Jensen Inequality

Jensen Inequality is a fundamental principle in mathematics and finance stating that for a convex function, the value of the function of an average is less than or equal to the average of the function values. In financial derivatives, this explains why the price of a convex asset, like a bond or an option, is higher than what a linear approximation would suggest.

Because the price-yield curve is convex, fluctuations in yield result in a higher average price than the price at the average yield. This inequality is the theoretical foundation for the convexity adjustment in pricing forward rates and options.

It highlights the importance of volatility in determining the value of derivative contracts. When pricing instruments, assuming linearity ignores this inherent bias, leading to mispricing.

By incorporating the second-order effects defined by the curvature, analysts correct for this mathematical phenomenon. It is a cornerstone of quantitative finance, ensuring that models account for the asymmetric impact of market volatility.

Understanding this allows traders to identify mispriced assets that do not correctly reflect the volatility-induced price premium.