Think Value

1. The starting point — stock price process

We assume that the stock price SSS follows a geometric Brownian motion: dS = μS*dt +σS*dz

where:

• μ = expected return (drift)
• σ = volatility
• dz = Wiener process increment (random noise)

find also: Geometric Brownian Motion of Stock Price

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2. Applying Ito’s Lemma to G=ln (S)

We want to find the process followed by G=ln⁡S

Ito’s Lemma tells us that if S follows a stochastic process dS=a*dt+b*dz, then for any function G(S,t):

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3. Compute the derivatives

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4. Substitute into Ito’s formula

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5. Interpretation

This equation shows that ln(⁡S) follows a generalized Wiener process with:

• drift = μ− (1/2)σ²
• variance rate = σ²

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6. Distribution over time

Integrate both sides from time 0 to T:

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7. Lognormal conclusion

Because ln⁡S_T​ is normally distributed, it follows that S_T​ itself is lognormally distributed:

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8. Key takeaways

• Stock returns (in log form) are normally distributed.
• Stock prices are lognormally distributed.
• The expected growth rate of prices is slightly less than μ because of the −(1/2)σ² (volatility drag). (μ -is arithmetic average of expected return, while “μ −(1/2) σ”, logarithmic, or geometric average growth of the stock price).
• The standard deviation of log prices grows with σ√T – It tells you how uncertain the future price is — in multiplicative (not additive) terms.

Adapted from:

Options, Futures & Other Derivatives, John C. Hull