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When Euler Discretization Is Exact for Geometric Brownian Motion

Article Quant Q&A · Author: quallenjäger
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Summary

The document clarifies a distinction in Euler discretization of geometric Brownian motion (GBM). Applying the logarithm to the process first yields a Brownian motion with drift, whose Euler time-step update is exact. This exactness applies to the log-price representation rather than directly to the original price process.

In the original coordinates, Euler discretization is not exact and can assign positive probability to negative prices, which GBM itself does not permit. The answer resolves the apparent conflict by specifying the coordinates in which exactness holds. It does not provide a detailed derivation or quantify discretization error, so its explanation is conceptual and focused on this limitation.

Key ideas

  • Taking logarithms of GBM yields Brownian motion with drift.
  • Euler discretization is exact for the log-price process.
  • Euler discretization in price coordinates is not exact.
  • The price-coordinate scheme can produce negative values with positive probability, unlike GBM.

Tags

StatisticsDerivatives pricingRisk management
Full text
# Euler discretization


# Euler discretization












I have been told that the Euler discretisation is exact for the GBM process.Is it true and how can I proof this? This would mean, for a GBM process, if I am increasing my discretisation step, the value is unaffected. However, in many application, the error for discretisation of GBM decreases with increasing steps. What is true?

## Answer by Mark Joshi (score 4, accepted)

https://quant.stackexchange.com/a/33163

It is exact if you shift to log coordinates first. In that case, you are discretising Brownian motion with drift i.e. $$ d\log S_t = (\mu-0.5\sigma^2 )dt + \sigma dW_t $$

It is definitely not exact in the original coordinates, since the probability of going negative is positive for this discretization.

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