Why the Black–Scholes Derivation Uses the Log Price

Summary

The note explains why a derivation for geometric Brownian motion applies Itô’s lemma to the logarithm of the asset price. The key benefit is that the original stochastic differential has a price-dependent coefficient, making direct integration awkward. Taking the logarithm transforms it into a process with constant drift and volatility, which can be integrated directly.

The document gives the resulting log-price dynamics, including the volatility adjustment to drift, as its mathematical illustration. It presents the transformation as a solution technique rather than as a claim that lognormality or normally distributed returns are required. The explanation is brief: it addresses this GBM equation, but does not compare alternative transformations or explain broader applications of Itô’s lemma.

Key ideas

• The price-dependent term in the GBM equation complicates direct integration.
• Applying Itô’s lemma to log price yields constant drift and volatility terms.
• The logarithm is used as a mathematical simplification in this derivation.

Tags

# Intuition behind Ln transformation of stock price when applying Ito lemma


# Intuition behind Ln transformation of stock price when applying Ito lemma












I am able to replicate steps and arrive to the option price using Black Scholes framework. Here however I am more interested to understand, at least intuitively, why the ln transformation of price process is performed (Ito lemma part) in the first place. Price process is already a function of time and Wiener process, so I wonder why do we need to apply another function (ln). I do not think it has to do with log normality of prices or normality of returns. I have seen such a transformation taking place in solution of other problems that were not related to GBM - BS framework.

Thanks,

## Answer by Quantuple (score 4, accepted)

https://quant.stackexchange.com/a/35773

This is merely a mathematical trick.

You cannot easily integrate $dS_t = S_t(\mu dt + \sigma dW_t)$ over time because the RHS depends on $S_t$.

Using Ito's lemma on the log price gets you: $d\ln(S_t) = \left(\mu-\frac{1}{2}\sigma^2\right) dt + \sigma dW_t$ which is straightforward to integrate.

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