Meaning
Mathematical theorems describe the relationship between the average of a function and the function of an average. In probability theory and financial modeling, Jensen’s inequality demonstrates that the expected value of a convex transformation of a random variable is greater than or equal to the convex transformation of that variable’s expected value. This relationship explains why volatility can systematically shift the average outcome of non-linear processes.
It forms the foundation for understanding options pricing and portfolio behavior under uncertainty.
Mathematical Foundation
Convex functions curve upward, meaning that a line segment between any two points on the graph lies above the graph itself. The proof of Jensen’s inequality relies on this geometric property to show that the average of the function’s outputs exceeds the function of the average input. This mathematical proof applies to any convex function regardless of the underlying probability distribution.
It provides a general tool for analyzing non-linear systems.
Financial Application
Option values benefit from increased underlying asset volatility because of their convex payoff profiles. Risk managers apply Jensen’s inequality to explain why a portfolio of options is worth more than the options evaluated at the average future price. This application assists in the pricing of complex derivatives and structured products.
It ensures that valuation models correctly account for the asymmetrical nature of market returns.
Volatility Impact
Portfolio returns are affected by the dispersion of individual asset performances. The asymmetry of compound growth rates illustrates Jensen’s inequality in long-term investing, where high volatility reduces the geometric mean return relative to the arithmetic mean. This impact forces investors to focus on volatility drag when constructing portfolios.
It clarifies why reducing volatility can increase long-term wealth even if the average return remains constant. Portfolio managers use this relationship to justify the costs of diversification and hedging, which reduce variance to preserve the compound growth rate over multiple investment cycles.
