Why Volatility Lowers Geometric Growth in the Black–Scholes Model

Summary

The document asks whether the negative half-variance term in the logarithm of a geometric Brownian motion price process is related to volatility drag when converting arithmetic returns into geometric returns. It supplies the Black–Scholes stochastic differential equation and its closed-form solution, highlighting the adjustment to the drift that appears in the exponent.

The material is a conceptual question rather than a completed explanation: it offers no derivation, empirical evidence, or answer. The connection to learn from is that the log-price drift differs from the arithmetic drift because of the curvature of the logarithm, an effect that also underlies volatility drag in compounded returns. The precise relationship depends on the return convention and model assumptions; the document itself does not explore those qualifications or discuss option pricing despite the model’s name.

Key ideas

• The geometric Brownian motion solution contains a negative half-variance adjustment in log-price growth.
• The question connects this adjustment to the loss in compounded growth associated with return variability.
• The document poses the relationship but does not provide a derivation or answer.
• Any interpretation depends on distinguishing arithmetic drift from expected log growth.

Tags

# Is there a relation between the so-called volatility drag and the sigma term in Black-Scholes' model?


# Is there a relation between the so-called volatility drag and the sigma term in Black-Scholes' model?












The closed-form solution of Black Scholes Dynamics $dS_t=S_t(\mu dt +\sigma dW_t$) is $$S_t=S_0 e^{(\mu -\sigma ^2/2) t+\sigma dW_t}.$$

The $-\sigma^2/2$ term is quite similar to the volatility drag when transforming an arithmetic return to a geometric return. Are there any relationship between the two?

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