Jensen’s Inequality

In considering the broad practical applications of convexity and optimization, many are intrigued by the prospect of developing improved models for use in global financial markets. As machine learning and algorithmic trading become an increasingly important presence in quantitative finance, knowledge of the underlying programs is essential to understanding the complex global marketplace. Among other industry paradigms, Jensen’s inequality is widely relied upon in supporting these interactions, and linking volatility and convexity to asset pricing . Developed by Danish mathematician Johan Jensen in 1906, this renowned theorem links the value of a convex function of an integral to the integral of the convex function. The inequality fundamentally states that the convex transformation of a mean is less than or equal to the mean after convex transformation, and that concave transformation is greater than or equal to the mean after concave transformation.

Applied to any two points, the theorem states that the weighted mean secant line of convex function $[tf(x_1)+(1-t)~f(x_2)]$ is above the graph of the function. The graph function is then defined as the convex function of weighted means $[f (t~{x_1}+(1-t)x_2)]$.This is expressed in probability theory as a random variable $X$ and convex function $\varphi$.

$\mathbb{E} [\varphi (X)] \geq \varphi \mathbb{E}[X]$

There is a disparity in the expectation of a non-linear function of a random variable and the function of the expectation. If the function is convex, then the former is greater. The pricing of publicly traded securities options effectively behaves as a non-linear function; the true value of the option is therefore quantified by the difference between the expectations surrounding the function and the function of the expectation. The parameters of volatility are unknown, prompting risk managers and quantitative analysts to conduct sensitivity analyses surrounding volatility, fluctuations in the underlying price of the asset, systemic market factors, and duration measures. Trading strategies which assume deterministic patterns of random events frequently result in highly volatile performance, leading to options mispricing.

The Taylor series expansion can be used to demonstrate the expected value of the function $\varphi (X)$ and the inequality between the greater lefthand side of the equation, illustrating the convexity $\gamma$ and variance (volatility) of the underlying asset.

For the intents and purposes of illustration, suppose a hypothetical scenario in which a basic market asset $x$ , such as shares in a fund designed to mimic the NASDAQ index, has a maturity/investment horizon of time $t_m$, denoted hereafter as $m$.

$\mathbb E[\varphi (x_m)] \geq \varphi[\mathbb E (x_m)]$

Assume that the fund share is presently trading at a normalized value, and that it will follow a randomized future path until $t_m$. For the sake of this example, randomness can be simulated as the asset being worth one of ten possible values $(x_1,x_2,~\dots~,x_{10})$ at $t_m$.

The expected value of the function $\varphi (x_m)$ is given by the squared estimates of the asset value at the maturity $t_m$.

$\mathbb E [\varphi (x_m)]=\frac {x_1^2,x_2^2,~\dots~,x_{10}^2} {10}$

The function of the expected future value $\varphi [\mathbb E(x_m)]$ can be expressed by the following:

${\varphi} [{\mathbb E}(x_m)]=({\frac {x_1, x_2,~\dots~,x_{10}} {10}})^2$

Any such combination of potential future values $x_m$ at $t_m$ yields $\mathbb E [\varphi (x_m)] \geq \varphi [\mathbb E(x_m)]$. This concept is particularly useful in options pricing, and is of many inputs frequently used in determining the fair value of an option depending upon the expiration date and the price of the underlying asset. Assuming an option strike price constant $k$ and an expiration date $t_m$, the expected value of the option function $\mathbb E [\gamma (x_m)]$ can be found though a similar equation, such as a call option illustrated below.

$\mathbb E[\gamma (x_m)]=\frac {max({x_1}-k,0)+max({x_2}-k,0)+~\dots~+max({x_{10}}-k,0)} {10}$

The function of the expected future option value can be expressed as:

$\gamma [\mathbb E(x_m)]=max([\frac {x_1+x_2~\dots~+x_{10}} {10}] {-k,0})$

Appropriately priced values are constrained by Jensen’s inequality $\mathbb E [\gamma (x_m)] \geq \gamma [\mathbb E(x_m)]$ for any combination $x_m$ at $t_m$.

An efficient market will price assets exhibiting random walk style future value expectations of this nature with an appropriate initial value at the present time period $t_0$ which is discounted from the future value $\varphi (x_m)$ at a rate consistent with the expected volatility of the contract and underlying asset.

In practice, $x_m$ is naturally not confined to a finite number of potential values such as ten, but rather an infinite number of values, given that $x_m$ follows a random walk. When financial firms structure and introduce a call option such as the one simulated, the contract $\gamma (x)$ simulated generally follows a pricing model $\gamma (x_{t=0})={e^{-rm}} \mathbb E^p [\varphi (x_m)]$, where $x_p$ is the confidence level or probability of the expectation $\mathbb E$. Phrased alternatively, convexity or higher order terms are added to the mean expected value these functions to assign the financial derivative instrument a fair value at inception $t_0$.

As applied mathematician Paul Wilmott noted in Derivatives: The Theory and Practice of Financial Engineering (Wiley & Sons 1998), higher order convexity terms $\gamma$ capture convexity, randomness, and variance in all financial derivatives.

$\mathbb E[\varphi (x_m)]=\varphi [\mathbb E(x_m)] + \gamma$

$max([\frac {x_1=x_2+~\dots~+x_3} {10}] -k,0)$

The time value adjustment in these option pricing models is effectively convexity. The common Black-Scholes model incorporates this concept by omitting interest rates and the first order derivative by expressing $\gamma$ as negative. Those interested in gaining a better understanding of options trading and derivatives pricing may wish to test this theory by attempting the types of simulations above with random sets of $x_m$.