The price of a path-independent derivative is the expected discounted payoff under the risk-neutral measure — which is a double integral of the payoff function against the joint density over the region where the payoff is positive. This note develops three incarnations of the same idea: volume (the integral of a height function over a planar region), average value (volume divided by area), and probability (the integral of a density over the event region). All three are applications of the same double integral machinery, now deployed over general regions.
Learning objectives. By the end of this note you should be able to:
- compute the volume under and above a planar domain using a double integral;
- compute the volume of the solid bounded between two surfaces and ;
- find the average value of over a domain;
- interpret the double integral as a probability when the integrand is a joint density.
In this lesson
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- Volume under a Surface
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- Volume between Two Surfaces
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- Average Value of a Function
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- Double Integrals as Probabilities
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- Pitfalls and Sanity Checks
Before you start
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