Chapter 2: FAQs 97
What is Jensen’s Inequality and What
is its Role in Finance?
Short Answer
Jensen’s Inequalitystates^1 that iff(·) is a convex func-
tion andxis a random variable then
E[f(x)]≥f(E[x]).
This justifies why non-linear instruments, options, have
inherent value.Example
You roll a die, square the number of spots you get, you
win that many dollars. For this exercisef(x)isx^2 ,a
convex function. SoE[f(x)] is 1+ 2 + 9 + 16 + 25 + 36 =
91 divided by 6, so 15 1/6. ButE[x]is31/2sof(E[x])
is 12 1/4.Long Answer
A functionf(·)isconvexon an interval if for everyx
andyin that interval
f(λx+(1−λ)y)≥λf(x)+(1−λ)f(y)
for any 0≤λ≤1. Graphically this means that the line
joining the points (x,f(x)) and (y,f(y)) is nowhere lower
than the curve. (Concave is the opposite, simply−fis
convex.)Jensen’s inequality and convexity can be used to explain
the relationship between randomness in stock prices
and the value inherent in options, the latter typically
having some convexity.Suppose that a stock priceSis random and we want
to consider the value of an option with payoffP(S). We(^1) This is the probabilistic interpretation of the inequality.