Excursion to Financial Engineering: Volatility Invariance in the Black–Scholes Model

Abstract

It is well-known that the diffusion invariance principle conveys an important insight for financial economics: when (logarithmic) Itô processes are used as models of stock prices, the drift coefficient of the logarithmic price process is interpreted as a measure for the expected return, and its diffusion coefficient is interpreted as a measure for the volatility. In this context, the diffusion invariance principle asserts roughly that under an equivalent change of probability measure, only the expected returns will be affected, but not the volatilities. In particular, a price process will have the same volatility under the real-world probability measure as under an equivalent risk-neutral (i.e. arbitrage-free) probability measure; changing the probability measure corresponds to changing the expected return (and vice versa).