A call payoff is convex, so its expected payoff can exceed the payoff at the expected terminal price. Jensen's inequality makes that comparison precise. This lesson proves the inequality, applies it to a call and to concave utility, and derives the arithmetic–geometric mean inequality. The result supports the Programme's payoff comparison; it does not by itself supply a risk-neutral option price.
Learning objectives. By the end of this note you should be able to:
- define convexity and concavity of a function, geometrically and algebraically;
- state and prove Jensen's inequality for a differentiable convex function;
- apply Jensen's inequality to option payoff convexity and to risk-averse (concave) utility;
- derive a classical inequality (e.g. AM GM) as a corollary of Jensen's inequality.
In this lesson
- Convexity
- Jensen's inequality
- Financial applications
- A classical corollary: AM GM
Before you start
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