## Jensen’s Inequality

Key words: Convexity adjustment, Convex function, Non-linearity in derivatives, time value of an option

Inequality generalizes the statement that a secant line of a convex/concave function is above/below the graph. This was proved by Danish mathematician in 1906.

Concept of convexity and Jensen’s inequality are intricately linked. Generally convexity can be used to interpret derivative pricing.

Convexity refers to non-linearity in a financial model and the convexity adjustment concerns the values of some derivative payoffs which are not paid at their underlying fixing time.
This is simply the case with Interest rate swaps where the fixing is in-arrears or if we assume volatility to be stochastic then we need to adjust an option’s value by Vomma and Volga greeks which are second order derivatives with respect to volatility.

The inequality also suggests that the time value of an option has to be positive, i.e. the option price has to be higher than its intrinsic value.
Also, it can be said that the expected value of the option is higher than simply taking the expected future value of the underlying and inputting it into the option payout function.
This is exactly what Jensen’s Inequality says:

The expected value of a convex function is higher than the function of the expected value
E[f(x)] ≥ f[E(x)] If f (x) is concave, f”(x)<0, the inequality is reversed

Example

The exponential function f(x)=e^x is convex. Let x be 1 with probability 0.5 or (-1) with the same probability.

E(x) = 0.5*(−1) + 0.5*1 = 0

f[E(x)]= e^0 = 1

f(e^1) = 2.7183, f(e^(-1)) = 0.3679

E[f(x)] = 0.5*2.7183 + 0.5*0.3679 = 1.543

E[f(X)] > f(E[X]) or 1.543 > 1

The average of E[f(x)] is on the red secant line connecting those points while f[E(x)] is below the chord. This is a graphical explanation of Jensen’s inequality.

There is another explanation that is easy to understand

Roll a dice with the following payoffs, where x is number of dotes:

Payoff = x^2. Clearly, the payoff is a convex function.

E[x] = (1+2+3+4+5+6)/6 = 3.5
f(E[x]) = 3.5*3.5 = 12.25
E[f(x)] = (1 + 4 + 9 + 16 + 25 + 36)/6 = 15.17
E[f(x)] > f(E[x]) or 15.17>12.25

Payoff =2*x. Clearly, the payoff is a linear function.

E[x] =(1+2+3+4+5+6)/6=3.5
f(E[x]) =2*3.5 = 7
E[f(x)] =2* (1+2+3+4+5+6)/6 = 7
E[f(x)] = f(E[x]) or 7 = 7