Log and Price Forms of a Mean-Reverting Process

Summary

The document asks whether a mean-reverting stochastic process written for the logarithm of an asset price can instead be expressed directly in price form. The response says both representations can be used and points out a practical advantage of the log formulation: it exposes a linear relationship that can be estimated with linear regression tools.

The discussion is brief and does not derive the transformation or compare the statistical assumptions and behavior of the two forms. In particular, the proposed price equation is not independently checked in the answer, so readers should not treat the response as a full validation of that equation. The main takeaway is that choosing a log representation can simplify estimation, while the underlying model specification still needs to be examined.

Key ideas

• A mean-reverting model can be represented using log prices or prices, subject to the model's transformation assumptions.
• The log representation makes a linear relationship available for regression-based estimation.
• The response does not derive or validate the proposed exponential price equation.

Tags

# Mean reversion formula in log normal or exponential form?


# Mean reversion formula in log normal or exponential form?












The formula for the mean reversion model in log normal form:

$x=\ln(S)$

$x_{i+1} = x_i + [a(m-x_i)-\frac{1}{2}\sigma^2] dt + \sigma \sqrt{dt} \epsilon$

Can this formula be written in exponential form?

$S(i+1)= S(i)\exp([a(m-S(i))-\frac{1}{2}\sigma^2] dt + \sigma \sqrt{dt}\epsilon)$

Is there any reason why we would use the log normal form?

Thanks,

Alex

## Answer by M. Jeunesse (score 1, accepted)

https://quant.stackexchange.com/a/29995

No, you can use both forms.

By using the first relationship, you can identify a linear relationship and use linear regression tools.

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