Why Lognormal Stock Prices Do Not Directly Yield the Black–Scholes PDE

Summary

The question asks whether the lognormal distribution for a stock price can be used to derive the Black–Scholes partial differential equation. The response clarifies that the stated log-price process is an assumption within the Black–Scholes framework, rather than a derivation of the pricing equation. The key next step is constructing a replicating portfolio; that argument leads to the PDE and is identified as the standard route to the Black–Scholes formula.

The exchange is conceptual and does not provide the replicating-portfolio calculation, boundary conditions, or a worked derivation. It therefore helps distinguish a model assumption about stock-price dynamics from the no-arbitrage reasoning used to obtain the pricing equation, but it is not a self-contained tutorial. Readers need a fuller derivation to see how the hedge and its rebalancing produce the PDE, and the discussion does not address limitations of the Black–Scholes assumptions in market practice.

Key ideas

• A lognormal stock-price process is an assumption of the Black–Scholes framework.
• That distributional assumption alone does not derive the pricing PDE.
• The standard derivation proceeds by constructing a replicating portfolio.
• The exchange points to the usual derivation but does not work through its steps or assumptions.

Tags

# How to derive the Black Scholes partial differential equation from a stock log-normal distribution?


# How to derive the Black Scholes partial differential equation from a stock log-normal distribution?












Is there a way to go from this

$$\ln S_t=\ln S_0+(\mu-\sigma^2/2)t+\sigma W_t $$

$$\ln S_t\sim N[\ln S_0+(\mu-\sigma^2/2)t, (\sigma^2)t]$$

To the Black-Scholes partial differential equation?

## Answer by SRKX (score 0)

https://quant.stackexchange.com/a/32602

What you wrote in your question is simply one of the assumptions of the Black-Scholes framework.

The whole interesting part about getting to the Black-Scholes PDE (i.e. constructing a replicating portfolio) comes next and is the most common derivation of the Black-Scholes formula, which you can find the derivation on the Wikipedia page.

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