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Why Drift Assumptions Change Bachelier Call Strike Derivatives

Article Quant Q&A · Author: Jay
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Summary

The document concerns taking a Bachelier call option’s partial derivative with respect to strike and finding a mismatch between two reference results. The resolution is that the sources use different stochastic differential equations: one includes a drift term and the other does not. Since the assumed process affects the distribution used in the option valuation, it can also affect derived expressions and their comparison.

The post offers a useful model-checking lesson: when formulas appear inconsistent, compare the underlying assumptions and dynamics before assuming an algebra or differentiation error. However, it provides no derivation, explicit option formula, strike derivative, or quantitative example. It identifies drift as the source of the discrepancy in this particular comparison, but does not establish which convention is appropriate for other Bachelier implementations. Readers should align the process specification and pricing assumptions before comparing results.

Key ideas

  • A Bachelier call’s strike derivative depends on the assumptions used to define the underlying process.
  • A drift term in the stochastic differential equation can explain a mismatch between reference formulas.
  • Check model dynamics and conventions before treating differing results as a calculus error.
  • The document identifies the discrepancy but does not provide the derivative or a full derivation.

Tags

OptionsDerivatives pricingStatistics
Full text
# Bachelier call option derivative w.r.t strike


# Bachelier call option derivative w.r.t strike












I tried to take the partial derivative of the Bachelier call function w.r.t. strike price K (eqn 2.2 here), but my result is not lining up with what is shown on page 43 here.

## Answer by Jay (score 1)

https://quant.stackexchange.com/a/64092

I figured it out. the second source had a drift component in the SDE that i didnt have.

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.