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Lognormal Asset Modeling

The lognormal solution
St=S0exp⁡((μ−σ22)t+σWt)S_t = S_0\exp\left(\left(\mu - \tfrac{\sigma^2}{2}\right)t + \sigma W_t\right)

Solving the SDE gives the price in closed form. The minus sigma-squared over two is the correction that keeps the expectation honest.

Log returns normal, so prices lognormal. This is the standard model, and it is worth being precise about what it gets right and what it gets wrong.

Select the lognormal and raise the volatility. The bulk slides left while the right tail stretches, so the median falls below the mean and most paths end below average. That gap is the drift correction, and it is why a fund can have a positive expected return and a majority of losing outcomes.

What it gets right

Positivity. St>0S_t > 0 always, since it is an exponential. A normal model for prices would assign positive probability to a negative price, which is not merely unrealistic but breaks the mathematics of returns.

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