Why prices are modelled as lognormal
Question
Explain why the lognormal distribution plays such a central role in quantitative finance.
Short answer
Because compounding is multiplicative. A price is a product of many small growth factors, so its logarithm is a sum and the central limit theorem makes it approximately normal, i.e. the price is lognormal. That keeps prices positive, makes returns scale with the price level, and is exactly the distribution produced by geometric Brownian motion, which underlies Black–Scholes and its closed-form option prices.
Adapted from Wilmott, Frequently Asked Questions in Quantitative Finance, FAQ 46, p.209; question and answer rewritten by SelfMadeQuant.com.
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- Continuous Random Variables and Probability Density Functions
- The Normal Distribution
- Cumulative Distribution Functions
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