What is the basis for assumption that asset prices follow a log normal distribution? Then how is it transformed to say that asset return follows a normal distribution? How this relationship between normal and log normal distribution is derived and when to use one vs the other, especially w.r.t. Black Scholes Models?


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In the Black Scholes (1973) model, the stock price is assumed to follow a geometric Brownian motion $\mathrm{d}S_t=S_t\mu \mathrm{d}t + S_t \sigma \mathrm{d}W_t$. If you solve the SDE, $(S_t)$ is log-normally distributed for every $t$.

Alternative, you can model the returns by a normal distribution and then take the exponential function to obtain the stock price (for positivity). You can see this transformation from Itô's Lemma.

A geometric Brownian motion is firstly very simple and allows for closed-form solutions of the prices of many derivates. It also satisfies naturally desirable properties such as the Markov property. But also recall the Binomial model in which the stock price converges to a log normal distribution as the amount of steps increases.

Nonetheless, there is plenty of empirical evidence that stock returns are in fact not normally distributed and this assumption is to strong. Models incorporating jumps and stochastic volatility aim to improve capturing real life dynamics. In real life, asset returns are skewed and have fat tails. Keep in mind, that the Black Scholes model is one of the first and simplest models.


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