Why Diffusion Price Variance Scales with Time

Summary

The document asks why a one-step price variance in an option-pricing model is written as the squared current price times volatility squared times the time increment. It relates the formula to a time-homogeneous diffusion: volatility is independent of time and depends only on the underlying state. Under that modeling condition, variance accumulates proportionally with elapsed time over the step.

The response is only a brief conceptual pointer and links to an external explanation rather than providing a derivation. It cautions that the same proportionality cannot simply be assumed for models whose volatility is time-dependent. The discussion does not specify a fuller stochastic process, assumptions about discretization, or alternative variance formulas, so it serves as a narrow explanation of the stated relationship rather than a general treatment of volatility models.

Key ideas

• The stated variance relationship assumes a time-homogeneous diffusion.
• In that setting, volatility does not depend explicitly on time and depends on the underlying state.
• The response warns that the proportionality may fail when volatility varies with time.
• The document provides no detailed derivation or comparison of alternative models.

Tags

# How underlying asset price variance is connected with time


# How underlying asset price variance is connected with time












I'm dealing with option pricing models and there is a statement that says the variance of underlying asset price is propotional with time $𝑉𝑎𝑟(𝑆_{𝑚+1})=𝑆_𝑚^2𝜎^2Δ𝑡$ where $\Delta t = \frac{T}{N}$ and $\sigma$ is volatility. How this equality can be explained/proved?

## Answer by Xman (score 1)

https://quant.stackexchange.com/a/46018

Exactly what @Alex C said. It's the time homogeneous diffusion proprety. You can't state such an argument in models where volatility is no longer time homogeneous ( that's being time independant and depending only on the underlyings).

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