercise boundary (B0 and B_inf)? The B0 boundary can be questionable when cost of carry is the same as risk free rate (r=b for options on non dividend-paying equity underlier). If cost of carry equals risk free rate, I will have to assume a small value for r-b to get the boundary B0.

https://core.ac.uk/download/pdf/30824897.pdf

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.