Portfolio risk depends on how asset returns move together. Marginal distributions describe each asset separately; a joint distribution describes their combined scenarios. This lesson derives covariance and portfolio variance, shows why zero correlation does not imply independence, and uses Jensen's inequality to compare a convex payoff with the payoff at the mean price.
Learning objectives. By the end of this note you should be able to:
- define joint PMFs/PDFs, recover marginals by summing/integrating, and test independence by factorisation;
- compute and , and expand ;
- explain why uncorrelated does not imply independent, with a finance-facing counterexample pattern;
- state and apply Jensen's inequality to a convex option payoff.
In this lesson
- Joint laws and marginals
- Covariance and correlation
- Correlation is not dependence
- Jensen's inequality and a pricing reading
- Covariance of linear combinations
Before you start
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